Wireless Algorithms, Systems, and Applications: 5th International Conference, WASA 2010, Beijing, China, August 15-17, 2010. Proceedings (Lecture ... Computer Science and General Issues)

Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris ...

Author: Gopal Pandurangan | V. S. Anil Kumar | Gu Ming | Yunhao Liu | Yingshu Li

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Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen TU Dortmund University, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany

6221

Gopal Pandurangan V. S. Anil Kumar Gu Ming Yunhao Liu Yingshu Li (Eds.)

Wireless Algorithms, Systems, and Applications 5th International Conference, WASA 2010 Beijing, China, August 15-17, 2010 Proceedings

13

Volume Editors Gopal Pandurangan Nanyang Technological University Singapore E-mail: [email protected] V. S. Anil Kumar Virginia Tech Blacksburg, VA, USA E-mail: [email protected] Gu Ming Tsinghua University Beijing, China E-mail: [email protected] Yunhao Liu Hong Kong University of Science and Technology Kowloon, Hong Kong E-mail: [email protected] Yingshu Li Georgia State University Atlanta, GA 30303, USA E-mail: [email protected]

Library of Congress Control Number: 2010931794 CR Subject Classification (1998): F.1, F.2, D.1, D.2, D.4, C.2, C.4, H.4 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues ISSN ISBN-10 ISBN-13

0302-9743 3-642-14653-8 Springer Berlin Heidelberg New York 978-3-642-14653-4 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. springer.com © Springer-Verlag Berlin Heidelberg 2010 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper 06/3180

Preface

Over the past decade, signﬁcant advances in wireless communication and computing technologies have led to the proliferation of reliable and ubiquitous infrastructure and infrastructureless wireless networks all over the world, as well as a diverse range of new applications, such as mobile social networking, and the surveillance and protection of critical infrastructures and environments. At the same time, these applications have raised new challenges ranging from the theoretical foundations of these systems, algorithms and protocol design, security and privacy to rigorous and systematic design and evaluation methodologies and new architectures for next-generation systems. The annual International Conference on Wireless Algorithms, Systems, and Applications (WASA) provides a forum for researchers and practitioners worldwide to exchange ideas, share new ﬁndings, and discuss challenging issues for the current and next-generation wireless networks. Past WASA conferences were held in Xian (2006), Chicago (2007), Dallas (2008), and Boston (2009). WASA 2010, the 5th WASA conference, took place at the Beijing Wenjin International Hotel in Beijing during August 15–17, 2010. Each submission was reviewed by at least three Program Committee members, who in some cases were assisted by external referees. Following a rigorous review process, 29 (19 regular and 10 short) papers were selected for presentation at the conference. The best paper award was given to the paper titled “Approximate Optimization for Proportional Fair AP Association in Multi-rate WLANs” by Wei Li, Yong Cui, Shengling Wang, and Xiuzhen Cheng. Four workshops were also organized along with WASA 2010: the Workshop on the Security of Wireless and Ad-hoc Networks (SWAN) 2010, the Workshop on Data Management and Network Control in Wireless Networks (DMNC), the First Workshop on Radar and Sonar Sensor Networks (RSSN), and the First Workshop on Compressive Sensing for Communications and Networking (CSCN). Eighteen papers from these workshops also appear in these proceedings. We thank the respective workshop organizers for their eﬀorts in organizing these workshops and contributing to the success of the WASA 2010 conference. We thank all the authors for submitting their papers to the conference. We also thank all the members of the Program Committee and external referees for their help in completing the reviewing process, especially under the tight time constraints. We are grateful to the members of the Steering Committee for their involvement, encouragement, and help throughout this process.

VI

Preface

Finally, many other people contributed to the success of WASA 2010 directly and indirectly. Even though their names cannot be listed here because of space limitation, we owe them our gratitude. August 2010

Gopal Pandurangan V.S. Anil Kumar Gu Ming Yunhao Liu Yingshu Li

Organization

Honorary General Chair Sun JiaGuang

Tsinghua University, China

General Co-chairs Gu Ming Yunhao Liu

Tsinghua University, China Hong Kong University of Science and Technology, China

Program Committee Co-chairs Gopal Pandurangan Anil Vullikanti

Nanyang Technological University, Singapore and Brown University, USA Virginia Tech, USA

Registration Co-chair Min Song

Old Dominion University, USA

Local Arrangements Co-chairs Zhiguo Wan Jizhong Zhao

Tsinghua University, USA Xi’An Jiaotong University, China

Publicity Co-chairs Jen-Yeu Chen Qilian Liang

National Dong-Hwa University, Taiwan University of Texas at Arlington, USA

Workshop Co-chairs Costas Busch Xiuzhen Cheng

Louisiana State University, USA George Washington University, USA

Publication Co-chair Yingshu Li

Georgia State University, USA

VIII

Organization

Steering Committee Peng-Jun Wan Xiuzhen Cheng Wei Zhao Ty Znati

Illinois Institute of Technology, USA (Chair) The George Washington University, USA University of Macau, China National Science Foundation, USA

Program Committee Members John Augustine Nanyang Technological University, Singapore Costas Busch Louisiana State University, USA Jiannong Cao Hong Kong Polytechnic University, China Jen-Yeu Chen National Dong-Hwa University, Taiwan Susan Cheng George Washington University, USA Yong Cui Tsinghua University, China Sajal Das NSF and University of Texas - Arlington, USA Amitabha Ghosh University of Southern California, USA Sukumar Ghosh University of Iowa, USA Seth Gilbert EPFL, Switzerland Chuanhe Huang Wuhan University, China Anura Jayasumuna Colorado State University, USA Krishna Kant Intel and NSF, USA Maleq Khan Virginia Tech, USA Bhaskar Krishnamachari University of Southern California, USA Wonjun Lee Korea University, Korea Deying Li Renmin University of China, China Minming Li City University of Hong Kong, China Benyuan Liu University of Massachusetts - Lowell, USA Wei Lou Polytechnic University of Hong Kong, China Madhav Marathe Virginia Tech, USA Frederique Oggier Nanyang Technological University, Singapore Marimuthu Palaniswami University of Melbourne, Australia Christos Papadopoulos Colorado State University, USA Srinivasan Parthasarathy IBM Research, USA Sriram Pemmaraju University of Iowa, USA S.S. Ravi SUNY at Albany, USA Michael Segal Ben Gurion University, Israel Yi Shi Virginia Tech, USA Violet Syrotiuk Arizona State University, USA Jian Tan Ohio State University, USA Bharadwaj Veeravali National University of Singapore, Singapore Peng-Jun Wan Illinois Institute of Technology, USA Amy Wang Tsinghua University, China Qing Wang IBM Research, China Kui Wu University of Victoria, Canada Xinbing Wang Shanghai Jiaotong University, China

Organization

IX

Workshop Committee on the Security of Wireless and Ad-hoc Networks Costas Busch Bo Sheng Haodong Wang Hui Chen Xiuzhen Cheng Guevara Noubir Chiu C. Tan Lei Xie Kai Xing Shuhui Yang

Louisiana State University, USA (General Chair) Northeastern University, USA (Workshop Program Co-chair) Virginia State University, USA (Workshop Program Co-chair) Virginia State University, USA George Washington University, USA Northeastern University, USA College of William and Mary, USA Nanjing University, China University of Science and Technology of China, China Purdue University at Calumet, USA)

Workshop Committee on Data Management and Network Control in Wireless Networks Jinshu Su Wei Cheng Nan Zhang Hongyang Chen Tingjian Ge Mikyung Kang Murat Kantarcioglu Yujun Liu Wei Peng Guangming Song Chiu C. Tan Lingyu Wang Kai Xing Mira Yun

National University of Defense Technology, China (General Chair) The George Washington University, USA (Workshop Program Co-chair) The George Washington University, USA (Workshop Program Co-chair) University of Tokyo, Japan University of Kentucky, USA ISI, University of Southern California, USA University of Texas at Dallas, USA Academy of Armored Forces Engineering, China National University of Defense Technology, China Southeast University, China College of William and Mary, USA Concordia University, Canada University of Science and Technology of China, China The George Washington University, USA

X

Organization

Workshop Committee on Radar and Sonar Sensor Networks Jing Liang Qingchun Ren Scott C.-H. Huang Ting Jiang Qilian Liang Sherwood W. Samn Lingming Wang Xinsheng Xia Liang Zhao Zheng Zhou

University of Texas at Arlington, USA (Workshop Program Co-chair) Microsoft, Seattle, USA (Workshop Program Co-chair) City University of Hong Kong, Hong Kong Beijing University of Posts and Telecommunications, China University of Texas at Arlington, USA Air Force Research Laboratory/RHX, San Antonio, USA iBiquity Digital Corporation, Basking Ridge, New Jersey, USA Tellabs Inc, New Jersey, USA Airvana Inc., Chelmsford, Massachusetts, USA Beijing University of Posts and Telecommunications, China

Workshop Committee on Compressive Sensing for Communications and Networking Jing Liang Dechang Chen

Qilian Liang Xiuzhen Cheng Ting Jiang Qingchun Ren Sherwood W. Samn LingmingWang Xinsheng Xia Liang Zhao Zheng Zhou

University of Texas at Arlington, USA (General Chair) Uniformed Services University of the Health Sciences, USA (Workshop Program Co-chair) University of Texas at Arlington, USA (Workshop Program Co-chair) George Washington University, USA Beijing Univerisity of Posts and Telecommunications, China Microsoft, Seattle, USA Air Force Research Laboratory/RHX, Texas, USA iBiquity Digital Corporation, Basking Ridge, New Jersey, USA Tellabs Inc., New Jersey, USA Airvana Inc., Chelmsford, Massachusetts, USA Beijing University of Posts and Telecommunications, China

Organization

External Referees Dilum Bandara Andrew D. Berns Wei Cheng Dulanjalie Dhanapala Fei Huang

Yuan Le Jia Liu Guanhong Pei Mohan Raj Sasanka Roy

Sponsoring Institution Tsinghua University

Vaishali Sadaphal Sushant Sharma Amin Teymorian Yan Wu Zhao Zhao

XI

Table of Contents

Topology Control and Coverage Arbitrary Obstacles Constrained Full Coverage in Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Haisheng Tan, Yuexuan Wang, Xiaohong Hao, Qiang-Sheng Hua, and Francis C.M. Lau

1

Heuristic Algorithms for Constructing Connected Dominating Sets with Minimum Size and Bounded Diameter in Wireless Networks . . . . . . Jiguo Yu, Nannan Wang, and Guanghui Wang

11

Energy-Eﬃcient Algorithm for the Target Q-coverage Problem in Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hui Liu, Wenping Chen, Huan Ma, and Deying Li

21

Approaching the Optimal Schedule for Data Aggregation in Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pei Wang, Yuan He, and Liusheng Huang

26

Theoretical Foundations Approximate Optimization for Proportional Fair AP Association in Multi-rate WLANs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wei Li, Yong Cui, Shengling Wang, and Xiuzhen Cheng

36

Minimum CDS in Multihop Wireless Networks with Disparate Communication Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lixin Wang, Peng-Jun Wan, and Frances Yao

47

Minimum Edge Interference in Wireless Sensor Networks . . . . . . . . . . . . . . Trac N. Nguyen, Nhat X. Lam, D.T. Huynh, and Jason Bolla Maximum Weighted Independent Set of Links under Physical Interference Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiaohua Xu, Shaojie Tang, and Peng-Jun Wan

57

68

Energy-Aware Algorithms and Protocol Design A QoS-Guaranteed Energy-Eﬃcient Packet Scheduling Algorithm for WiMax Mobile Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hung-Cheng Shih and Kuochen Wang

75

XIV

Table of Contents

Minimum Energy Cost k -barrier Coverage in Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huiqiang Yang, Deying Li, Qinghua Zhu, Wenping Chen, and Yi Hong On the Performance of Distributed N -Cooperation Power Allocation via Diﬀerential Game in Cognitive Radio System . . . . . . . . . . . . . . . . . . . . . Shunxi Gao, Long Zhang, Suqin Fan, Wei Huang, Qiwu Wu, and Yu Deng Energy-Eﬃcient Restricted Greedy Routing for Three Dimensional Random Wireless Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minsu Huang, Fan Li, and Yu Wang

80

90

95

Wireless Sensor Networks and Applications Adaptive Energy and Location Aware Routing in Wireless Sensor Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hong Fu, Xiaoming Wang, and Yingshu Li

105

Utilizing Temporal Highway for Data Collection in Asynchronous Duty-Cycling Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tao Chen, Deke Guo, Honghui Chen, and Xueshan Luo

110

The Impact of Reader to Tag Collision on RFID Tag Identiﬁcation . . . . . Yiyang Zhao, Weijun Hong, S.C. Cheung, and Shufang Li

115

A Desynchronization Tolerant RFID Private Authentication Protocol . . . Qingsong Yao, Yong Qi, Ying Chen, and Xiao Zhong

120

Study of Joint Routing and Wireless Charging Strategies in Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zi Li, Yang Peng, Wensheng Zhang, and Daji Qiao

125

Page Size Optimization for Code Dissemination in Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wei Dong, Xi-bin Zhao, and Min Xi

136

Dynamic Routing Algorithm for Priority Guarantee in Low Duty-Cycled Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Guodong Sun and Bin Xu

146

Applications and Experimentation Heterogeneity of Device Contact Process in Pocket Switched Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ye Tian and Jiang Li

157

Table of Contents

XV

Delay Minimization of Tree-Based Neighbor Discovery in Mobile Robot Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heejun Roh, Kyunghwi Kim, and Wonjun Lee

167

Two-Stage Target Locating Algorithm in Three Dimensional WSNs under Typical Deployment Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lei Mao, Junzhao Du, Hui Liu, Deke Guo, Xing Tang, and Ning Wei

172

Scheduling and Channel assignment Interference Analysis for FH-Based Multi-radio Wireless Mesh Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Davis Kirachaiwanich and Qilian Liang Interference-Aware Gossiping Scheduling in Uncoordinated Duty-Cycled Multi-hop Wireless Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xianlong Jiao, Wei Lou, Xiaodong Wang, Junchao Ma, Jiannong Cao, and Xingming Zhou

182

192

A Game Theoretic Approach to Multi-radio Multi-channel Assignment in Wireless Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Devu Manikantan Shila, Yu Cheng, and Tricha Anjali

203

PAPR Analysis for SOFDM and NC-SOFDM Systems in Cognitive Radio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xue Li, Chi Zhou, Xiangqian Zhou, Zhiqiang Wu, and Bing Xie

209

Coding, Information Theory and Security Application of Compressed Sensing for Secure Image Coding . . . . . . . . . . Gesen Zhang, Shuhong Jiao, and Xiaoli Xu

220

Eﬃcient Wireless Broadcasting Using Onion Decoding . . . . . . . . . . . . . . . . Pei Wang, Qunfeng Dong, Mingjun Xiao, and Liusheng Huang

225

A Spectrally Eﬃcient Anti-Jamming Technique Based on Message Driven Frequency Hopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lei Zhang, Jian Ren, and Tongtong Li

235

Security of Wireless and Ad-Hoc Networks Secure RFID Application Data Management Using All-Or-Nothing Transform Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Namje Park and Youjin Song

245

Prevention of Wormhole Attacks in Mobile Ad Hoc Networks by Intrusion Detection Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ming-Yang Su and Kun-Lin Chiang

253

XVI

Table of Contents

Data Management and Network Control in Wireless Networks A Publicly Veriﬁable Encryption Scheme with Short Public/Private Keys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yujun Liu, Yonggang Cui, and Limin Liu

261

Algorithm on Self-organization of Wireless or Connectionless Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhen Shen, Sheng Qiang, and Dong-yun Yi

266

A Strongly Partitioned Operating System Model for Data Link Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiaoming Tang, Yuting Zhao, Yinjuan Li, and Yujun Liu

274

Twin Hybrid ElGamal Encryption over Signed Quadratic Residue Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yonggang Cui and Yujun Liu

282

Extra Slot Allocation for Fair Data Collection in the Slot-Based Grid Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Junghoon Lee and Gyung-Leen Park

287

An Eﬃcient Multipath Existence Checking Scheme for Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feng Wei, Yingchang Xiang, and Bowu Zhang

291

Data Collection Scheme for Two-Tier Vehicular Sensor Networks . . . . . . . Junghoon Lee and Mikyung Kang

295

Radar and Sonar Sensor Networks Energy Eﬃcient Water Filling Ultra Wideband Waveform Shaping Based on Radius Basis Function Neural Networks . . . . . . . . . . . . . . . . . . . . Weixia Zou, Bin Li, Zheng Zhou, and Shubin Wang An Introduction to Bayesian Techniques for Sensor Networks . . . . . . . . . . Bin Liu Fuzzy C-Means Clustering Based Robust and Blind Noncoherent Receivers for Underwater Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . Bin Li, Zheng Zhou, Weixia Zou, and Shubin Wang Research on Enhanced Spectrum Eﬃciency for BWA Networks . . . . . . . . Xu-hui Wang and Cheng-lin Zhao

299 307

314 322

Table of Contents

XVII

Compressive Sensing for Communications and Networking Improved Channel Estimation Based on Compressed Sensing for Pulse Ultrawideband Communication System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dejian Li, Zheng Zhou, Feng Zhao, Weixia Zou, and Bin Li Compressive Sensing Using Singular Value Decomposition . . . . . . . . . . . . . Lei Xu and Qilian Liang The Wideband Spectrum Sensing Based On Compressed Sensing and Interference Temperature Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ting Jiang and Shijun Zhai

330 338

343

The Applications of Compressive Sensing to Radio Astronomy . . . . . . . . . Feng Li, Tim J. Cornwell, and Frank De hoog

352

Compressive Sensing for Autoregressive Hidden Markov Model Signal . . . Ji Wu, Qilian Liang, and Zheng Zhou

360

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

365

Arbitrary Obstacles Constrained Full Coverage in Wireless Sensor Networks Haisheng Tan1 , Yuexuan Wang2 , Xiaohong Hao2 , Qiang-Sheng Hua1 , and Francis C.M. Lau1 1 2

Department of Computer Science, The University of Hong Kong, Pokfulam, Hong Kong, China Institute for Theoretical Computer Science, Tsinghua University, Beijing, 100084, China

Abstract. Coverage is critical for wireless sensor networks to monitor a region of interest and to provide a good quality of service. In many application scenarios, full coverage is required, which means every point inside the region (excluding the obstacles) must be covered by at least one sensor. The problem of using the minimum number of sensors to achieve full coverage for an arbitrary region with obstacles is NP-hard. Most existing coverage methods, such as contour-based ones, simply place sensors along the boundaries to cover the holes that are near the obstacles and the region boundary. These methods are ineﬃcient especially when the obstacles or the region are irregular. In this paper, based on computational geometry, we design a full coverage method, which accurately ﬁnds the uncovered holes and places sensors eﬃciently for both the regular and irregular obstacles and regions. Speciﬁcally, we show that the more irregular the obstacles and the region are, the more sensors our method can save. Keywords: Wireless sensor networks, Coverage, Obstacles, Computational Geometry.

1

Introduction

Wireless sensor networks (WSNs) have been applied extensively in military and civilian applications, and health care, for such purposes as environmental monitoring, intrusion detection, cancer monitoring, and smart agriculture [1]. Coverage is one of the fundamental issues in WSNs. It is measured by how well the region can be monitored and certain services can be provided. Paper [3] studies how to place disks to fully cover a plane. The authors prove that it is asymptotically optimal, in terms of the number of disks used, to place disks

The work presented in this paper was supported in part by Hong Kong RGC-GRF grants (7136/07E and 714009E), the National Basic Research Program of China Grant Nos. 2007CB807900, 2007CB807901, the National Natural Science Foundation of China Grant Nos. 60604033,60553001, and the Hi-Tech research and Development Program of China Grant No. 2006AA10Z216.

G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 1–10, 2010. c Springer-Verlag Berlin Heidelberg 2010

2

H. Tan et al.

Fig. 1. The optimal placement pattern on a plane

Fig. 2. Coverage of a sensor in a ﬁnite rectangle with an obstacle (the gray area is covered)

on the vertices of equilateral triangles (Figure 1). Several other optimal deployment patterns have also been proposed to achieve full coverage and k-connectivity (k ≤ 6) on a plane in [4]. The Art Gallery problem [14] studies how to guard a gallery with the minimum number of cameras, where a location is guarded as long as there is line-of-sight from a camera. In this paper, in addition to the constraint of the limited sensing radius (rs ), the coverage of a sensor (s) is also aﬀected by obstacles and the region boundary (Figure 2). The problem of using the minimum number of sensors to achieve full coverage for a region with obstacles has been proved to be NP-hard [5]. Paper [2] categorizes the approaches for coverage into three groups: force based [6,7], grid based [4,8,13], and computational geometry based [10,11,12]. The authors of [7] propose a virtual force algorithm (VFA) to enhance an initial random placement over a region with obstacles. Their method extremely depends on the sensors’ mobility and energy. In [9], an eﬃcient algorithm is designed for a √robot to place sensors amid obstacles. Paper [12] deploys sensors with distance 2rs along the boundaries of obstacles and the region. Then Delaunay triangulation is applied to determine the positions for additional sensors for full coverage. The whole region is divided into single-row and multi-row regions in [13]; besides full coverage, the authors also guarantee network connectivity. In order to cover the holes, they deploy sensors along the boundaries of obstacles and the region with a constant distance based on a relationship between rs and the sensor’s communication radius rc . To handle areas near obstacles and the region boundary, existing methods for stationary sensors [9,12,13] would simply place sensors along the boundaries with constant distances. We call these methods contour-based. To achieve full coverage, contour-based methods are eﬀective for simple regular regions and obstacles, such as where the boundaries are long straight lines. However, for obstacles and regions that are arbitrarily irregular, such as when there are comb-shaped boundaries, these methods become ineﬃcient. Without considering the speciﬁc shapes of boundaries, they may not produce full coverage unless additional sensors are placed at each turning point on the boundaries. In fact, even the best results by contour-based methods that place sensors with dynamic distances can be ineﬃcient (Figure 3). In this paper, in view of the ineﬃciency of existing deployment methods when handling irregular boundaries, we consider the shapes of boundaries explicitly. Using computational geometry techniques, our algorithms can ﬁnd the holes accurately and cover them eﬃciently. Our method performs excellently for both regular and irregular obstacles and regions, including some extremely irregular

Arbitrary Obstacles Constrained Full Coverage in Wireless Sensor Networks

(a)

(b)

3

(c)

Fig. 3. Coverage near a part of the boundary: (a) the contour-based placement with a constant distance: 5 sensors, (b) the best for the contour-based methods: 2 sensors, (c) the optimal placement: 1 sensor

ones. The rest of the paper is organized as follows: in Section 2, we deﬁne the models and the problem; Section 3 describes our placement method; Section 4 gives the analysis and experimental results; Section 5 discusses some variations of our method; Section 6 concludes the paper and points out some future work.

2

Problem Definition

The region of interest, denoted as A, is a 2D ﬁnite area with an arbitrary boundary and contains arbitrary obstacles. There are no isolated subregions, and the obstacles have no holes inside themselves. The area that needs to be covered is the region minus the obstacles. Both the region and obstacles are modeled as simple polygons of ﬁnite sizes on the 2D plane. The sensors are stationary and homogenous with a ﬁxed sensing radius rs . Here we assume the binary sensor model: the sensing range of s is a disk centered at s with a radius of rs ; a point is covered by a sensor if it is within a distance of rs and there is a line of sight from the sensor. The coverage of a sensor with obstacles, Cov(s), may be a partial disk (Figure 2). If there are n obstacles inside A, and the obstacle i occupies an area Obsi , the total area occupied can be denoted as O = ∪i∈[1,n] Obsi . And Cov(s) is deﬁned as: Cov(s) = {u ∈ A − O| u − s ≤ rs , ku + (1 − k)s ∈ A − O, ∀k ∈ [0, 1]} Full coverage for a sensing region A means every point in the area A−O must be within Cov(s) of at least one sensor s. Our sensor placement problem is deﬁned as placing the minimum number of sensors in a ﬁnite region with arbitrary boundary and obstacles to achieve full coverage.

3

Sensor Placement Algorithms

To achieve full coverage for a region A, our method works in four procedures: 1) to deploy a regular pattern over the region; 2) to ﬁnd the uncovered holes; 3) to partition the holes into triangulations; and 4) to place sensors to cover the holes. In the following, we explain these procedures in details. 3.1

Optimal Regular Pattern Deployment (ORPD)

We ﬁrstly deploy the regular pattern in Figure 1, which is optimal for covering a plane, for the region A disregarding obstacles and the region boundary. The

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starting position can be randomly chosen at (x, y) ∈√A. The sensors are placed at (x, y) as well as its six neighbors with distance 3rs in the pattern. Note that we assume a sensor can be ‘placed’ inside an obstacle or outside A in this procedure. For each sensor, as long as it is in A, sensors are added at its neighbor locations. We continue doing this until every neighbor of the sensors inside A is occupied by one sensor. We use three lists Lobs , Lin and Lout to keep track of the sensors placed inside obstacles O, inside A − O, and outside A respectively. Figure 4 gives an example.

Fig. 4. The region after Fig. 5. Finding a hole the ORPD: the white area (gray areas are hole(s)) is A − O

3.2

Fig. 6. Holes are found and merged (gray areas are holes)

Finding Holes

After the ORPD procedure, there may be some subareas near an obstacle or the region boundary, called holes, that are not covered, which is due to: 1) the lack of sensors: some sensors are placed into obstacles or outside the region, such as s1 and s2 in Figure 4; 2) blocking: some sensors’ coverage is blocked by obstacles or the region boundary, such as s3 and s4 . The arbitrary shapes of the obstacles and the region make it diﬃcult to cover these holes eﬃciently. Our method overcomes this by ﬁrstly accurately ﬁnding the holes. The ORPD procedure can be regarded as a tessellation of regular hexagons with edge rs inscribed in the circles (Figure 1). Therefore, we model the sensing range of a sensor s by the regular hexagon, and denote it as H(s). Then, it is easy to ﬁnd the holes caused by the lack of sensors. For each sensor deployed inside obstacles or outside A, the hole of s, hole(s) is the overlap, Hin (s), between H(s) and A − O. It is not as easy to calculate the holes caused by blocking (Figure 5). For a sensor deployed inside A − O, we also ﬁrst calculate Hin (s). If Hin (s) is nonempty, holes may exist inside it. We denote the perimeter of an area as P (area). P (Hin (s)) consists of PH (s), the blue lines in Figure 5, and PA (s), the red lines excluding the intersection points with PH (s). Note that PH (s) is a part of P (H(s)), and PA (s) is a part of P (A − O). Next, we draw directed lines from s to each vertex v of Hin (s) and extend them until reaching P (H(s)). The line → − sv crosses P (Hin (s)) at points p1 , p2 , ..., pn (n > 0) listed in increasing distances → from s. The segment of − sv inside H(s) is denoted as (s, p1 , p2 , ..., pn ). We set p0 = s and obtain Theorem 1 for ﬁnding the vertices of hole(s). Theorem 1. Given (s, p1 , p2 , ..., pn ), for ∀k ∈ [1, n), pk is a vertex of hole(s) if and only if one of the following conditions is satisfied:

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1) pk+1 ∈ PH (s) 2) pk+1 ∈ PA (s) AND pk +p2 k+1 ∈ Hin (s) p +p 3) pk−1 ∈ PA (s) AND k 2 k−1 ∈ Hin (s) 4) pk−1 ∈ PA (s) AND pk+1 ∈ PA (s) For k = n, pn is a vertex of hole(s) if and only if pn ∈ PH (s) and pn−1 = s. → Proof. As each − sv stops at P (H(s)), we can get k = n ⇐⇒ pk ∈ P (H(s)), and k ∈ [1, n) ⇐⇒ pk ∈ PA (s). P (H(s)) − PH (s) is inside obstacles or outside A. Therefore, pn is a vertex of hole(s) if and only if pn ∈ PH (s) and pn−1 is not s. As for k ∈ [1, n), the theorem is proved by enumerating all the cases of the positions of pk , pk−1 and pk+1 as follows: pk PA (s) PA (s) PA (s) PA (s) PA (s) PA (s)

pk−1 s s s PA (s) PA (s) PA (s)

pk+1 pk is a vertex of hole(s) PH (s) True P (H(s))−PH (s) False PA (s) True iﬀ pk +p2 k+1 ∈ Hin (s) PH (s) True P (H(s))−PH (s) True iﬀ pk +p2 k−1 ∈ Hin (s) PA (s) True

As the perimeter of hole(s) can only consist of parts of P (Hin (s)) and segments → on each directed line − sv, we have Theorem 2: →i , i = Theorem 2. Connecting the vertices of hole(s) along P (Hin (s)) and − sv 1, 2, ..., m, where m is the number of vertices in Hin (s), we get hole(s). For a single sensor, hole(s) may comprise a set of subholes. We merge the subholes that share at least a point. For all the sensors placed, we continue merging their holes, and denote the ﬁnal set of holes as HOLE. According to our ﬁnding process, we can see that the holes have good properties: 1) narrow in width: the width of the hole for a single s is no larger than √ rs . After merging, each part of a hole in HOLE has a width no larger than 3rs ; otherwise there must be some sensors inside the hole, which is a contradiction; √ 2) accurate in size: if the sensing range of s is simulated as H(s) of area 3 2 3 rs2 as we do now, HOLE is exactly the uncovered area. Actually the range is a disk √ of area πrs2 . Then hole(s) is at most (π − 3 2 3 )rs2 larger than the real uncovered area in the sensing range of s. Algorithm 1 implements the whole procedure, and Figure 6 gives an example. 3.3

DT-Based Partition of Holes

To cover a hole in HOLE, we partition it into triangles whose edges are no longer than rs , so that a sensor can be placed at any of the three vertices of a triangle. We achieve this by the following three steps: Step 1: Partitioning the long edges. If an edge of the hole is longer than rs , rls − 1 points are added to partition it evenly, such as AB in Figure 7. We treat the newly added points as vertices of the hole.

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Algorithm 1. Finding Holes Input:the region A, obstacles O, Lin , Lout and Lobs Output: the holes HOLE 1: For each item s ∈ Lout ∪ Lobs do 2: hole(s) = H(s) ∩ (A − O) 3: if hole(s) = ∅, HOLE = HOLE ∪ hole(s) 4: End For 5: For each item s ∈ Lin do 6: Hin (s) = H(s) ∩ (A − O) 7: If Hin (s) = ∅, Then 8: Finding the vertices of hole(s) /* Theorem 1 */ Compute and merge subholes to get hole(s); update HOLE /* Theorem 2 */ 9: End If 10: End For 11: For each pair of items hole1 and hole2 in HOLE 12: if hole1 ∩ hole2 = ∅, hole = hole1 ∪ hole2 and update HOLE 13: End For

Step 2: Applying Delaunay triangulation. We apply DT over the vertices of the hole, and get a triangulation of the hole. Here DT is chosen because 1) it is eﬃcient (O(nlogn) time, where n is the number of vertices), and 2) the minimum angle of all the triangles is maximized thus avoiding skinny triangles [14]. Based on the advantage of DT and the property that our holes are narrow, the triangulation will add no or just a few long edges that need to be handled in next step. Step 3: Partitioning triangles with an edge longer than rs , such as CD in Figure 7. Also, to avoid skinny triangles, we always partition the longest edge among all the triangles. This step stops until all the edges are bounded by rs and we ﬁnally get a partition, Thole , over the hole. The partition for a hole from Figure 6 is illustrated in Figure 7. After handling all the holes in HOLE, we get a set of partitions, Tholes , as described in Algorithm 2. 3.4

Placing Sensors to Cover Holes

Based on the partition, full coverage is equivalent to satisfying the requirement that at least one vertex of each triangle in Tholes is placed a sensor. For each Thole in Tholes , we ﬁrst compute its dual graph, D(Thole ), which has a node v for every triangle t(v) in Thole . Two nodes u and v form an edge if t(v) and t(u) share at least a point. If D(Thole ) is a tree, it is 3-colorable. When performing DFS (Depth-First Search) on the tree, for each v being visited except the root, there must be one or two (when t(v) shares only a vertex with the triangle visited in the last step) free nodes uncolored in t(v). However, in our case, D(Thole ) may be a graph with cycles and not 3-colorable. In order to ensure that every triangle has three diﬀerent colored vertices, we allow a vertex to be ‘doubly colored’. When performing DFS on the graph, for the last node v in a cycle, the free vertex in t(v) has been colored. We check whether the three vertices of t(v) have had three diﬀerent colors. If yes, we go on to visit the next node directly; if no, we append the absent color to a colored free vertex and make it doubly colored,

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Algorithm 2. DT-based partition of Holes Input: the set of holes HOLE Output: a partition Tholes over HOLE 1: For each hole in HOLE do 2: Initialize an array Lp , a partition Thole and a sorted array Le empty 3: Lp = Lp ∪ vertices of hole 4: For each edge e of hole do /* Step 1 */ add length(e) − 1 points evenly on e and append the points to Lp rs End For /* Step 2 */ 5: Do DT over Lp and store the triangulation of hole to Thole 6: Insert edges in Thole longer than rs to Le in decreasing order of lengths /* Step 3 */ 7: While(Le is nonempty) do 8: add length(e) − 1 points evenly on the ﬁrst item e and remove it rs 9: For each triangle containing e do connect the points added to the vertex that is not on e; update Thole 10: if there are edges added longer than rs , insert them to Le 11: End For 12: End While 13: Tholes = Thole ∪ Tholes 14: End For

such as the vertex D in Figure 7. After coloring, we choose the smallest group of the vertices with the same color to place our sensors. Because for each cycle in D(Thole ), there is at most one vertex in Thole doubly colored, we have Theorem 3: Theorem 3. For a hole h, if it has n vertices in Th and c cycles in D(Th ), n+c 3 sensors can always fully cover it. After placing sensors to cover all the holes, together with the sensors in Lin , we can get full coverage of the region A (Figure 8).

Fig. 7. Partition a hole into triangles and color the vertices.

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Fig. 8. Full coverage: red Fig. 9. Cover a rectangle of and green points are sen- long edges: red and green sors points are sensors

Analysis and Experiments

Because of the arbitrary obstacles and the region, to analyze the bounds on the number of sensors for full coverage, we need to use parameters from both the input

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region and our placement algorithms. For the region A, nv is the number of vertices on P (A − O), and the length of each edge i on P (A − O) is leni . For the holes in HOLE, the number of vertices located on {PH (s)|hole(s) = ∅} is nvph , and = ∅} is nvpa . The number of sensors to the number of vertices on {PA (s)|hole(s) cover HOLE is nch . k is the number of vertices to divide long edges after applying DT over HOLE. Recall that there are no or at most a few edges longer than rs after DT, so k is small. m is the total number of cycles in the dual graphs of Tholes . m is also small and closely related to the shapes and positions of obstacles and the region boundary. The size of Lin is nin , and the number of vertices in Tholes is nvt . Then the number of sensors to achieve full coverage of A is: N = nin + nch ≤ nin + (nvt + m)/3 / ∗ T heorem 3 ∗ / i nvph + nvpa + i ( len rs − 1) + k + m = nin + 3 leni nvph + nv + i ( rs − 1) + k + m ≤ nin + / ∗ nvpa ≤ nv ∗ / 3 area(A − O) area(A − O) √ N≥ = area(H(s)) 3 3rs2 /2

(1) (2) (3) (4)

nin is the asymptotically optimal number of sensors to cover the region excluding i HOLE by the ORPD procedure. nvph approximates to nvpa + i ( len rs − 1) in most cases, and for irregular obstacles and regions, nvph can be much i smaller. So, the upper bound of the sensors for holes is nearly 23 (nv + i ( len rs − 1)). However, for contour-based methods, the sensors placed along boundaries can i − 1), where r is the constant placement distance not larger reach nv + i ( len r √ than 2rs . We carried out experiments to validate the above analysis and to show the effectiveness of our approach for both regular and irregular cases. The example in Figure 8 illustrates the eﬀectiveness of our approach in handling comb-shaped boundaries. Figure 9 is to cover a rectangle, which illustrates the handling of long straight lines. Moreover, we conducted simulations for four types of regions (Figure 10) using three types of sensors with diﬀerent rs . Figure 11 compares our results with the contour-based ones’. ‘Contour-based-1’ is to ﬁrst place sensors along the boundaries, and then add extra sensors for full coverage via a nearoptimal placement. ‘Contour-based-2’ applies the ORPD procedure and then places sensors along the boundaries to cover the holes. When executing ORPD in ’Contour-based-2’ and in our method, we try diﬀerent locations for the ﬁrst sensor, and choose the one leading to the fewest sensors for full coverage. Greedily, we place the ﬁrst sensor r2s away from the longest boundary and rotate the regular pattern to fully cover the longest boundary. We can see that the more irregular the obstacles and region are, the more sensors our method can save.

5

Discussion

We have considered only opaque obstacles which neither allow sensors to be deployed inside nor allow the sensing signal to pass through. There exist other

Arbitrary Obstacles Constrained Full Coverage in Wireless Sensor Networks

(a)

(b)

(c)

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(d)

Fig. 10. Four types of regions: (a) a regular region and obstacles (b) an indoor-like region (c) an outdoor-like region (d) an extremely irregular region and obstacles

Fig. 11. Comparison of the contour-based methods with ours in above 4 regions

kinds of obstacles such as transparent obstacles (ponds are examples) which do not allow sensors placed inside but allow signals to pass through. When the obstacles and region boundary are all transparent, the holes appearing after the ORPD procedure are only those due to the lack of sensors. We can compute Hin (s) for each s ∈ Lout ∪ Lob . For each nonempty Hin (s), we need to add at most 5 sensors to cover it [5]. Setting n = |{s ∈ Lout ∪ Lob |Hin (s) = ∅}|, we have the upper bound of the sensors needed as nin + 5n. Note that the upper bound in [5] is incorrect because it sums up each Hin (s) without considering their positions which are as crucial as their sizes. An example to refute their upper bound is a w × l rectangle with w → 0. In practice, due to the inherent uncertainty of a sensor’s sensing ability, the probabilistic sensor model is more reasonable than the binary model [7]. As full coverage is achieved in our placement, for any point in A − O, there must be at least one sensor s within a distance rs . Therefore, our coverage probability for any β point is not smaller than e−λre , where λ, re and β are parameters in the model. If a higher probability is required, we need to use a virtual sensing radius r instead β of rs , where rs − re ≤ r < rs , so that the probability e−λ(r+re −rs ) is guaranteed. For arbitrary obstacles, the eﬀective coverage of a sensor can be inﬁnitesimal. The number, shapes, sizes as well as relative positions of obstacles make it hard to design and analyze algorithms for full coverage. In practice, obstacles are rarely extremely irregular. Therefore, we may choose to study coverage for a limited number of special sizes and shapes of obstacles. Moreover, instead of full coverage, we may also choose to ignore holes smaller than a certain predeﬁned threshold.

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Conclusion and Future work

In this paper, we propose an eﬃcient full coverage method for a sensing region with arbitrary boundary and obstacles. We achieve this by carefully studying how the obstacles and the region boundary block coverage. The holes arising from an optimal regular pattern deployment are eﬃciently and accurately calculated. Algorithms based on Delaunay triangulation and 3-coloring are designed to cover these holes. Compared with other methods, the proposed method can achieve full coverage while dramatically saving sensors especially when the region and obstacles are irregular. Besides the extension mentioned in the previous section, future work can be carried out in many directions, such as multiple connectivity and coverage with obstacles, 3D coverage and the surface coverage [15] with obstacles, coverage by mobile and heterogeneous sensors, etc.

References 1. Wu, J.: Handbook on Theoretical & Algorithmic Aspects of Sensor. In: Ad Hoc Wireless, and Peer-to-Peer Networks. Auerbach Publication, US (2006) 2. Aziz, N.A.A., Aziz, K.A., Ismail, W.Z.W.: Coverage Strategies for Wireless Sensor Networks. World Academy of Science, Engineering and Technology 50 (2009) 3. Kershner, R.: The Number of Circles Covering a Set. American Journal of Mathematics 61, 665–671 (1939) 4. Bai, X., Xuan, D., Yun, Z., Lai, T.H., Jia, W.: Complete Optimal Deployment Patterns for Full-Coverage and k-Connectivity (k ≤ 6) Wireless Sensor Networks. In: Proc. of ACM MobiHoc 2008 (2008) 5. Shyam, M., Kumar, A.: Obstacle Constrained Total Area Coverage in Wireless Sensor Networks. CoRR abs/1001.4753 (2010) 6. Howard, A., Poduri, S.: Potential Field Methods for Mobile-Sensor- Network Deployment. In: Bulusu, N., Jha, S. (eds.) Wireless Sensor Networks A System Perspective. Artech House, London (2005) 7. Zou, Y., Chakrabarty, K.: Sensor deployment and target localization based on virtual forces. In: IEEE INFOCOM 2003 (2003) 8. Shen, X., Chen, J., Wang, Z., Sun, Y.: Grid Scan: A Simple and Eﬀective Approach for Coverage Issue in Wireless Sensor Networks. IEEE International Communications Conference, volume: 8, pp. 3480-3484. (2006) 9. Chang, C.Y., Chang, C.T., Chen, Y.C.: Obstacle-Resistant Deployment Algorithms for Wireless Sensor Networks. IEEE Trans. on Veh. Tech. (2009) 10. Wang, G., Cao, G., Porta, T.L.: Movement-Assisted Sensor Deployment. In: IEEE INFOCOM 2004, vol. 4, pp. 2469–2479 (2004) 11. Megerian, S., Koushanfar, F., Potkonjak, M., Srivastava, M.: Worst and Best-Case Coverage in Sensor Networks. IEEE Trans. on Mob. Comput. 4(1), 84–92 (2005) 12. Wu, C.H., Lee, K.C., Chung, Y.C.: A Delaunay Triangulation based method for wireless sensor network deployment. Computer Communications 30 (2007) 13. Wang, Y.C., Hu, C.C., Tseng, Y.C.: Eﬃcient Placement and Dispatch of Sensors in a Wireless Sensor Network. IEEE Trans. on Mob. Comput. 7(2) (2008) 14. Berg, M., Cheong, O., Kreveld, M.V., Overmars, M.: Computational geometry: algorithms and applications, 3rd edn. Springer Press, Heidelberg (2008) 15. Zhao, M.C., Lei, J., Wu, M.Y., Liu, Y., Shu, W.: Surface Coverage in Wireless Sensor Networks. In: IEEE INFOCOM 2009, pp. 109–117 (2009)

Heuristic Algorithms for Constructing Connected Dominating Sets with Minimum Size and Bounded Diameter in Wireless Networks Jiguo Yu1, , Nannan Wang1 , and Guanghui Wang2 1

School of Computer Science, Qufu Normal University, Ri-zhao, Shandong, 276826, P. R. China [email protected], [email protected] 2 School of Mathematics, Shandong University, Ji-nan, Shandong, 250100, P.R. China [email protected]

Abstract. Connected dominating set (CDS) is widely used in wireless networks as a virtual backbone for communication and routing between nodes. In order to measure the quality of a CDS, most researches in this area focus on reducing the size of a CDS, neglecting other important metrics, such as diameter between two communication parties. This paper considers the diameter as a quality factor for CDS construction, and develops two new heuristic algorithms. In particular, the CDS computed by the first algorithm has constant ratios 9 and 3 for its size and diameter, respectively. And that of the second algorithm has constant ratios 5 + ln 10 and 2. Both theoretical analysis and simulation show out the performance of the algorithms.

1

Introduction

Wireless ad-hoc networks are multi-hop, self-organizing autonomous networks, do not rely on any existing or predeﬁned network infrastructure. And terminal nodes randomly dispose. Wireless sensor networks are decentralized distributed systems. Numerous sensors are densely disposed in the monitor region with a random manner. The sensors are used for collecting physical parameters such as light intensity, sound, or temperature. Wireless networks including wireless adhoc networks and wireless sensor networks have been attracting more and more attentions in the recent years and they are being used in a variety of military and civil applications such as battle ﬁelds, disaster recoveries, conferences, concerts, environmental detections, health applications and so on.

The work is supported by NNSF (60373012, 10871119) of China, NSF (ZR2009GM009, ZR2009AM013), Promotional Foundation (BS2009DX024) for Middle-aged or Young Scientists, EDRP (J07WH05) of Shandong Province, DRF and UF (XJ0609) of QFNU. The corresponding author.

G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 11–20, 2010. c Springer-Verlag Berlin Heidelberg 2010

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An undirected graph G = (V, E) is usually used to represent a wireless network. V is the set of mobile hosts in the network and E represents all the links in the network. It is assumed that all the hosts are deployed in a 2-D plane and their maximum transmission range R are the same. Thus the resultant topology graph of the network is modeled as an undirected Unit Disk Graph (UDG) [3]. Otherwise, it is a general graph [13]. In the context of graph theory, we call a host as a node. A link between nodes u and v exists if and only if their distance is at most R. Although a wireless network has no physical backbone, many researchers have proposed the concept of virtual backbone infrastructure [5], which organizes a hierarchical structure of ordinary nodes to achieve scalability and eﬃciency. Finding a connected dominating set (CDS) of a given graph is an eﬀective way for constructing a virtual backbone. In general, a dominating set (DS) of a graph G = (V, E) is a subset V ⊂ V such that each node in V − V is adjacent to at least one node in V . A connected dominating set (CDS) is a dominating set whose induced subgraph is connected. The nodes in the CDS are called the dominators. Other nodes in V are called dominatees. With the help of the CDS, the routing is easier and can adapt quickly to the topology changes, and only the nodes in the CDS need to maintain the routing information. Then the search space for the route is reduced only within the CDS. The construction of a minimum-sized CDS in an UDG is proved to be NP-hard [7]. In order to measure the quality of a CDS, the size of the CDS is the primary concern. However, there exist other important metrics that are also needed to be taken into account. In addition to size, this paper also considers diameter as a metric to evaluate the constructed CDS. The diameter of a given connected graph is the length of the longest shortest paths between a pair of nodes in the graph. Mohammed et al. [12] pointed out that in wireless networks, the probability of message transmission failure often increases when a message is sent through a longer path. The rest of the paper is organized as follows. Section 2 reviews some related works. Section 3 presents our two heuristic algorithms for constructing a CDS, and theoretical analysis shows out the performance of the algorithms. Section 4 gives the results of simulation. Section 5 concludes this paper.

2

Related Works

The idea of using a CDS as a virtual backbone for routing was proposed in [6]. Since then many eﬀorts [1, 2, 4, 8, 11, 12] have been made to design approximations or heuristics for MCDS in wireless networks which focus on devising algorithms to produce a small approximation ratio. However, all of the above works only consider the size of a CDS. The diameter of a CDS which is also an important parameter should be another primary concern when constructing a CDS. In [12], the diameter of a CDS is taken into account. However, the performance ratio of the algorithm is not given. Kim et al. proposed a heuristic algorithm (CDS-BD) for constructing a CDS with bounded diameters in wireless networks [10]. This paper improves the CDS-BD algorithm and develops two heuristics for constructing connected dominating set with minimum size and bounded

Heuristic Algorithms for Constructing Connected Dominating Sets

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diameter. It also proves that our algorithms have constant performance ratios for the size and diameter of the constructed CDS.

3

Heuristic Algorithms

In order to develop algorithms for wireless networks and give mathematical proofs of their correctness and performance, appropriate models are needed [13]. Let V ⊂ 2 be a set of nodes in the 2-dimensional plane. The Euclidean graph G = (V, E) is called unit disk graph (UDG) if any two nodes are adjacent if only if their Euclidean distance is at most 1. That is, for arbitrary u, v ∈ V , it holds that {u, v} ∈ E ⇔ |u, v| ≤ 1. In other words, two nodes are adjacent if and only if they are within each other’s transmission range. For simplicity, in the following selection of dominators, we use node rank as the deterministic criterion. Deﬁne the rank of node u to be an ordered pair of (du , idu ), where du is the node degree and idu is the node ID of u. We say that a node u with rank (du , idu ) has a higher order than a node v with rank (dv , idv ) if du > dv , or du = dv and idu < idv . The new CDS construction algorithms have two phases. First to construct an MIS for the graph G. Then to connect the MIS. The detailed process of the phase 1 is as follows. ——————————————————————————————————– Phase 1: M IS(G) (G = (V, E)) ——————————————————————————————————– /* ﬁnd an MIS in the following way: */ 1. Choose a root r ∈ V (i.e. using a leader selection algorithm); 2. Compute the hop distance from r to each node (i.e. using breadth ﬁrst search); 3. Let Vk = {y ∈ V |Hopdist(r, y) = k}, kmax is the maximum k, and Gk is the subgraph of G induced by Vk ; /* Hopdist(x,y) is the hop distance from node x to node y over the shortest path between them. */ 4. Find a maximal independent set(MIS) Ik of Gk for all odd k. Let I be the Ik ; union of Ik for all odd k, thus I = k∈odd

5. Color all nodes of I in black and color every node adjacent to a black in gray; 6. Return I. —————————————————————————————————— Lemma 1. For any unit disk graph G, the size of a maximal independent set is at most 3 ∗ opt + 3, where opt is the size of a minimum connected dominating set. This lemma has been proved in [14], and the bound is tight. Theorem 1. I returned from phase 1 is an MIS of the whole graph G.

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Proof. From phase 1, we can see that I =

Ik , that is, I is the union of

k∈odd

Ik for all odd k. Thus I dominates all the nodes of Vk . Furthermore, phase 1 colors every node adjacent to a black in gray, thus all the nodes of Vk−1 are gray. When the phase 1 ﬁnishes, each node in V is either black or gray, dominated by I. Therefore, I is an MIS of G. When the phase 1 ends, an MIS I of G is obtained. The next phase is to ﬁnd the least connectors to connect the nodes in the MIS to form a CDS of G. The main diﬃculty in lowering the approximation ratio between CDS and MCDS lies in the reduction of connectors used to connect the nodes in the MIS [9]. There are two methods to connect the nodes in I obtained from the phase 1. —————————————————————————————————— Phase 2: CDS(G) (M IS(G), G = (V, E)) —————————————————————————————————— Method 1: /* connect I into a CDS in the following way: */ 1. Suppose k is odd; 2. for k = 1 to kmax do 3. if Ik has i (i ≥ 2) black nodes 4. for m = 2 to i do 5. if there are m black nodes in Ik which have common parent nodes in Vk−1 6. then color the parent node y with highest rank in black; 7. else Each of the m black nodes selects a parent node u with the highest rank in Vk−1 , and colors u in black. Suppose v is u’s parent node with the highest rank in Vk−2 , color v in black; 8. end for 9. else if Ik has only one black node x 10. then color y ∈ Vk−1 in black, which is the neighbor of x with the highest rank; Suppose z is y’s parent node with the highest rank in Vk−2 , color z in black; 11. end if 11. end for 12. Add all black nodes into C; 13. Return C. Method 2: /* connect I into a CDS in the following way: */ 1. Initialize the set of blue nodes S as ∅; 2. while there are more than one connected black-blue components exist do /*A black-blue component is a connected component of the sub-graph induced only by black and blue nodes without considering connections between blue nodes.*/ 3. Choose the vertex v that connects the maximum number of black-blue components, change it’s color from gray to blue;

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15

4. And set S = S ∪ {v}; 5. end while 6. Return the blue node set S; 7. Denote C as the set of black and blue nodes in V , thus C = I ∪ S; 8. Return C. —————————————————————————————————— Theorem 2. C returned from method1 in phase 2 is a CDS of G. Proof. After phase 1, all the nodes should be black or gray, and all the black nodes (set I) form a dominating set. Now we need to prove that C formed by adding more black nodes to I is connected. As we all know, if a graph is connected, then every node (except the root r) has a path to the root node. Therefore, in order to show C is connected, we now prove by induction that there exists a path from every black node x ∈ V /r to the root r. Let x be a black node in V2i−1 . When i = 1, x is connected to r, since r is the only parent node of x. Assume for i ≤ k, all the black nodes are connected to r. For a black node x ∈ V2(k+1)−1 , we color y ∈ V2k black where y is the highest rank parent node of x’s parent nodes in V2k . Furthermore, y is gray before becoming black, dominated by I2k−1 , that is, y is dominated by a black node z in V2k−1 , thus x is connected with z. Owing to for i ≤ k, all the black nodes are connected to r, thus z is connected to r. Therefore, the black node set C is a CDS of G. Theorem 3. C returned from method2 in phase 2 is a CDS of G. Proof. After phase 1, an MIS I of G is obtained, then I is also a dominating set of G. In the method2, a CDS is constructed by selecting blue nodes to connect the nodes in the MIS I. In this method, we select the node that could connect the maximum number of connected black-blue components together each time, and color the node in blue. A black-blue component is a connected component of the sub-graph induced only by black and blue nodes, without considering connections between blue nodes. Using this heuristic algorithm, we can select the least blue nodes to connect the nodes in I. Finally, the set C of black and blue nodes is a CDS of G. Thus, phase 1 and method1 of the phase 2 constitute the ﬁrst algorithm for CDS, and phase 1 and method2 of the phase 2 constitute the second algorithm. The following analyzes the performance ratios for the size and diameter of CDS constructed by the two algorithms. Theorem 4. Let set C is the CDS constructed by the first heuristic algorithm, opt is the size of a minimum connected dominating set of G. Then |C| ≤ 9 ∗ opt + 7. Proof. The main idea of the algorithms is ﬁrst to construct an MIS I of G, then to select connectors for connecting the nodes in I to form a CDS. Therefore, set

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C includes the nodes of I and connectors, thus |C| = |I| + |connectors|. From lemma 1, we know that |I| ≤ 3 ∗ opt + 3. Now we consider the size of connectors. If there are α black node groups in I having common parent nodes, and m is the average size of these groups. Then we have |connectors| ≤ 2(α + |I| − mα − 1) = 2(|I| − (m − 1)α − 1),

(1)

|C| = |I| + |connectors| = 3|I| − 2(m − 1)α − 2.

(2)

Suppose the worst happens, that is, none of the black nodes in I has common parent nodes. Then, α = 0. Therefore, |C| = 3|I| − 2.

(3)

Furthermore, |I| ≤ 3 ∗ opt + 3, therefore, |C| ≤ 9 ∗ opt + 7.

(4)

Lemma 2. Let S is the set of blue nodes obtained from method2 in phase 2, opt is the size of a minimum connected dominating set for an Unit Disk Graph G. Then |S| ≤ (2 + ln 10) ∗ opt. This lemma has been proved in [15]. Theorem 5. Let set C is the CDS constructed by the second heuristic algorithm, opt is the size of a minimum connected dominating set of G. Then |C| ≤ (5 + ln 10) ∗ opt + 3. Proof. From the method 2 in the phase 2, we know |C| = |I| + |S|. From the lemma 1, we have |I| ≤ 3 ∗ opt + 3, and from the lemma 2, we have |S| ≤ (2 + ln 10) ∗ opt. Therefore, |C| = |I| + |S| ≤ (5 + ln 10) ∗ opt + 3.

(5)

Theorem 6. In the first algorithm, every black node in Vk is away from root r within a distance of at most Tk , where Ti =

1 3 3 i − , i ∈ (1, 3, 5, · · ·), Ti = i − 1, i ∈ (2, 4, 6, · · ·). 2 2 2

(6)

Proof. Let Di be the distance within which a black node in Vi is away from r. According to the method1, if k is an odd number, then the maximum number of introduced black nodes is 2. Therefore, the distance between a node at level k to a node at level k − 2 is at most 3, i.e. Dk ≤ Dk−2 + 3. Therefore, D1 = 1, D3 ≤ D1 + 3, . . . , Dk ≤ Dk−2 + 3

(7)

Heuristic Algorithms for Constructing Connected Dominating Sets

17

By solving these inequalities, we have: Dk ≤ 3( k2 − 12 ) + D1 = 32 k − 12 .

(8)

If k is an even number, one black node at level k is adjacent to a black node at level k − 1. Thus, Dk ≤ Dk−1 + 1 = 32 (k − 1) −

1 2

+ 1 = 32 k − 1.

Since Ti is the maximum Di , the theorem is true.

(9)

Theorem 7. In the second algorithm, every black node in Vk is away from root r within a distance of at most Tk , where Ti = i, i ∈ (1, 3, 5, · · ·).

(10)

And every blue node in Vk is away from root r within a distance of at most Tk , where Ti = i, i ∈ (2, 4, 6, · · ·). (11) Proof. Let Di be the distance within which a black node in Vi is away from r. According to the method2, if k is an odd number, then the maximum number of introduced connectors is 1. Therefore, the distance between a node at level k to a node at level k − 2 is at most 2, i.e. Dk ≤ Dk−2 + 2. Therefore, D1 = 1, D3 ≤ D1 + 2, . . . , Dk ≤ Dk−2 + 2

(12)

By solving these inequalities, we derive: Dk ≤ k − 1 + D1 = k.

(13)

k is an even number if one blue node is at level k. And one blue node at level k is adjacent to a black node at level k − 1. Thus, Dk ≤ Dk−1 + 1 = k − 1 + 1 = k. Since Ti is the maximum Di , the theorem is true.

(14)

Theorem 8. Let D∗ be the diameter of any optimal CDS of a graph G, then the diameter of CDS C constructed by our first algorithm satisfies: Diam(C) ≤ 3D∗ + 5, while the diameter of CDS C constructed by our second algorithm satisfies: Diam(C) ≤ 2D∗ + 4. Proof. Let p denote the longest shortest-path from a node u to a node v in graph G. According to the deﬁnition of diameter, we have |p| = Diam(G). From our algorithms, we can see that a tree based on breadth ﬁrst search is built ﬁrst. Thus, it is obvious that Diam(G) ≥ kmax . The diameter of a CDS is at least Diam(G)-2, i.e. D∗ ≥ Diam(G) − 2. Since there exists a path p between u and v, which only includes nodes in C except u and v. If u and v are dominatees, we have D∗ ≥ |p | − 2.

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Meanwhile, |p | ≥ Diam(G). Therefore, D∗ ≥ |p | − 2 ≥ Diam(G) − 2 ≥ kmax − 2

(15)

According to the theorem 6, the maximum distance from any black node to r is 3 1 2 kmax − 2 . Thus, Diam(C) ≤ 2( 23 kmax − 12 ) = 3kmax − 1 (16) = 3(kmax − 2) + 5 ≤ 3D∗ + 5 According to the theorem 7, the maximum distance from any node in C to r is kmax . Thus, Diam(C) ≤ 2kmax = 2(kmax − 2) + 4 = 2D∗ + 4.

(17)

4

Simulation

This section shows the results of simulations and evaluates the performance of the algorithms through comparing with the size and diameter of CDS constructed by our two algorithms, CDS-1 and CDS-2, and two centralized algorithms in [10], CDS-BD-C1 and CDS-BD-C2. Simulations are carried out under an ideal network environment. There is no consideration of node’s movement and channel

(a) r= 1 5

(b) r= 2 0

(c) r= 2 5

(d) r= 3 0

Fig. 1. CDS size comparison when the transmission range of nodes is (a)15, (b)20, (c)25, (d)30.

Heuristic Algorithms for Constructing Connected Dominating Sets

( a) r = 1 5

( b) r = 2 0

(c ) r= 2 5

( d) r = 3 0

19

Fig. 2. CDS diameter comparison when the transmission range of nodes is (a)15, (b)20, (c)25, (d)30.

collision. All simulations are implemented in C++. In order to generate a random network, n nodes are randomly distributed in a 100 × 100 2D space. We need to set two parameters. One is the number of nodes in the network, n (ranging from 0 to 100 by 5). The other one is the transmission range of each node, r (varying among 15, 20, 25 and 30). For each ﬁxed number of nodes, we perform the simulation for 200 times and compute the average value. Figure 1 shows the comparison of the four algorithms in terms of CDS size when the maximum transmission range of nodes is 15, 20, 25 and 30. No matter how many the transmission range of nodes, the CDS constructed by CDS-2 has the smallest size, while CDS-BD-C1 obtains the largest CDS size. And the results of CDS-1 and CDS-BD-C2 are close, but that of the CDS-BD-C2 is slightly better than CDS-1. Moreover, as the number of nodes in the network and the transmission range are increasing, CDS-2 is more better than other three algorithms. Figure 2 shows the comparison of the four algorithms in terms of CDS diameter when the maximum transmission range of nodes is 15, 20, 25 and 30. CDS-2 always generates CDS with the shortest diameter, regardless of the number of nodes or transmission range. As the transmission range of nodes is increasing, the diameter of CDSs constructed by the four algorithms is reducing.

5

Conclusion

This paper studies on constructing a minimum CDS with bounded diameter in wireless networks, and proposes two heuristic algorithms. First to construct a maximal independent set MIS for the graph G, then to select connectors to connect the nodes in MIS using two heuristic rules for obtaining a CDS. Theoretical

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analysis and simulation results prove that the two algorithms have good performances on CDS size and diameter. In particular, the second one has better performance ratios than other related algorithms in the literature.

References 1. Alzoubi, K.M., Wan, P.J., Frieder, O.: Message-optimal connected dominating sets in mobile ad hoc networks. In: Proceedings of the 3rd ACM International Symposium on Mobile Ad Hoc Networking and Computing (June 2002) 2. Cardei, M., Cheng, M.X., Cheng, X., Du, D.-Z.: Connected domination in ad hoc wireless networks. In: International Conference on Computer Science and Informatics (2002) 3. Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit Disk Graphs. Discrete Mathematics 86, 165–177 (1990) 4. Cheng, X., Ding, M., Du, H., Jia, X.: Virtual backbone construction in multi hop ad hoc wireless networks. Wireless Communications and Mobile Computing 6(2), 183–190 (2006) 5. Das, B., Bharghavan, V.: Routing in ad hoc networks using minimum connected dominating sets. In: ICC 1997, Montreal, Canada (June 1997) 6. Ephremides, A., Wieselthier, I., Baker, D.: A design concept for reliable mobile radio networks with frequency hopping signaling. Proc. IEEE 75(1), 56–70 (1987) 7. Garey, M.L., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco (1979) 8. Guha, S., Khuller, S.: Approximation algorithms for connected dominating sets. Algorithmica 20, 374–387 (1998) 9. Islam, K., Akl, S.G., Meijer, H.: A Constant Factor Localized Algorithm for Computing Connected Dominating Sets in Wireless Sensor Networks. In: 2008 14th IEEE International Conference on Parallel and Distributed Systems, pp. 559–566 (2008) 10. Kim, D., Li, Y., Zou, F., Du, D.-Z.: Constructing Minimum Connected Dominating Sets with Bounded Diameters in Wireless Networks. IEEE Transactions on Parallel and Distributed System 20(2), 147–157 (2009) 11. Marathe, M.V., et al.: Simple heuristics for unit disk graphs. Networks 25, 59–68 (1995) 12. Mohammed, K., Gewali, L., Muthukumar, V.: Generating quality dominating sets for sensor networks. In: Proceedings of the Sixth International Conference on Computational Intelligence and Multimedia Applications, pp. 204–211 (August 2005) 13. Schmid, S., Wattenhofer, R.: Algorithmic models for sensor networks. In: Proc. of the 14th International Workshop on Parallel and Distributed Real-Time Systems (2006) 14. Vahdatpour, A., Dabiri, F., Moazeni, M., Sarrafzadeh, M.: Theoretical Bound and Practical Analysis of Connected Dominating Set in Ad Hoc and Sensor Networks. In: Proceedings of the 22nd International Symposium on Distributed Computing, pp. 481–495 (2008) 15. Zou, F., Li, X., Kim, D., Wu, W.: Construction of Minimum Connected Dominating Set in 3-Dimensional Wireless Network. In: Li, Y., Huynh, D.T., Das, S.K., Du, D.-Z. (eds.) WASA 2008. LNCS, vol. 5258, pp. 134–140. Springer, Heidelberg (2008)

Energy-Eﬃcient Algorithm for the Target Q-coverage Problem in Wireless Sensor Networks Hui Liu, Wenping Chen, Huan Ma, and Deying Li Key Laboratory of Data Engineering and Knowledge Engineering, MOE School of Information, Renmin University of China, China [email protected]

Abstract. In this paper we study the target Q-coverage (TQC) problem where each target needs to be covered by diﬀerent numbers of sensors. We try to ﬁnd a collection of Q-covsets which satisfy the coverage quality requirement to maximize the network lifetime. We ﬁrst prove that the problem is NP-Hard. Then we design a greedy algorithm to eﬃciently compute the Q-covsets. Finally, simulation results are presented to verify our approach. Keywords: target Q-coverage, network lifetime, greedy algorithm, wireless sensor network.

1

Introduction

Target coverage is a fundamental problem in wireless sensor networks for environment monitoring and surveillance purposes. In general, sensor nodes are powered by limited battery resources. To maximize the network lifetime, many algorithms were proposed to organize sensors into a number of subsets, such that each set completely covers all the targets, and scheduling the time to make these subsets activated successively, such that only one set is active at any time instant [1]-[3]. In [4], the authors studied the maximizing lifetime for multi-attributes of target coverage problem in wireless sensor networks. [6] [7] studied k-coverage problem with the assumption that a sensor can monitor only one target at a time. In practical applications, individual targets are usually associated with diﬀerent importance. According to the importance of every target, diﬀerent quality of coverage is required, that is to say, each target needs diﬀerent numbers of sensors to cover to satisfy its QoS requirement. This problem is called target Q-coverage problem which is ﬁrst deﬁned in [8]. The authors [8] proposed the column general based approach to solve the problem, but time complexity of the approach may not be a polynomial time. In [9], a simpler heuristic algorithm for the target Q-coverage problem is proposed. [5] studied target coverage in directional sensor network where each target has diﬀerentiated coverage quality requirement.

Corresponding author.

G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 21–25, 2010. c Springer-Verlag Berlin Heidelberg 2010

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In this paper, we also study target Q-coverage problem. The rest of this paper is organized as follows. Section 2 describes the target Q-coverage problem. We prove it is NP-hard and give a greedy algorithm to solve this problem in Section 3. Section 4 presents the simulation results for comparing our algorithm with the HESL algorithm in [9].

2

Network Model and Problem Description

Suppose there are a number of targets {a1 , a2 , · · · , am } with known locations deployed in a plane, while a great number of sensors {s1 , s2 , · · · , sn } are randomly scattered around the targets and continuously observe (cover) them. Each sensor si has its sensing radius ri . A target aj can be sensed by a sensor si if and only if the distance between aj and si is at most si ’s sensing radius. Suppose sensor si has a initial lifetime Ei . We use an integer vector Q = {q1 , q2 , · · · , qm } to express coverage requirement of targets, where qj is the minimum number of sensors which simultaneously cover target aj . To extend the network lifetime and meet coverage demand, we use the sensing scheduling that alternates the sensors between active state and sleep state and guarantees that each target can be covered by at least the requirement number of sensors. Definition 1. Target Q-coverage (TQC) problem: Given m targets and n sensors in an energy constrained wireless sensor network, how to schedule at least qj (qj ≥ 1) sensors at any time cover aj target (j = 1, · · · , m) such that the network lifetime is maximized. Definition 2. A subset D of sensors is called Q-covset if it can satisfy the coverage requirement, i.e., for each target aj , there are at least qj sensors in D which can cover aj . Our goal is to try to ﬁnd a collection of Q-covsets and assign the lifetimes of Q-covsets to maximize the network lifetime.

3

The Greedy Algorithm

In this section, we ﬁrst prove that the TQC problem is NP-hard. Theorem 1. The TQC problem is NP-hard. Proof: The maximum set covers (MSC)[1] is a special case of the TQC problem. This is because when qj = 1 for each target aj , the TQC problem becomes the MSC problem. Since the MSC problem is NP-hard [1], thus the TQC problem is NP-hard. In the following, we propose a greedy algorithm for the TQC problem: Greedy with the More Uncovered Targets (the GMUT algorithm). The main idea of the GMUT algorithm is follows:

Energy-Eﬃcient Algorithm for the Target Q-coverage Problem

23

Firstly, we form a Q-covset Dk (k = 1, · · · , K). Here we sort sensors by the decreasing number of covering uncovered targets(If there are some sensors with the same numbers of covering uncovered targets, we list these sensors in order of the decreasing available lifetime). Next we put the ﬁrst sensor su from the sort set to Dk and modify the values of correlated variables, such as the requirement of targets covered by it. Then repeat these steps till we form a Q-covset. Secondly, we allocate a time slot to every Q-covset Dk . tk is used to represent the time that each Q-covset is active. We set tk as the smaller value between the minimum available lifetime E[i] from Dk and a given small constant w. Next we update the correlated variables’ values, for example the sensors’ available lifetimes in Dk . At last, we continue to ﬁnd the next Q-covset till there are not enough sensors with available lifetimes to cover the targets. The greedy algorithms returns Knumber of Q-covsets: D1 , D2 , · · · , DK with lifetime t1 , t2 , · · · , tK . The GMUT algorithm is formally represented as follows:

Algorithm 1. GMUT-TQC(S, A, Ei , Q) 1: SET E[i] = Ei , i = 1, 2, · · · , n, K = 1 2: WHILE each target aj is covered by at least qj available sensors in S do 3: Stemp = S, Atemp = A, DK = ∅, Q[j] = qj , j = 1, · · · , m = ∅ do 4: WHILE Atemp 5: sort sensors by the decrement of the numbers of uncovered targets that sensors cover. If some sensors have the same numbers, give the priority to the sensor with maximum available lifetime. 6: Choose the ﬁrst sensor su ∈ Stemp from the sort set 7: let DK = DK ∪ {su }, Stemp = Stemp − {su } 8: FORALL target aj ∈ Atemp do 9: IF aj is covered by su then 10: Q[j] = Q[j] − 1 11: IF Q[j] == 0 then 12: Atemp = Atemp − {aj } 13: tK = min(minsi ∈DK E[i], w), T = T + tK 14: FORALL sensors si ∈ DK do 15: E[i] = E[i] − tK 16: IF E[i] == 0 then 17: Stemp = Stemp − {si } 18: K++ 19: Return K Q-covsets: D1 , D2 , · · · , DK with lifetime t1 , t2 , · · · , tK .

The input parameters of this algorithm are below: S as the set of sensors, A as the set of targets, Ei as the initial lifetime of the ith sensor and Q = {q1 , · · · , qm } as the targets’ coverage requirements. The set Stemp maintains the list of sensors that have the residual lifetimes greater than zero, thus these sensors can participate in additional Q-covsets. The set Atemp contains the targets that still have not be covered by the current Q-covsets Dk .

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The Performance Evaluation below shows our algorithm is more better than the HESL algorithm in [9]. This is because each time we select the sensor which covers the maximal number of uncovered targets and has higher residual lifetime to form Dk , thus the formed Q-covset Dk is with both less number and higher residual lifetime. However the HESL algorithm just chooses the sensor with the highest residual lifetime, therefore, its Q-covset may not be better in energyeﬃcient and its network lifetime may lesser than ours.

4

Performance Evaluation

In this section, we will compare our algorithm with the algorithm (HESL) in [9]. In our simulation, we deploy the sensor nodes and targets randomly in a region of 300m × 300m area. To evaluate the performance of our algorithm, we consider the following tunable parameters: the number of sensors N , the number of targets M and the sensing range of sensors R. We suppose each sensor has initial energy Ei randomly from 1 to 10, each target’s coverage requirement qj is selected randomly from 1 to 3, and the time slot w is 0.1. In each experiment, we run 100 times and compute its average value. In ﬁrst case, we set the sensing range of each sensor is 60m and the target number is ﬁxed 10. From Fig.1, we ﬁnd that the network lifetime increases as the number of sensors increasing. In second case, we set the sensing range of sensors is 60m and there are 200 sensors deployed randomly in the area. We can see that

180 Network Lifetime

120

HESL GMUT

90 60 30

150

HESL GMUT

120 90 60 30 0

0 60

70

80

90

5

100 110 120 130 140 150

Number of Sensors

Fig. 1. Network lifetime vs. N

10

15

20 25 30 35 Number of Targets

250 HESL GMUT

200 150 100 50 0 50

60

70

40

45

Fig. 2. Network lifetime vs. M

300 Network Lifetime

Network Lifetime

150

80

90 100 110 120 130 140 150 Sensing range

Fig. 3. Network lifetime vs. sensing range

50

Energy-Eﬃcient Algorithm for the Target Q-coverage Problem

25

the network lifetime decreases with the increasing of the number of targets from Fig.2. In third case, we have the parameters setting as that: 100 sensors and 15 targets, and from Fig.3 we ﬁnd that the network lifetime increases with the increasing of the sensing range. Obviously, our algorithm is much better than the HESL algorithm.

5

Conclusion

In this paper, we study the target Q-coverage problem. First prove the problem is NP-hard. Then a greedy algorithm is proposed to solve this problem. At last, we compare our greedy algorithm with the HESL algorithm and simulation results show that our algorithm is more better than the HESL algorithm.

Acknowledgment This paper was supported in part by Renmim University of China under Grants 10XNJ032 and 10XNG037.

References 1. Cardei, M., Thai, M.T., Li, Y., Wu, W.: Energy-eﬃcient target coverage in wireless sensor networks. In: INFOCOM (2005) 2. Cardei, M., Du, D.-Z.: Improving wireless sensor network lifetime through power aware organization. ACM Wireless Networks 11(3), 333–340 (2005) 3. Liu, H., Wan, P., Yi, C., Jia, X., Makki, S., Niki, P.: Maximal lifetime scheduling in sensor surveillance networks. In: INFOCOM (2005) 4. Wang, L., Li, D.-Y., Li, Z.: Maximizing lifetime for multi-attributes of target coverage problem in wireless sensor networks. In: CWSN (2009) 5. Yang, H., Li, D.-Y., Chen, H.: Coverage quality based target-oriented scheduling in directional sensor networks. In: ICC (2010) 6. Liu, H., Wan, P., Jia, X.: Maximal lifetime scheduling for sensor surveillance systems with K sensors to 1 target. IEEE Transactions on Parallel and Distributed Systems 17(12), 1526–1536 (2006) 7. Liu, H., Jia, X., Wan, P., Yi, C., Makki, S.K., Pissinou, N.: Maximizing lifetime of sensor surveillance systems. IEEE Transactions on Networking 15(4), 334–345 (2007) 8. Gu, Y., Ji, Y., Li, J., Zhao, B.: QoS-aware target coverage in wireless sensor network. Wireless Communications and Mobile Computing 9(12), 1645–1659 (2009) 9. Chaudhary, M., Pujari, A.-K.: Q-coverage problem in wireless sensor networks. In: Garg, V., Wattenhofer, R., Kothapalli, K. (eds.) ICDCN 2009. LNCS, vol. 5408, pp. 325–330. Springer, Heidelberg (2009)

Approaching the Optimal Schedule for Data Aggregation in Wireless Sensor Networks Pei Wang1 , Yuan He2 , and Liusheng Huang1 1

Department of Computer Science University of Science and Technology of China, China 2 Department of Computer Science Hong Kong University of Science and Technology, China

Abstract. Due to the large-scale ad hoc deployments and wireless interference, data aggregation is a fundamental but time consuming task in wireless sensor networks. This paper focuses on the latency of data aggregation. Previously, it has been proved that the problem of minimizing the latency of data aggregation is NP-hard [1]. Using maximum independent set and ﬁrst ﬁt algorithms, in this study we design a scheduling algorithm, Peony-tree-based Data Aggregation (PDA), which has a latency bound of 15R + Δ − 15, where R is the network radius (measured in hops) and Δ is the maximum node degree. We theoretically analyze the performance of PDA based on diﬀerent network models, and further evaluate it through extensive simulations. Both the analytical and simulation results demonstrate the advantages of PDA over the state-of-art algorithm in [2], which has a latency bound of 23R + Δ − 18. Keywords: wireless sensor networks, data aggregation, latency.

1

Introduction

A wireless sensor network (WSN) consists of a large number of sensor nodes which communicate with each other via their RF transceivers. Owing to the ad hoc deployments and self-organizing characteristics of sensors, WSNs are suitable for a wide variety of applications such as scientiﬁc observation, environmental monitoring, health care, and military surveillance [3,4,5,6,7,8,9]. In those applications, the operation of data aggregation is often used for querying information like event numbers. Shortening the latency of data aggregation is a fundamental requirement in WSNs, especially in real-time applications. In this paper, we focus on reducing the latency of data aggregation by designing a good schedule. This is a challenging issue mainly because of the intrinsic interference in WSNs. When two interfering signals are sent simultaneously, neither of them can be received correctly. Certainly, we can schedule the transmissions of data aggregation to avoid interference. The latency of data aggregation, however, will thus be increased inevitably. Without a good scheduling scheme, data aggregation will result in an impractically high latency. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 26–35, 2010. c Springer-Verlag Berlin Heidelberg 2010

Approaching the Optimal Schedule for Data Aggregation

27

Previously, it has been proved that the problem of minimizing the latency of data aggregation is NP-hard [1]. Many scheduling algorithms have been proposed to reduce the latency of data aggregation [1,2,10].Theoretically, Chen et al. in [1] prove that the problem of minimizing the latency of data aggregation is NP-hard. They also design an algorithm for data aggregation which has a latency bound of (Δ − 1)R, where Δ is the maximum node degree and R is the network radius measured in hops. Kesselman and Kowalski [10] design an algorithm which has a latency bound of O(log(N )). However, they assume that each node can learn the distance to the closest neighbor and has a special collision detection capability, although such conditions cannot always be guaranteed in WSNs. Among all the existing works, the proposal of Wan et al. in [2] is the closest to our work. They propose a scheduling algorithm based on maximum independent set, which has a latency bound of 23R+Δ−18. They mainly focus on a special scenario, however, in which the communication range equals the interference range. Wan’s results are the state-of-arts on scheduling algorithms of data aggregation. The objective of our study in this paper targets approaching the optimal schedule of data aggregation. We design a scheduling algorithm for data aggregation which has a latency bound of 15R + Δ − 15. Our proposed algorithm, Peonytree-based Data Aggregation, or PDA in short, works as follows. Initially, a data aggregation tree called peony tree is constructed by using maximum independent set and ﬁrst ﬁt algorithms. The data aggregation process mainly consists of two phases: local aggregation and global aggregation. In local aggregation, raw data at the leaf nodes are transmitted to their corresponding parent nodes for aggregation. In global aggregation, data gathered in local aggregation are further aggregated layer by layer in a bottom-up manner. The latency bounds of the two phases are Δ − 1 and 15R − 14, respectively. We also theoretically analyze the performance of PDA and deduce the latency bound based on a more generic network model, which does not assume equal communication and interference ranges. The rest of this paper is organized as follows. Section 2 introduces the terminology, models, and assumptions in this paper as well as the problem formulation. In Section 3 we elaborate the scheduling algorithm of data aggregation. Section 4 presents the proofs of correctness and theoretical performance analysis, followed by the simulations results in Section 5. We conclude the work in Section 6.

2 2.1

Preliminaries Network Model

We consider a WSN consisting of one sink node s and N sensor nodes. Every sensor node is equipped with only one radio. All nodes share a common wireless channel to communicate. We also assume omni-directional antennas and unit-disk model. A node is only able to communicate with nodes within the communicate range rc and be interfered by nodes within the interference range ri . A node u can receive data correctly if and only if there is exactly one node sending data in the interference range ri of u. Otherwise, collision happens and node u cannot receive data correctly. Our algorithm has no assumptions on rc

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and ri . For ease of presentation, we ﬁrst assume rc = ri when elaborating the algorithm in Section 3. We will remove this assumption in Section 4. As for the process of data aggregation, we partition time into slots. A node can either send or receive a data packet in a time slot. The latency of data aggregation in this paper refers to the number of time slots passed during the entire course of data aggregation. Consider a sending node set A and its corresponding receiving node set B. We say data can be aggregated from A to B in a time slot if and only if all nodes in B can receive data correctly when all nodes in A transmit data simultaneously. Now we formulate the problem of data aggregation. We use an undirected graph G = (V, E) to represent a WSN of N nodes, where V denotes the set of nodes and E denotes the set of edges. There is an edge between node u and node v if and only if the Euclidean distance between them is not greater than rc . A feasible data aggregation schedule is a sequence of node-disjoint sending i=K sets {S1 , S2 , ..., SK }, where Si = V − {s}. All nodes in Si can transmit data i=1

simultaneously without interference in the ith time slot. After all nodes in SK ﬁnish transmitting data in the Kth time slot, all data are aggregated to the sink node. That is, the latency of data aggregation with the above schedule is K. Denote the corresponding receiving set of Si by Ri . It can be seen that l=K Ri ⊆ ( Sl ) {s}. l=i+1

2.2

Independent Set and Concurrent Set

Node u and node v are independent if and only if they are not within the interference range of each other. If all nodes in X are independent from each other, we call node set X is an independent set. When node u and node v are not independent from each other, we call u and v cover each other. We will use independent set to select dominative nodes in the aggregation tree. Consider two links l1 : p → q and l2 : u → v where p, q, u and v are four nodes. Link l1 and link l2 are independent if and only if p is independent from v and q is independent from u. When link l1 and link l2 are independent, node p and node u can transmit simultaneously without interference. Let L = (VS , VR , f ) be a link set, where VS is the sending set, VR is the receiving set, and f is a mapping function from VS to VR . If all links in L are independent from each other, link set L is an independent link set. Thus, transmissions through the links in an independent link set can be executed simultaneously without interference. Here we call the sending set VS as a concurrent set. Later in this paper, we will use concurrent set to divide the nodes into multiple sets and correspondingly schedule data transmissions into diﬀerent time slots.

3

Peony-Tree-Based Data Aggregation (PDA)

PDA mainly consists of three components, the construction of peony tree, local aggregation, and global aggregation.

Approaching the Optimal Schedule for Data Aggregation

3.1

29

Construction of Peony Tree

We construct a tree called peony tree for data aggregation, which includes all the nodes in the network. First, an approximately maximum independent set is formed to cover all the nodes. We call nodes in the maximum independent set as dominative nodes. Some additional nodes are then selected to connect dominative nodes so that all dominative nodes are reachable from the sink node. The selected nodes are called connective nodes. Each of the other nodes in the network selects a neighboring dominative node as its parent, and the peony tree is constructed. A node which is neither a dominative node nor a connective node is called a white node. Therefore we have three types of nodes in total. Diﬀerent from the algorithm proposed in [2] which selects connective nodes arbitrarily, we try to select the minimum number of connective nodes to avoid collision and reduce the aggregation latency. 1. Dominative nodes selection. Initially, we construct a common breadth ﬁrst search tree TBF S rooted at the sink node s. Denote the depth of TBF S by R. All nodes in the network are thus divided into layers L0 , L1 , ..., LR , according to their depths in TBF S . Specially, L0 = {s}. We denote the maximum independent set by D. At the beginning, D = φ. To minimize the aggregation latency, dominative nodes in D are selected layer by layer in a top-down manner. For a node u in Li , if u is independent from all the other nodes in D, add u into D and remove u and its neighbors from all layers. Otherwise, remove u from Li . Repeat this procedure until Li is empty. When there is no node in Li , continue the procedure in Li+1 . After the procedure ﬁnishes at LR , the resulting set D is a maximum independent set, i.e. the dominative nodes are selected. 2. Determination of connective nodes and white nodes. The connective nodes and dominative nodes together constitute the backbone of the network. According to the selection of dominative nodes, a connective node may cover one or more dominative nodes. When connective nodes are selected arbitrarily [2], the number of connective nodes are often more than necessary, reducing the rate of data aggregation. Based on such observation, we select minimum number of connective nodes by using ﬁrst ﬁt algorithm. Denote the peony tree by T = (V, E }, where V is the node set and E is the edge set. We may divide the dominate nodes into layers D0 , D1 , ..., DR according to their hops to the sink node. For a dominative node u in Di , select a node v in Li−1 , which is a neighbor node of u. Node v is set as the parent of all the dominative nodes in Di which cover v. Remove all the children of node v from Di . Then a dominative node p in Di−1 which covers v is set as the parent of v. Repeat this procedure until Di is empty. When there is no node in Di , repeat the procedure in Di−1 . The procedure continues until no dominative node remains in D2 . At last, set s as the parent of all connective nodes in L1 . Thus all the connective nodes are selected.

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Those nodes which are neither dominative nodes nor connective nodes are set as white nodes. Denote the set of white nodes by W . For each white node w in W , a dominative node is selected as its parent, which has the lowest depth among all the dominative nodes covering w. Given the depth of TBF S as R, according to the construction of peony tree, there are both connective nodes and dominative nodes in each layer of TBF S , except in layer 0, layer 1, and Layer R. Speciﬁcally, sink node s is the unique dominative node in layer 0. There are no dominative nodes in layer 1. There are no connective nodes in layer R of TBF S , either. Consequently, the depth of the peony tree is 2(R − 2) + 1 + 1 + 1 = 2R − 1. 3.2

Local Aggregation

During the local aggregation phase, white nodes transmit the sensory data to their parents (the corresponding dominative nodes) for aggregation. Meanwhile, a dominative node takes aggregation operations when all its child nodes have transmitted data. Because transmissions on dependent links should transmit data in diﬀerent time slots, to avoid interference and minimize the latency in local aggregation, the set of white nodes W is divided into k node-disjoint maximum concurrent sets. Denote these node-disjoint maximum concurrent sets by U Wi (i = 1, 2, ..., k), where k is the number of the node-disjoint maximum concurrent subsets of W . Therefore, nodes in U Wi can transmit data without interference in the ith time slot. Then the latency of data aggregation in the phase of local aggregation is k. U Wi (i = 1, 2, ..., k) are iteratively determined by using ﬁrst ﬁt algorithm. The ﬁrst ﬁt algorithm for constructing node-disjoint maximum concurrent sets works as follows. Consider a link set L = (VS , VR , f ), where VS is sending node set, VR is its corresponding receiving node set, and f is a mapping function from VS to VR . Denote all maximum concurrent sets by U1 , U2 , ..., Uh , where h is the number of maximum concurrent sets. For each node u in VS , if link l : u → f (u) is independent from all links in {l : v → f (v) | v ∈ Ui }, add u into Ui and remove u from VS . When there is no more node to add into Ui , repeat the above procedure to form the next maximum concurrent set Ui+1 until no node remains in VS . 3.3

Global Aggregation

Global aggregation involves two types of transmissions. One is from a dominative node to its parent connective node, and the other is from a connective node to its parent dominative node. Global aggregation starts with the transmission from the dominative nodes in LR to their parent connective nodes in LR−1 . The above two types of transmissions run alternately until the data are aggregated from the connective nodes in layer 1 to the sink node. To avoid interference and minimize the latency of global aggregation, we divide dominative nodes and connective nodes in each layer of TBF S into nodedisjoint maximum concurrent sets. Denote them by U Di,1 , U Di,2 , ..., U Di,mi and

Approaching the Optimal Schedule for Data Aggregation

31

U Ci,1 , U Ci,2 , ..., U Ci,ni respectively, where mi is the number of maximum concurrent subsets of Di and ni is the number of maximum concurrent subsets of Ci . Specially, s is sink node and the unique one in layer 0, m0 = 0. Because there is no dominative nodes in layer 1, m1 = 0. Similarly, we have nR = 0. Similarly with local aggregation, we allocate diﬀerent time slots for diﬀerent maximum concurrent sets. Data are aggregated layer by layer in a bottom-up manner. A node transmits data after all its children transmit data. Due to the fact that the latency at local aggregation phase is k, we schedule all dominative nodes and connective nodes as follows. Connective nodes in U Ci,j transmit data l=i+1 in the ( (ml + nl ) + j + k)th time slot. Dominative nodes in U Di,j transmit l=R

l=i+1

data in the (

l=R

(ml + nl ) + ni + j + k)th time slot. We can imply that the

latency of global aggregation is

l=1 l=R

4

(ml + nl ) slots.

Discussion

In this section, we ﬁrst prove the latency bound of our algorithm based on the assumption rc = ri , and then we remove the constraint and deduce the latency bound based on a generic network model. 4.1

Correctness

Theorem 1. The latency bound of data aggregation in local aggregation is Δ−1. Proof. Due to the page limit, we skip the proof here. Theorem 2. The maximum number of dominative neighbors for a connective node is 5. Proof. Due to the page limit, we skip the proof here. Theorem 3. The maximum number of connective neighbors of a dominative node is 12. Proof. Due to the page limit, we skip the proof here. Theorem 4. The latency bound of data aggregation on a peony tree is 15R + Δ − 15. Proof. Recall that our schedule consists of local aggregation phase and global aggregation phase. As the latency bound of local aggregation phase is Δ − 1 according to Theorem 1, we analyze the latency bound of global aggregation phase before we get the latency bound of data aggregation on a peony tree. Recall that in a peony tree, dominative nodes are in the even layers while connective nodes are in the odd layers, so there are two transmission patterns

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in global aggregation. One is data transmission from a dominative node to a connective node, and the other is data transmission from a connective node to a dominative node. First, according to Theorem 2, a connective node has at most 5 dominative neighbors and one of them is its parent. So the number of dominative children of a connective node is at most 4. Consequently, the latency bound of data aggregation from dominative nodes in an even layer to connective nodes in the upper layer is 4. Second, according to Theorem 3, a dominative node has at most 12 connective neighbors and one of them is its parent node. So the number of connective children of a dominative node is at most 11. Consequently, the latency bound of data aggregation from connective nodes in an odd layer to dominative nodes in the upper layer is 11. Specially, the sink node is a dominative node without parent. The latency bound of aggregation from connective nodes in layer 1 to the sink node is 12. In the peony tree, the number of layers which contain connective nodes is R−1. The number of layers which contain dominative nodes except sink node s is also R − 1. Basing on the above analysis, the latency bound of global aggregation is 11(R − 2) + 12 + 4(R − 1) = 15R − 14. Therefore, the latency bound of data aggregation on a peony tree is (Δ − 1) + (15R − 14) = 15R + Δ − 15. 4.2

On Generic Model

Most existing works base their schemes on the assumption rc = ri , so far we also discuss based on this assumption. Indeed, our scheme is not constrained by the assumption, and in this section we remove it. As data aggregation is conducted layer by layer, nodes suﬀer interference only from nodes in the same layer when they transmit data simultaneously. As mentioned above, there are three types of transmissions during the course of data aggregation, below we will discuss separately. For simplicity, we normalize the communication range rc = 1 and the interference range ri = ρ. Denote the upper bound of the number of independent nodes in a radius of ρ by Γ (ρ). Before deducing the latency of data aggregation, we ﬁrst present two theorems related to Γ (ρ). Theorem 5. (Wegner Theorem [11]) The area of the convex hull of n (n ≥ 2) non-overlapping unit-radius circular disks is at least √ √ √ (1) 2 3(n − 1) + (2 − 3) 12n − 3 − 3 + π Theorem 6. Γ (ρ), the upper bound of the number of independent nodes in a radius of ρ, satisfies the condition that √ √ 2 3(Γ (ρ) − 1) + (2 − 3) 12Γ (ρ) − 3 − 3 + π ≤ (2ρ + 1)2 π (2)

Approaching the Optimal Schedule for Data Aggregation

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Fig. 1. The maximum number of independent node in a radius of ρ

Proof. Consider independent nodes in a disk of radius ρ centered at point o as shown in Fig. 1. Place a small disk with radius 0.5 centered at each independent node. Because the distance between any two independent nodes is greater than 1 (rc = 1), all the small disks are non-overlapping. According to Theorem 5, the area covers all the small disks of radius 0.5 is at least √ of the convex hull√which (2 3(Γ (ρ) − 1) + (2 − 3) 12Γ (h) − 3 − 3 + π)/4. Because all the small disks are in a big disk of radius h + 0.5 centered at point o, the convex hull which covers all the small disks is also in the big disk. The area of the convex hull is not greater than that of the big disk, so Γ (ρ) satisﬁes the inequation 2. Now we deduce the latency of data aggregation on generic model. 1. Transmissions from white nodes to dominative nodes. When ρ ≤ 1, a dominative node receives data from at most Δ − 1 white neighbors. Hence, the latency bound of transmissions from white nodes to dominative nodes is still Δ−1. When ρ > 1, a dominative node can be interfered by all the white nodes in the interference range, no matter whether or not they are neighbors. Thus the latency bound is aﬀected by the maximum number of white nodes in a radius of ρ. As there are at most Γ (ρ) dominative nodes in a radius of ρ and a dominative node has at most Δ − 1 white neighbors, the latency bound of transmissions from white nodes to dominative nodes is Γ (ρ)(Δ − 1). 2. Transmissions from dominative nodes to connective nodes. For a connective node u, there are at most Γ (ρ) dominative nodes in a radius of ρ and one of them is its parent. Consequently, the latency bound of transmissions from dominative nodes to connective nodes is Γ (ρ) − 1. 3. Transmissions from connective nodes to dominative nodes. Because a connective node connects dominative nodes in the adjacent layer, the number of connective nodes in a radius of ρ is not more than the number of dominative nodes in a radius of 1 + ρ. Hence, a dominative node has at most Γ (1 + ρ) connective neighbors. Meanwhile, one of those connective nodes is the parent of the dominative node. The latency bound of transmissions from connective nodes to dominative nodes is Γ (1 + ρ) − 1. Specially, the latency from connective nodes in layer 1 to the sink node is at most Γ (1 + ρ).

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Theorem 7. The latency bound of data aggregation in a peony tree is (Γ (ρ) + Γ (1 + ρ) − 2)(R − 2) + Γ (ρ)Δ + Γ (1 + ρ) − 1

(3)

Proof. Recall that there are no connective node in layer R and no dominative node in layer 1 of the tree. The latency from connective nodes in layer 1 to sink node is Γ (1 + ρ). So the latency bound of data aggregation on peony tree is Γ (ρ)(Δ − 1) + (Γ (ρ) − 1) + ((Γ (ρ) − 1) + (Γ (1 + ρ) − 1))(R − 2) + Γ (1 + ρ) = (Γ (ρ) + Γ (1 + ρ) − 2)(R − 2) + Γ (ρ)Δ + Γ (1 + ρ) − 1.

5

Performance Evaluation

In the simulations, we let all sensor nodes have the identical communication range rc and identical interference range ri , where rc = ri = 10m. Sensor nodes are randomly deployed into a 100m × 100m area for multiple times to generate diﬀerent network topologies. 20 diﬀerent topologies are generated with network sizes (measured by the number of nodes) varying from 400 to 2000. We compare PDA with the algorithm proposed by Wan et al. in [2] (denoted as WALG in short in the ﬁgures) by measuring the number of time slots to aggregate all the sensory data in a WSN. Figure 2(a) plots the number of connective nodes selected by PDA and WALG (note that the numbers of dominative nodes selected by the two algorithms are same, so we do not show them in the ﬁgures). We can see the number of connective nodes selected by PDA is much less than that of WALG. Also, WALG is slightly aﬀected by the network size, while PDA performs consistently with diﬀerent network sizes. The total aggregation latency by using PDA is about 20% shorter than that of using WALG, as shown in Fig. 2(b).

(a) Number of connective Nodes

(b) Total aggregation latency

Fig. 2. Comparison with diﬀerent network sizes

6

Conclusion

We study the issue of minimizing the latency of data aggregation in WSNs and propose a scheduling algorithm named PDA. PDA has a latency bound of

Approaching the Optimal Schedule for Data Aggregation

35

15R + Δ − 15, where R is the network radius (measured in hops) and Δ is the maximum node degree. To the best our knowledge, this latency bound is by far the lowest one. Moreover, we remove the constraint of equal communication and interference ranges used in most existing works and discuss the latency bound of PDA based on a generic network model. We compare PDA with the state-of-arts algorithm proposed in [2] through simulations and the results show that PDA signiﬁcantly outperforms existing designs.

References 1. Xujin, C., Xiaodong, H., Jianming, Z.: Minimum data aggregation time problem in wireless sensor networks. In: 1st International Conference on Mobile Ad-hoc and Sensor Networks, pp. 133–142. IEEE Press, Wuhan (December 2005) 2. Scott, C.-H.H., Peng-Jun, W., Chinh, T.V., et al.: Nearly constant approximation for data aggregation scheduling in wireless sensor networks. In: Proceedings of INFOCOM, pp. 366–372. IEEE Press, Anchorage (May 2007) 3. Ian, F.A., Weilian, S., Yogesh, S., Erdal, C.: Wireless sensor networks: a survey. Computer Networks 38(4), 393–422 (2002) 4. Mo, L., Yunhao, L.: Underground structure monitoring with wireless sensor networks. In: Proceedings of ACM/IEEE IPSN, pp. 69–78. ACM Press, Cambridge (April 2007) 5. Robert, S., Alan, M.M., Joseph, P., et al.: An analysis of a large scale habitat monitoring application. In: Proceedings of ACM SenSys, pp. 214–226. ACM Press, Baltimore (November 2004) 6. Mo, L., Yunhao, L.: Iso-map: Energy-eﬃcient contour mapping in wireless sensor networks. IEEE Transactions on Knowledge and Data Engineering (TKDE) 22(5), 699–710 (2010) 7. Mo, L., Yunhao, L., Lei, C.: Non-threshold based event detection for 3d environment monitoring in sensor networks. IEEE Transactions on Knowledge and Data Engineering (TKDE) 20(12), 1699–1711 (2008) 8. Kebin, L., Mo, L., Yunhao, L., et al.: Passive diagnosis for wireless sensor networks. In: Proceedings of ACM SenSys, pp. 113–126. ACM Press, Raleigh (November 2008) 9. Lufeng, M., Yuan, H., Yunhao, L., et al.: Canopy closure estimates with greenorbs: Sustainable sensing in the forest. In: Proceedings of ACM Sensys, pp. 99–112. ACM Press, Berkeley (November 2008) 10. Alexander, K., Dariusz, R.K.: Fast distributed algorithm for convergecast in ad hoc geometric radio networks. J. Parallel Distrib. Comput. 66(4), 578–585 (2006) ¨ 11. Wegner, G.: Uber endliche kreispackungen in der ebene. Studia Scientiarium Mathematicarium Hungarica 21, 1–28 (1986)

Approximate Optimization for Proportional Fair AP Association in Multi-rate WLANs* Wei Li1, Yong Cui2, Shengling Wang2, and Xiuzhen Cheng3 1

Beijing University of Posts and Telecommunications, Beijing, P.R.China [email protected] 2 Tsinghua University, Beijing, P.R.China {cy,slwang}@csnet1.cs.tsinghua.edu.cn 3 The George Washington University, Washington DC, USA [email protected]

Abstract. In this study, we investigate the problem of achieving proportional fairness via Access Point (AP) association in multi-rate WLANs. This problem is formulated as a non-linear program with an objective function of maximizing the total user bandwidth utilities in the whole network. It is NP-hard, and therefore effort in this paper is made to seek approximate solutions. We propose a centralized algorithm to derive the user-AP association via relaxation. Such a relaxation may cause a large integrality gap. Therefore a compensation function is introduced to guarantee that our algorithm can achieve at least half of the optimal solution in the worst-case scenario theoretically. Extensive simulation study has been reported to validate and compare the performances of our algorithms with those of the state-of-the-art. Keywords: AP association, bandwidth allocation, multi-rate WLANs, proportional fairness.

1 Introduction By default, each user in IEEE 802.11 WLANs associates with the AP that has the largest received signal strength indicator (RSSI). As typically users are not uniformly distributed among all APs, RSSI based approach may overload some APs while leave others carrying very light load or even being idle. This load unbalancing could result in unfair bandwidth allocation. Although the network is supposed to serve fairly at high performance, fairness and efficiency are often in conflict with each other. With the development of multi-rate WLANs, this problem has become even more challenging, as users with different bit rates intend to share the same WLAN. 802.11 MAC protocol provides equal long-term transmission opportunities to users. Therefore, users with the same frame size can achieve equal throughput (i.e. throughput-based fairness). However, in multi-rate WLANs, throughput-based fairness requires that users with lower bit rates occupy the channel for a longer time *

Supported by National Natural Science Foundation of China (no.60873252), National Major Basic Research Program of China (no.2009CB320503), and the US National Science Foundation (CNS-0831852).

G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 36–46, 2010. © Springer-Verlag Berlin Heidelberg 2010

Approximate Optimization for Proportional Fair AP Association in Multi-rate WLANs

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than those with higher bit rates, drastically reducing the network throughput [1]. To overcome this problem, time-based fairness is proposed such that each user can obtain an equal share of channel occupancy time. Recent research [1] has shown that timebased fairness outperforms throughput-based fairness in multi-rate WLANs. There are two fairness criteria that are widely used in network resource assignment. Max-min fairness [2] distributes resources as equally as possible among users. While proportional fairness [3], on the other hand, allocates bandwidth to users in proportion to their bit rates to maximize the sum of the bandwidth utility of all users. Proportional fairness has been utilized to effectively exploit the tradeoff between fairness and network performance [3], [4]. Fairness, load balancing, and AP selection are three interrelated dimensions of the resource management problem. Nevertheless, very few recent researches jointly consider these three dimensions in multi-rate WLANs. Though different algorithms have been proposed to achieve fairness [1], [5], they only consider the problem at a single AP instead of the whole network. Other approaches optimize the AP association for efficient resource assignment [6], but they do not consider fairness. Load balancing has been proposed to optimize the resource assignment in [7], but fairness again is not considered. It has been observed that jointly considering AP association and fairness can effectively improve the aggregated network throughput [4], [8] in WLANs, since users may reside in the overlapping coverage areas of multiple APs while each user is only associated with one AP at a time. Li et al. [4] propose two approximate AP selection schemes, cvapPF and nlapPF, for periodic offline optimization. Both cvapPF and nlapPF rely on relaxation and rounding to obtain an integral user-AP association. Bejerano et al. [8] demonstrate the strong correlation between fairness and load balancing, and propose a load balancing technique to obtain an optimal max-min fair bandwidth allocation. In this paper, we propose an algorithm for AP selection to achieve proportional fairness. In our system model, the resources at all APs are considered as a whole when allocating bandwidth fairly to users. With this network-wide fairness objective, load balancing is automatically considered. Our Non-linear Approximate Optimization for Proportional Fairness (NLAO-PF) is centralized and can be adopted periodically in real multi-rate WLANs applications. Since the non-linear optimization problem is NP-hard [4], NLAO-PF is decomposed into four steps to simplify the issue and improve the degree of approximation. Our problem formulation is motivated by the non-linear program in [4] but we adopt a completely different approach to relax the variables in our approximation algorithm design. By introducing a compensation function to the objective function to narrow down the gap caused by relaxation, the total utility of the bandwidth allocation via NLAO-PF is proved to be at least 1/2 of the optimal in the worst case. Our comparison-based simulation study indicates that NLAO-PF performs well when the users are distributed randomly and uniformly in the network. Moreover, the performance is even better when users are distributed in a hot spot area. The rest of the paper is organized as follows. Our system model is introduced in Section 2. The algorithm is detailed in Sections 3. After presenting the evaluation results in Section 4, we conclude this paper in Section 5.

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2 System Model Our network topology models an IEEE 802.11 based multi-rate WLAN network that consists of multiple APs. Each AP has the same limited coverage area and serves users in its area. Overlapping coverage areas of adjacent APs may exist. The union of the coverage areas of all APs forms the network coverage area. We assume that each AP transmits messages with the same power as defined by IEEE 802.11. We further assume that each user is covered by at least one AP, and each AP has at least one associated user. The notations and definitions to be utilized are summarized in Table 1. Table 1. Notations Symbol A Ai n U m

Jij gij pj N0 wi bi rij xij X tij T

Semantics The set of all access points (AP). The set of APs that can be associated with (cover) user i. n=|A|, the number of APs. The set of all users m=|U|, the number of users. The SINR of user i when associated with AP j. The channel gain from AP j to user i. The transmission power of AP j. The receiver noise power. The weight of user i. The effective bandwidth allocated to user i. The effective bit rate between user i and AP j. The association coefficient between user i and AP j. The 0-1 user-AP association matrix. The transmission time between user i and AP j. The transmission time allocation matrix.

Table 2. The Relationship between SINRs and Rates

Jij (dB) rij (Mbps)

6-7.8 6

7.8-9 9

9-10.8 12

10.8-17 18

17-18.8 24

18.8-24 36

24-24.6 48

24.654

As we have known, a user in an overlapping coverage area will be interfered by other APs. The effective bit rate of a user in an 802.11 network is determined by the experienced SINR of the user. More precisely, let γij denote the SINR of user i when associated with AP j. We have, g ij p j (1) γ ij = . ∑ g ik p k + N 0 k∈ Ai ∩ k ≠ j

Further, the relationship between the effective bit rates and the SINR ranges in the 802.11 network is shown in Table 2 [9]. It is assumed that the network is saturated such that all APs are busy all the time and all users always have data ready to send. We will consider a unit of time in which the network is stable, with no new user joins and no current user leaves. This means that under our consideration the total transmission time of an AP is equal to 1. Each AP assigns transmission times to users in accordance with proportional fairness. A user is allowed to choose only one AP within the unit time.

Approximate Optimization for Proportional Fair AP Association in Multi-rate WLANs

39

We formulate the AP association problem based on proportional fairness to a nonlinear program. In our system model, the resources at all APs are considered as a whole when allocating bandwidth to users. With this network-wide fairness objective, load balancing is automatically taken into account. We intend to maximize the total utility of the user bandwidth, which is defined to be the sum of logarithms of the bandwidths allocated to all users [3]. Since the effective bandwidth of user i is bi = ∑nj=1xijtij rij , we obtain the following optimization formulation:

max ∑im=1 wi log(∑nj =1 xij t ij rij ) s.t. 1≤ i ≤ m,∑nj =1xij =1 ,

(a)

1≤ j ≤ n,∑im=1xijtij =1 ,

(b)

1≤i ≤ m,1≤ j ≤n, xij ∈{0,1} ,

(c)

1≤ i ≤ m,1≤ j ≤ n,t ij ∈[0,1] .

(d)

(2)

Eq. (2) is referred as a Non-linear Program (NLP). Note that our objective function considers the weights of users, which reflects their priorities in a real network. The constraint (a) indicates that each user can associate with one and only one AP; the constraint (b) requires that the total transmission time of each AP j is equal to 1; the constraint (c) assures that xij is a binary variable that is equal to 1 if and only if user i associates to AP j; and the constraint (d) specifies the range of the variable tij. We can prove that NLP is NP-hard by slightly adapting the reduction procedure proposed in [4]. Note that this problem formulation is motivated by [4] but our approach to solving the problem via relaxation, as elaborated in Section 3, is fundamentally different and totally novel.

3 The NLAO-PF Algorithm Since NLP is NP-hard, we propose an approximation algorithm Non-Linear Approximate Optimization for Proportional Fairness (NLAO-PF) outlined in Alg.1, to simplify the issue and improve the degree of approximation. Algorithm 1. NLAO-PF 1. {t’ij}= solve r-NLP {{wi}, {rij}}. 2. Get fractional solution {x’ij}= solve c-NLP {{wi}, {rij}, {t’ij}}. 3. Get integral solution {xij} by a rounding process {{t’ij}, {x’ij}}. 4. Redistribute transmission time to obtain {tij} and calculate {bi}.

The basic idea of NLAO-PF is to relax the binary variable xij such that each user is allowed to associate with multiple APs within a unit time. This relaxation may result in a large integrality gap [10]. To overcome this problem, we modify the objective function of NLP by adding a compensation function g(X,T) in NLAO-PF, which is defined as follows. Definition 1. The compensation of user i on AP j is defined by wi xijtij log(rij ) , if rij>0;

0, otherwise. Thus the compensation of user i to all APs can be expressed

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by wi (∑nj =1xij t ij log(rij )) . Therefore the compensation function g(X,T) can be defined correspondingly as follows: g ( X ,T ) = ∑im=1wi (∑ nj=1xij t ij log(rij )) .

(3)

This compensation function is introduced to improve the lower bound of our algorithm to effectively narrow down the integrality gap caused by relaxation. The steps of NLAO-PF are detailed in the following subsections. 3.1 Relaxed Optimization Program

The first step of NLAO-PF is to solve the following relaxed optimization problem to obtain an optimal {t’ij}. max

∑i =1wi log(∑ j =1t ij′ rij )+∑i =1wi (∑ j =1t ij′ log(rij )) m

n

m

n

s.t. 1≤ i ≤ m,∑nj =1tij′ ≤1 ,

(a)

1≤ j ≤ n,∑im=1tij′ =1 ,

(b)

1≤ i ≤ m,1≤ j ≤ n,t ij′ ∈[0,1] .

(c)

(4)

Eq. (4) is referred as the relaxed Non-linear Program (r-NLP). Compared with Eq. (2), r-NLP replaces tij by t’ij, sets xij =1, and includes the compensation function in its objective function. The constraint (a) indicates that the total transmission time of user i with all APs cannot surpass 1; the constraint (b) requires that the total transmission time of each AP is equal to 1, which means that all APs are saturated in the unit time; and the constraint (c) defines the range of the variable t’ij. Obviously, the optimal solution for {t’ij} from Eq. (4) can be found in polynomial time [4]. 3.2 Fractional Association

After solving r-NLP, we obtain the transmission time {t’ij}. Now we take {t’ij} as the input, and get the fractional user-AP association {x’ij}. Because of the requirements for solving convex programs, we change the linear equality constraint of NLP to a linear inequality constraint in the following problem formulation, which does not change the solution value. max ∑im=1wi log(∑nj =1xij′ tij′ rij ) + ∑im=1wi (∑nj =1xij′ tij′ log(rij )) s.t. 1≤ i ≤ m,∑ nj =1xij′ > 0 ,

(a)

1≤ j ≤ n,∑im=1xij′ t ij′ =1 ,

(b)

1≤ i ≤ m,1≤ j ≤ n, xij′ ≥ 0 .

(c)

(5)

Eq. (5) is referred as the complemented Non-linear Program (c-NLP). Its objective function is designed to approximate the optimal solution to NLP. The constraint (a) indicates that a user should connect with at least one AP; the constraint (b) forces the total transmission time of each AP be equal to 1; and the constraint (c) defines the

Approximate Optimization for Proportional Fair AP Association in Multi-rate WLANs

41

range of x’ij for the case of fractional association. Note that here we take {t’ij} obtained from Eq. (4) as the input to c-NLP and obtain the optimal association {x’ij} for c-NLP given {t’ij}. We can prove that the gap introduced by our relaxation procedure is bounded. Let f ( X ,T ) = ∑im=1wi log(∑nj=1xijtij rij ) , h( X ,T ) = f ( X ,T )+ g ( X ,T ) .Then the objective functions of r-NLP and c-NLP become h(X=1, T) and h(X,T), respectively. Correspondingly, h(X=1, T’) and h(X’,T’) are the solutions obtained from r-NLP and c-NLP, respectively. Theorem 1. Let f(X*,T*) be the optimal solution to NLP. Then f(X*,T*)≤h(X’,T’)≤2 f(X*,T*). Proof. With

m ∑i=1xij′ tij′ =1 (constraint (b) in Eq. (5)) and rij ≥ 1, f(X’,T’)≥g(X’,T’)≥0.

Thus h(X’,T’) ≤2f(X’,T’) ≤2f(X*,T*). Note that f(X*,T*) is also feasible to c-NLP from the relationship between NLP and c-NLP. Thus we have, f(X*,T*) ≤f(X=1,T’)+g(X=1,T’)=h(X=1,T’)≤h(X’,T’), where the last inequality holds true from the fact that h(X=1, T’) is feasible to c-NLP. ■ 3.3 Optimization Program Rounding

In this step, we use the rounding algorithm proposed in [11] to obtain an integral association matrix X. That is, we fix the time allocation {t’ij} and replace the fractional association {x’ij} by a 0-1 variable {xij} that encodes the desired association of users to APs. The description of the rounding scheme is as follows. First, we construct a bipartite graph G(x)=(U,V,E), where the set U represents the users in the network, and the set V consists of AP nodes denoted by V={vjk: j=1,…,n,

⎡

⎤

k=1,…,Qj }, with Qj = ∑i=1xij . This means that each AP may have multiple nodes in m

V. The edges in G(x) are constructed in the following way. If Qj≤1, there is only one node vj1 corresponding to AP j. For each x’ij>0, add an edge e(ui, vj1) to E, and set x’(ui, vj1)=x’ij, where x’(e) is the fractional association weight of the corresponding user and AP. Otherwise, find the minimum index ik such that ∑iik=1xij′ ≥ k . For i=ik-1 +1,…, ik

－1 and x’ >0, add edges e(u , v ) and set x’(u , v )=x’ . ij

i

jk

ik −1 ′ i =ik −1 +1x (ui ,v jk ) .

edge e(ui, vjk) and set x′(ui ,v jk ) =1−∑

i

If ∑

jk

ik ′ i =1 xij

ij

For i=ik, add the

> k , add the edge e(ui,

vj(k+1)) and set x′(ui ,v j(k +1) ) = ∑iik=1x′(ui ,v jk )−k . The profit of each edge e(ui, vjk) in E is defined to be wi log(t ij′ rij ) .

Second, we find a maximum-profit matching M(x) that matches each user node to an AP node in G(x). For each edge e(ui, vjk) in M(x), schedule user i on AP j and set xij=1. Set other xij’s to be 0. Since the fractional association {x’ij} specifies a fractional matching, such a maximal matching does exist and it determines the integral association {xij}. More details can be found in [11]. Note that {t’ij} and {x’ij} are computed from r-NLP and c-NLP, respectively. The rounding scheme constructs an integral assignment {xij}. We denote this integral solution as f (Xa,T’), which is also feasible to NLP. We have

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Theorem 2. f (Xa,T’) ≥ f (X*,T*)/2. Proof. Note that {xij} is obtained by employing the rounding scheme proposed by Shmoys and Tardos in [11], which proves the following property: f(Xa,T’)≥f(X’,T’). Thus, f(Xa,T’)≥f(X’,T’)≥[f(X’,T’)+g(X’,T’)]/2=h(X’,T’)/2≥f(X*,T*)/2, where the last ■ inequality holds from Theorem 1. 3.4 Transmission Time Redistribution

Since the user-AP association changes after rounding, we need to redistribute the transmission time. This is the last step of NLAO-PF, in which we assign transmission times to users according to proportional fairness. Theorem 3. Let {xij} be the integral user-AP association coefficients obtained from the rounding procedure outlined in Section 3.3. Given {xij}, the unique optimal transmission time assigned to user i by AP j according to proportional fairness

is tij = xij wi (∑m k =1xkj wk ) . Proof. (a) First, we consider the case of a single AP. Assume that the number of users covered by the AP is m. Since the objective function of Eq. (2) is the sum of logarithms, maximizing the total utility of the user bandwidth (Eq. (2)) is equivalent to maximizing Eq. (6): m m m m ∏i =1(tij rij )wi = ∏i =1(ti1ri1)wi = ∏i =1(ti1) wi ∏i =1(ri1)wi .

(6)

Note that {ri1} is the set of optimization constant. Therefore maximizing Eq. (6) is equivalent to maximizing Eq. (7):

t11 t11)(t t21 t21)"(t t

" " ∏i=1(ti1)wi = (t 11 21 1tm 1" m m1) .

m

w1

Since ∑im=1ti1 =1 ,

Eq.

(7)

w2

is

maximized

(7)

wm

if

and

only

if if t11 w1

t 21

1 (¦ (∑m k =1wk ) . (b) Now we consider the case of multiple APs. Let xkj be a 0-1 variable denoting

w2

t m1 wm

m i 1 wi ) . Thus we have ti1 = wi

the association coefficient between user k and AP j. Then ∑m k =1xkj wk is the sum of the weights of all users associated to AP j. With a similar analysis as that of case (a), the optimal transmission time given {xij} can be calculated by Eq. (8). tij = xij wi (∑m k =1xkj wk ) .

(8)

We conclude that given {xij}, our transmission time assignment based on proportional fairness is unique and optimal. ■ The solution obtained from our algorithm NLAO-PF can be denoted as f (Xa,Ta). Based on Theorems 2 and 3, we have f (Xa,Ta) ≥ f (Xa,T’) ≥ f (X*,T*)/2. That is, the approximate solution obtained from NLAO-PF is no less than half of the optimal solution of NLP.

Approximate Optimization for Proportional Fair AP Association in Multi-rate WLANs

43

4 Evaluation For ease of comparison, we employ the same simulation settings as those in [4], which are detailed as follows. We place a total of 20 APs on a 5 by 4 grid, with each AP on a grid point. The coverage area of each AP is set to 150 meters and the distance between two adjacent APs is set to 100 meters. We arrange 50~300 users to simulate different levels of network loads. Assume that the transmission power of each AP is 20dBm [12]. The coverage area of the network is the union of the coverage areas of all APs. Assume that all users have the same weight. There are two types of user distributions under our consideration: (1) users are randomly and uniformly distributed within the coverage area of the network; (2) users are randomly positioned in a circle-shaped hotspot with a radius of 100 meters near the center of the 20-AP network. We employ a simple wireless channel model in which the effective bit rate only depends on the experienced SINR. For simplicity, we adopt the values commonly advertised by 802.11a which is shown in Table 2. The channel gain is modeled by the following equation,

gij = sij dij−4 ,

(10)

Table 3. Data Analysis on Different Algorithms Case

Uniform

Hotspot

Algorithm FraOp NLAO-PF cvapPF SSF LLF FraOp NLAO-PF cvapPF SSF LLF

Ave.(Mbps) 4.82 4.79 4.19 4.82 3.87 4.06 4.05 3.55 2.04 3.17

(a) uniform case

Var.(Mbps) 3.42 3.78 3.75 9.12 3.36 1.07 1.23 1.53 14.29 1.93

Fairness 0.87 0.86 0.87 0.71 0.85 0.94 0.93 0.93 0.39 0.85

Total utility 129.18 128.01 117.32 120.18 109.53 119.51 118.73 108.01 38.12 92.64

(b) hotspot case

Fig. 1. The user bandwidth of different AP association algorithms

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where sij is a log-normally distributed shadowing factor, and dij is the distance between user i and AP j. Shadowing factors are generated according to the Viterbi model [13], with E(sij)=0dB and σ(sij)=10dB. The receiver noise power N0 is set to -80dBm. Fairness is quantified by the Jain’s fairness index [14], which is defined as follows,

∈

J = (∑im=1bi2 ) (∑im=1bi )2 ,

(11)

where larger values of J [0,1] indicate a better fairness. In proportional fairness, larger values of the total utility, which reflects the tradeoff between the aggregated throughput and fairness, indicate a better performance. During our simulation, the network topology structure and the effective bit rates are assumed to be steady. There is no backhaul capacity limitation. We compare the performances of the following algorithms: (1) NLAO-PF; (2) cvapPF [4]; (3) Strongest Signal First (SSF); (4) Least Load First (LLF) [8]; (5) Since the problem of AP association based on proportional fairness is NP-hard, NLAO-PF and cvapPF only obtain approximate solutions. For comparison purpose, we use the result obtained from r-NLP without the compensation function g(X,T) as a benchmark and call it FraOp. We have performed extensive simulations by varying the number of users and obtained qualitatively similar results. Thus in this subsection we only report the results for the 200-user case. The statistics of achieved bandwidth, Jain’s fairness index, and total bandwidth utility of different algorithms are presented in Table 3. First Fig.1 plots the achieved per-user bandwidth in Mbps vs. user index, with the users sorted by their bandwidths in increasing order. In the uniform case, SSF achieves a little bit higher average bandwidth, demonstrating a much larger variance in the bandwidth allocation and a poorer fairness. However, in the hotspot case where users reside in the vicinity of certain APs, leading to a more intensive competition for resources and a more imbalanced network load. Obviously, SSF aggravates the extent of load imbalance and enlarges the bandwidth allocation variances without considering fairness. It is also difficult for LLF to enhance the user bandwidth effectively because it only takes into account load balancing but ignores the user rate. On the other hand, in these two cases, NLAO-PF and cvapPF can both achieve proportional fairness, but NLAO-PF outperforms cvapPF in terms of bandwidth allocation with a value closer to FraOp. The fairness index of these two algorithms is almost the same. Moreover, the total utility of NLAO-PF is 99% of that of FraOp, while that of cvapPF is about 90%. Therefore, we conclude that NLAO-PF outperforms cvapPF, SSF and LLF. Second, the number of users vs. AP index is shown in Fig.2, with the APs sorted by their users in increasing order. The number of users associated with an AP is considered as the load metric. In these two cases, both NLAO-PF and LLF perform better than the other two algorithms in terms of load balancing. However, from Table 3, we observe that LLF has a lower average bandwidth and a poorer fairness than NLAO-PF, since it ignores the user rate. It can be concluded that NLAO-PF offers a more effective tradeoff between the bandwidth allocation and load balancing. Besides per-user bandwidth, we also compare the aggregated throughput of all algorithms. Fig.3 plots the aggregated throughput in Mbps vs. the number of users. In the uniform case, although SSF achieves a higher aggregated throughput, it reduces the level of fairness in bandwidth allocation. Among the remaining three algorithms

Approximate Optimization for Proportional Fair AP Association in Multi-rate WLANs

(a) uniform case

45

(b) hotspot case

Fig. 2. The load of different AP association algorithms with 200 users

(a) uniform case

(b) hotspot case

Fig. 3. The aggregated throughput of different AP association algorithms

(except FraOp), NLAO-PF obtains much higher aggregated throughputs than the other two algorithms, especially in the hotspot case. On the other hand, compared with cvapPF, NLAO-PF has a higher approximate degree, and its optimization results are more prominent. What is more, in the hotspot case, advantages of our algorithm are even more significant.

5 Conclusion The widespread of multi-rate WLAN applications makes the network management more complex and critical. Fairness and AP association are two hot issues. In multirate WLANs, some users may get starved if fairness is not carefully considered. In this paper, we investigate how to optimize user-AP association to achieve proportional fairness, and propose a new AP association algorithm termed NLAO-PF for this purpose. Although the problem of AP association for proportional fairness is NP-hard, NLAO-PF obtains a result that is guaranteed to be at least half of the optimal solution via a compensation function. Simulations confirm that our scheme can achieve proportional fairness in bandwidth allocation and enhance the aggregated throughput effectively. Moreover, in the hotspot case, the advantage of our algorithm is even more significant.

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References 1. Tan, G., Guttag, J.: Time-based Fairness Improves Performance in Multi-rate WLANs. In: The USENIX Annual Technical Conf., Boston (2004) 2. Bertsekas, D., Gallager, R.: Data Networks, pp. 524–529. Prentice-Hall, Englewood Cliffs (1987) 3. Kelly, F.P.: Charging and rate control for elastic traffic. European Transactions on Telecommunications 8(1), 33–37 (1997) 4. Li, L., Pal, M., Yang, Y.R.: Proportional Fairness in Multi-rate Wireless LANs. In: IEEE INFOCOM, Phoenix, pp. 1004–1012 (2008) 5. Bredel, M., Fidler, M.: Understanding Fairness and its Impact on Quality of Service in IEEE 802.11. In: IEEE INFOCOM, Rio de Janeiro, pp. 1098–1106 (2009) 6. Abusubaih, M., Wolisz, A.: An Optimal Station Association Policy for Multi-Rate IEEE 802.11 Wireless LANs. In: 10th ACM Symposium MSWiM, pp. 117–123 (2007) 7. Rahul, H., Edalat, F., Katabi, D., Sodini, C.: Frequency-Aware Rate Adaptation and MAC Protocols. In: 15th ACM International Conf. MobiCom, Beijing, pp. 193–204 (2009) 8. Bejerano, Y., Han, S.-J., Li, L.E.: Fairness and load balancing in wireless LANs using association control. IEEE/ACM Trans. Networking 15(3), 560–573 (2007) 9. High-speed Physical Layer in the 5 GHz Band, IEEE Standard 802.11a (1999) 10. Azar, Y., Epstein, A.: Convex Programming for Scheduling Unrelated Parallel Machines. In: 30th ACM symposium Theory of computing, Baltimore, pp. 331–337 (2005) 11. Shmoys, D.B., Tardos, E.: An approx algorithm for the generalized assignment problem. Math. Program 62(3), 461–474 (1993) 12. Radio Resource Measurement of Wireless LANs, IEEE Standard 802.11k (2008) 13. Viterbi, A.J.: CDMA: Principles of Spread Spectrum Communication, pp. 185–186. Addison-Wesley, New York (1995) 14. Jain, R., Chiu, D.-M., Hawe, W.R.: A quantitative measure of fairness and discrimination for resource allocation in shared computer system. Digital Equipment, Technical report, DEC-TR-301 (1984)

Minimum CDS in Multihop Wireless Networks with Disparate Communication Ranges Lixin Wang1 , Peng-Jun Wan1 , and Frances Yao2 1 2

Illinois Institute of Technology, Chicago IL 60616, USA City Univesrsity of Hong Kong, Kowloon, Hong Kong

Abstract. Connected dominating set (CDS) has a wide range of applications in mutihop wireless networks. The Minimum CDS problem has been studied extensively in mutihop wireless networks with uniform communication ranges. However, in practice the nodes may have diﬀerent communication ranges either because of the heterogeneity of the nodes, or due to interference mitigation, or due to a chosen range assignment for energy conservation. In this paper, we present a greedy approximation algorithm for computing a Minimum CDS in multihop wireless networks with disparate communications ranges and prove that its approximation ratio is better than the best one known in the literature. Our analysis utilizes a tighter relation between the independence number and the connected domination number.

1

Introduction

Connected dominating set (CDS) has a wide range of applications in multihop wireless networks (cf. a recent survey [2] and references therein). It plays a very important role in routing, broadcasting, and connectivity management in wireless ad hoc networks. Consider a multihop wireless network with undirected communication topology G = (V, E). A CDS of G is a subset U ⊂ V satisfying that each node in V \ U is adjacent to at least one node in U and the subgraph of G induced by U is connected. A minimum CDS (MCDS) of G is a CDS of G with the smallest size. The problem of computing a MCDS in a multihop wireless networks with uniform communications ranges has been intensively studied in the literature. This problem is NP-hard [3], and a number of distributed algorithms for constructing a small CDS in wireless ad hoc networks have been proposed in [1,5,7,8] among others. However, in practice the nodes may have diﬀerent communication ranges either because of the heterogeneity of the nodes, or due to interference mitigation, or due to a chosen range assignment for energy conservation. In this paper, we assume all the nodes V lie in an Euclidean plane, and each node v has a communication radius rv . The communication topology of such a network is deﬁned

This work was supported in part by NSF of USA under grants CNS-0831831 and CNS-0916666, by the RGC of Hong Kong under Project No. 122807, and by the National Basic Research Program of China Grant 2007CB807900, 2007CB807901.

G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 47–56, 2010. c Springer-Verlag Berlin Heidelberg 2010

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by a graph G = (V, E) in which there is an edge between two nodes u and v if and only if they are within each other’s communication range. By proper scaling, we assume that the smallest communication radius is one and the largest communication radius is R. Under the assumption that G is connected, any pair of nodes are apart by a distance of at most n − 1 and consequently we always assume that R ≤ n − 1. MCDS in multihop wireless networks with disparate communication ranges have been studied in [6] and [9]. Thai et al. [6] applied the approximation algorithm given in [7] for MCDS in multihop wireless networks with uniform communication ranges to compute a CDS in a multihop wireless network with disparate communication ranges. The approximation bound of this algorithm involves the relation between the independence number α (the size of a maximum independent set) and connected domination number γc (the size of a minimum connected dominating set) of the communication topology. It was shown in [6] that α ≤ 10 logg R γc √

where g = 1+25 is the such a bound on α, an approxima golden ratio. With tion bound 10 logg R + 2 + log 10 logg R was derived in [6]. Xing et al. [9] targeted at obtaining a tighter approximation bound of the same approximation algorithm. claimed (in Theorem 3.1 in [9]) a tighter upper bound bound 5 They 4 6 + 8 32 logg R γc on α. However, their proof of Theorem 3.1 in [9] contains a critical error, which has no apparent ﬁx. An explanation of this error is included in the appendix of this paper. Thus, the improved approximation bound based on the above bound of α in [9] becomes baseless. In this paper, we ﬁrst derive an improved upper bound on the number of independent nodes in the neighborhood of any node. For any R ≥ 1, let R∗ = 5 + 8 logg R . We show that the number of independent nodes in the neighborhood of any node is at most R∗ . Based on this upper bound, we then prove a tighter upper bound (R∗ − 1) γc + 1 on α. Thus, the approximation bounds of the approximation algorithms presented in [6] and [9] can be improved accordingly. We will adapt the two-phased greedy approximation algorithm presented in [8] to multihop wireless networks with disparate communication ranges, and show that its approximation ratio is at most R∗ + ln (R∗ − 2) + 1. The remaining of this paper is organized as follows. In Section 2, we present an improved upper bound on the independence number α in terms of the connected domination number γc . In Section 3, we analyze the approximation bound of a two-phased greedy approximation algorithm for MCDS adapted from an algorithm originally proposed in [8] for computing MCDS with uniform communication radii. In Section 4, we summarize the paper and discuss future studies for potential improvements. Throughout this paper, D(u, r) denotes the closed disk of radius r centered at u. The Euclidean distance between two nodes u and v is denoted by uv. The cardinality of a ﬁnite set S is denoted by |S|.

Minimum CDS in Multihop Wireless Networks

2

49

Independence Number vs. Connected Domination Number

In this section, we present an improved upper bound on the independence number α in terms of the connected domination number γc . Theorem 1. α ≤ (R∗ − 1) γc + 1. Theorem 1 follows from the lemma below by using the same argument as in [7]. Lemma 1. Suppose that I is an independent set of nodes adjacent to a node u. Then |I| ≤ R∗ . The rest of this section is devoted to the proof of Lemma 1. Consider an arbitrary node u ∈ V and an independent set I of nodes adjacent to a node u. Let I1 be the set of nodes in I lying in the closed disk of radius g centered at u, and for each j ≥ 2 let Ij = v ∈ I : g j−1 < uv ≤ g j . From [4] we have |I1 | ≤ 12. The following lemma on |Ij | for j ≥ 2 was proved in [9]. Lemma 2. For any j ≥ 2, |Ij | ≤ 9. We shall further prove the following lemma on |Ij ∪ Ij+1 | for j ≥ 2. Lemma 3. For any j ≥ 2, |Ij ∪ Ij+1 | ≤ 16.

These two lemmas together imply Lemma 1 immediately. If logg R is odd, then R −1)/2

(logg

logg R

|I| = j=1 Ij ≤ |I1 | + |I2i ∪ I2i+1 | i=1 ≤ 12 + 16 · logg R − 1 /2 = 8 logg R + 4 < R∗ .

If logg R is even, then

R /2−1

logg

logg R

|I| = j=1 Ij ≤ |I1 | + |I2 | + |I2i−1 ∪ I2i | i=2 ≤ 12 + 9 + 16 logg R /2 − 1 = 8 logg R + 5 = R∗ .

So, Lemma 1 holds in either case. Next, we prove Lemma 3 by using a subtle angular arguement. Fix a j ≥ 2. We begin with the following two simple geometric lemmas, whose proofs are omitted due to the space limitation. Lemma 4. Suppose that v and w are two distinct nodes in Ij satisfying that uv ≥ uw. Then, ∠wuv > 36 ◦ . In addition, for any acute angle α, 1. if uv ≤ 2g j−1 cos α, then ∠wuv > α; g 2. if uw ≥ 2g j−1 cos α, then ∠wuv > arccos 4 cos α.

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Lemma 5. Suppose that w ∈ Ij and v ∈ Ij+1 . For any acute angle α, 2

g 1. if uw ≥ 2g j−1 cos α, then ∠wuv > arccos 4 cos α; j 2. if uv ≤ 2g cos α, then ∠wuv > arccos (g cos α).

By Lemma 2, |Ij | ≤ 9. We present some necessary conditions for |Ij | = 9 in the lemma below. Lemma 6. Suppose that Ij consists of nine nodes v1 , v2 , · · · , v9 sorted in the increasing order of the distances from u. Then 1. uv9 ≥ 2g j−1 cos 39 ◦ and uv1 ≤ 2g j−1 cos 58.6 ◦ ; 2. uv8 ≥ 2g j−1 cos 39.8 ◦ and uv2 ≤ 2g j−1 cos 58.2 ◦; 3. uv7 ≥ 2g j−1 cos 43.2 ◦ and uv3 ≤ 2g j−1 cos 56.29 ◦. Proof. We will use the following fact multiple times in this proof: Suppose that I is a subset of ﬁve nodes in Ij . Then, among ﬁve consective sectors centered at u formed by the ﬁve nodes in I , at least one of them does not contain any other node in Ij . This is because |Ij \ I | = 4 < 5 and hence at least one of those ﬁve sectors does not contain any node in Ij \ I . (1) We prove the ﬁrst part of lemma by contradiction. Assume to the contrary that either uv9 < 2g j−1 cos 39 ◦ or uv1 > 2g j−1 cos 58.6 ◦. Then either uvi < 2g j−1 cos 39 ◦ for all 1 ≤ i ≤ 9 or uvi > 2g j−1 cos 58.6 ◦ for all 1 ≤ i ≤ 9. In either case, the angle separation of any two nodes in Ij at u is greater than 39 ◦ by Lemma 4. If uv5 < 2g j−1 cos 50 ◦ , let vi and vk be the two nodes in {v1 , v2 , · · · , v5 } such that the sector vi uvk centered at u does not contain any other node in Ij . Then by Lemma 4, ∠vi uvk > 50 ◦ . So, the total of the nine consecutive angles at u formed by the nodes in Ij is greater than 8 · 39 ◦ +50 ◦ = 362 ◦ > 360 ◦, which is a contradiction. Next we assume uv5 ≥ 2g j−1 cos 50 ◦ . Let vi and vk be the two nodes in {v5 , v6 , · · · , v9 } such that the sector vi uvk centered at u does not contain any other node in Ij . Then by Lemma 4, ∠vi uvk > 51 ◦ . So, the total of the nine consecutive angles at u formed by the nodes in Ij is greater than 8 · 39 ◦ +51 ◦ = 363 ◦ > 360 ◦, which is also a contradiction. Therefore, the ﬁrst part of the lemma holds. (2) We prove the second part of the lemma by contradiction. Assume to the contrary that either uv8 < 2g j−1 cos 39.8 ◦ or uv2 > 2g j−1 cos 58.2 ◦. We ﬁrst claim that there exists a node va ∈ Ij such that the angle separation of any two nodes in Ij \ {va } at u is greater than 39.8 ◦. Indeed, if uv8 < 2g j−1 cos 39.8 ◦ , then uvi < 2g j−1 cos 39.8 ◦ for all 1 ≤ i ≤ 8 and hence the claim holds for a = 9 by Lemma 4. If uv2 > 2g j−1 cos 58.2 ◦, then uvi > 2g j−1 cos 58.2 ◦ for all 2 ≤ i ≤ 9 and hence the claim holds for a = 1 by Lemma 4. So, our claim is true. Note that the angle separation between any two nodes in Ij is greater than 36 ◦ . If uv5 < 2g j−1 cos 50 ◦ , let vi and vk be the two nodes in {v1 , v2 , · · · , v5 } such that the sector vi uvk centered at u does not contain any other node in Ij . Then by Lemma 4, ∠vi uvk > 50 ◦ . So, the total of the nine consecutive angles at u formed by the nodes in Ij is greater than

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51

2 · 36 ◦ +6 · 39.8 ◦ +50 ◦ = 360. 8 ◦ > 360 ◦, which is a contradiction. Next we assume uv5 ≥ 2g j−1 cos 50 ◦ . Let vi and vk be the two nodes in {v5 , v6 , · · · , v9 } such that the sector vi uvk centered at u does not contain any other node in Ij . Then by Lemma 4, ∠vi uvk > 51 ◦ , which similarly leads to a contradiction. So, the second part of the lemma holds. (3) We prove the third part of the lemma by contradiction. Assume to the contrary that either uv7 < 2g j−1 cos 43.2 ◦ or uv3 > 2g j−1 cos 56.29 ◦ . We claim that there exist two nodes va , vb ∈ Ij such that ∠va uvb > 58.2 ◦ and the angle separation at u of any two nodes in I = Ij \ {va , vb } is greater than 43.2 ◦ . Indeed, if uv7 < 2g j−1 cos 43.2 ◦ , then uvi < 2g j−1 cos 43.2 ◦ for all 1 ≤ i ≤ 7 and hence the angle separation at u of any two nodes in Ij \ {v8 , v9 } is greater than 43.2 ◦ by Lemma 4. By part (2), we have uv8 ≥ 2g j−1 cos 39.8 ◦, which implies that ∠v8 uv9 > 58.2 ◦ by Lemma 4. Thus the claim holds with a = 8 and b = 9. Similarly, if uv3 > 2g j−1 cos 56.29 ◦, then uvi > 2g j−1 cos 56.29 ◦ for all 3 ≤ i ≤ 9 and hence the angle separation at u of any two nodes in Ij \ {v1 , v2 } is greater than 43.2 ◦ by Lemma 4. By part (2), we have uv2 ≤ 2g j−1 cos 58.2 ◦ , which implies ∠v1 uv2 > 58.2 ◦ by Lemma 4. Thus the claim holds with a = 1 and b = 2. Therefore, the claim holds in either case. Note that the angle separation between any two nodes in Ij is greater than 36 ◦ . If the sector va uvb centered at u does not contain any node in I , then the total of the nine consecutive angles at u formed by the nodes in Ij is greater than 2 · 36 ◦ +6 · 43.2 ◦ +58.2 ◦ = 389. 4 ◦ > 360 ◦ , which is a contradiction. So, we assume that the sector va uvb centered at u contains at least one node in I . Then, the total of the nine consecutive angles at u formed by the nodes in Ij is greater than 4 · 36 ◦ +5 · 43.2 ◦ = 360.0 ◦, which is also a contradiction. So, the third part of the lemma follows.

Now are ready to prove Lemma 3. Assume to the contrary that |Ij ∪ Ij+1 | = l ≥ 17. Let Ij ∪ Ij+1 = {vi : 1 ≤ i ≤ l} where v1 , v2 , · · · , vl are sorted in the increasing order of the distances from the node u. By Lemma 2, we have max {|Ij | , |Ij+1 |} ≤ 9. Thus, max {|Ij | , |Ij+1 |} = 9 and min {|Ij | , |Ij+1 |} ≥ 8 as l ≥ 17. We consider two cases: Case 1: |Ij | = 9. Then |Ij+1 | ≥ 8. By Lemma 6, we have uv7 ≥ 2g j−1 cos 43.2 ◦ . Let J = {v7 , v8 , v9 }. Then the angle separation between any two nodes in J at u is greater than 56.29 ◦. We further consider two subcases: Subcase 1.1: There exist two nodes va , vb ∈ J such that the sector va uvb centered at u does not contain any node in Ij (see Fig. 1(a)). Let vi and vk be the two nodes in Ij+1 such that the sector vi uvk contains va and vb but does not contain any other node in Ij+1 , and vi , va , vb and vk are in the clockwise direction with respect to u. Then min{∠vk uvb , ∠va uvi } > 26 ◦ by Lemma 5(1). Thus, ∠vk uvb + ∠vb uva + ∠va uvi > 2 · 26 ◦ +56.29 ◦ = 108.29 ◦ .

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vi

v’7 v’8 v7

va

v8

u

v’’ 7

u

vb v’’ 8 v9 v’9

vk v’’ 9 (a)

(b)

Fig. 1. Figure for Case 1: (a) a sector va uvb for a, b ∈ {7, 8, 9} does not contain any node in Ij+1 ; (b) every sector for va uvb for each a = 7, 8 and 9 contains a node in Ij+1 .

Hence, the total of the |Ij+1 | consecutive angles at u formed by the nodes in Ij+1 is greater than 7 · 36 ◦ +108.29 ◦ = 360. 29 ◦ > 360.0 ◦ , which is a contradiction. Subcase 1.2: For any two nodes va , vb ∈ J, the sector va uvb centered at u contains at least one node in Ij+1 (see Fig. 1(b)). For each a = 7, 8 and 9, let va , va ∈ Ij+1 satisfying that va is the only node contained in the sector va uva centered at u among all the nodes in Ij+1 ∪ J. Then by Lemma 5(1) and Lemma 6, we have ∠v7 uv7 + ∠v8 uv8 + ∠v9 uv9 > 2 · (26 ◦ +31.5 ◦ +32.5 ◦) = 180 ◦ . Hence, the total of the |Ij+1 | consecutive angles at u formed by the nodes in Ij+1 is greater than 5 · 36 ◦ +180 ◦ = 360.0 ◦, which is a contradiction. Case 2: |Ij | = 8. Then |Ij+1 | = 9. By Lemma 6, we have uv11 ≤ 2g j cos 56.29 ◦. Let J = {v9 , v10 , v11 }. By Lemma 4, the angle separation between any two nodes in J at u is greater than 56.29 ◦ . We further consider two subcases: Subcase 2.1: There exist two nodes va , vb ∈ J such that the sector va uvb centered at u does not contain any node in Ij (see Fig.2(a)). Let vi and vk be the two nodes in Ij such that the sector vi uvk contains va and vb but does not contain any other node in Ij , and vi , va , vb and vk are in the clockwise direction with respect to u. Then min{∠vk uvb , ∠va uvi } > 26 ◦ by Lemma 5(2). Thus, ∠vk uvb + ∠vb uva + ∠va uvi > 2 · 26 ◦ +56.29 ◦ = 108.29 ◦ .

Minimum CDS in Multihop Wireless Networks

vi

v11

u

v’10

v’11

va

53

v10

u

v’’ 10

v’’ 11 vk

vb

v’9 v’’ 9 v9

(a)

(b)

Fig. 2. Figure for Case 2: (a) a sector va uvb for a, b ∈ {9, 10, 11} does not contain any node in Ij ; (b) every sector for va uvb for each a = 9, 10 and 11 contains a node in Ij .

Hence, the total of the 8 consecutive angles at u formed by the nodes in Ij is greater than 7 · 36 ◦ +108.29 ◦ = 360. 29 ◦ > 360.0 ◦ , which is a contradiction. Subcase 2.2: For any two nodes va , vb ∈ J, the sector va uvb centered at u contains at least one node in Ij (see Fig.2(b)). For each a = 9, 10 and 11, let va , va ∈ Ij satisfying that va is the only node contained in the sector va uva centered at u among all the nodes in Ij ∪ J. Then by Lemma 5(2) and Lemma 6, we have ∠v9 uv9 + ∠v10 uv10 + ∠v11 uv11 > 2 · (26 ◦ +31.5 ◦ +32.5 ◦) = 180 ◦ .

Hence, the total of the 8 consecutive angles at u formed by the nodes in Ij is greater than 5 · 36 ◦ +180 ◦ = 360.0 ◦, which is a contradiction. Thus, in every case we have reached a contradiction. So, we must have |Ij ∪ Ij+1 | ≤ 16. This completes the proof of Lemma 3.

3

Greedy Approximation Algorithm for MCDS

In this section, we present a greedy algorithm adapted from the two-phased greedy approximation algorithm originally proposed in [8] for computing a CDS in a multihop wireless network with uniform communication ranges to multihop wireless networks with disparate communication ranges. The greedy algorithm consists of two phases. The ﬁrst phase selects a maximal independent set (MIS) I of G. Speciﬁcally, we construct an arbitrary rooted spanning tree T of G, and select an MIS I of G in the ﬁrst-ﬁt manner in the breadth-ﬁrst-search ordering in T . The second phase selects a set C of connectors to interconnect I . For

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any subset U ⊆ V \ I, f (U ) denotes the number of connected components in G [I ∪ U ]. For any U ⊆ V \ I and any w ∈ V \ I, the gain of w with respect to U is deﬁned to be f (U ) − f (U ∪ {w}). The second phase greedily selects C iteratively as follows. Initially C is empty. While f (C) > 1, choose a node w ∈ V \ (I ∪ C) with maximum gain with respect to C and add w to C. When f (C) = 1, then I ∪ C is a CDS. Let C be the output of the second phase. We have the following bound on |C|. Lemma 7. |C| ≤ (ln (R∗ − 2) + 2) γc . The proof of the above lemma is similar to that in [8] and is omitted due to the space limitation. From Theorem 1 and Lemma 7, we obtain the following bound on the size of the CDS output by the greedy algorithm. Theorem 2. |I ∪ C| ≤ (R∗ + ln (R∗ − 2) + 1) γc + 1.

4

Discussion

The relation between the independence number α and the connected domination number γc plays a key role in deriving the approximation bounds of various twophased greedy approximation algorithms adapted for MCDS of multihop wireless networks with disparate communication ranges [6] [8] [9]. In this paper, we ﬁrst proved that α ≤ (R∗ − 1) γc + 1, where R∗ = 5 + 8 logg R for any R ≥ 1. From this relation, we then derived an approximation bound R∗ + ln (R∗ − 2) + 1 of the two-phased greedy approximation algorithm adapted from [8]. This approximation bound is better than the known ones obtained in [6] and [9]. Tighter relation between α and γc may be derived with more sophisticated analyses. A possible approach of obtaining tighter relation between α and γc is to develop a tighter bound on the number of independent nodes that can be packed in the neighborhood of a pair of adjacent nodes. An attempt along this approach has been made in [9], but the argument in [9] contains a critical error. However, we do believe that this approach is very promising to achieve tighter relation between α and γc .

References 1. Alzoubi, K.M., Wan, P.-J., Frieder, O.: Message-Optimal Connected Dominating Sets in Mobile Ad Hoc Networks. In: ACM Mobihoc (2002) 2. Blum, J., Ding, M., Cheng, X.: Applications of Connected Dominating Sets in Wireless Networks. In: Du, D.-Z., Pardalos, P. (eds.) Handbook of Combinatorial Optimization, pp. 329–369. Kluwer Academic Publisher, Dordrecht (2004) 3. Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit Disk Graphs. Discrete Mathematics 86, 165–177 (1990) 4. Fodor, F.: The densest packing of 13 congruent circles in a circle. Beitrage Algebra Geom. 44(2), 431–440 (2003)

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5. Li, Y.S., Thai, M.T., Wang, F., Yi, C.-W., Wan, P.-J., Du, D.-Z.: On Greedy Construction of Connected Dominating Sets in Wireless Networks. Wiley Journal on Wireless Communications and Mobile Computing 5(8), 927–932 (2005) 6. Thai, M.T., Wang, F., Liu, D., Zhu, S., Du, D.-Z.: Connected dominating sets in wireless networks with diﬀerent communication ranges. IEEE Transactions on Mobile Computing 6(7), 721–730 (2007) 7. Wan, P.-J., Alzoubi, K.M., Frieder, O.: Distributed Construction of Connected Dominating Set in Wireless Ad Hoc Networks. ACM/Springer Mobile Networks and Applications 9(2), 141–149 (2004); A preliminary version of this paper appeared in IEEE INFOCOM 2002 (2002) 8. Wan, P.-J., Wang, L., Yao, F.: Two-Phased Approximation Algorithms for Minimum CDS in Wireless Ad Hoc Networks. In: IEEE ICDCS 2008, pp. 337–344 (2008) 9. Xing, K., Cheng, W., Park, E.K., Rotenstreich, S.: Distributed Connected Dominating Set Construction in Geometric k-Disk Graphs. In: IEEE ICDCS 2008, pp. 673–680 (2008)

Appendix In this appendix, we explainthe error in the proof of Theorem 3.1 in [9] which claimed that α ≤ 4 65 + 8 32 logg R γc , where α is the independence number and γc is the connected domination number. Let S be any maximum independent set of G and OP T be any MCDS of G. Then |I| = α and |OP T | = γc . In the proof of this theorem in [9], the γc nodes u1 , u2 , · · · , uγc in OP T are sorted in radius-decreasing order. Let 1 denote the number of nodes that are adjacent to u1 . For any 2 ≤ j ≤ γc , let j denote the number of nodes that are adjacent to uj but none of nodes u1 , u2 , · · · , uj−1 . Consider a spanning tree T of G[OP T ], the subgraph of G induced by OP T . The nodes in OP T are classiﬁed into two types. A node ui ∈ OP T is of the ﬁrst type if and only if ui has the smallest index among itself and all of its neighbors in T . Then, Lemma 3.2 in [9] claimed that if a node ui ∈ OP T is of the second type, then i ≤ 4 + 7 logg R . The proof of Lemma 3.2 in [9] contains a critical error in bounding the number of independent nodes that can be packed in the neighborhood of two adjacent nodes. For each node v and each j ≥ 2, Aj (v) denotes the annulus centered at v of inner radius g j−1 and outer radius g j . Suppose that u and v are a pair of adjacent nodes with ru ≥ rv = uv. Let j ≥ 2 and Ij (v) ⊂ Aj (v) be an independent set of nodes adjacent to v but not adjacent to u. Then Lemma 3.2 in [9] claimed that |Ij (v)| ≤ 7. This claim is incorrect. An instance illustrated in Fig. 3 shows a packing of nine independent nodes adjacent to the node v but not adjacent to the node u in an annulus Aj (v) with j ≥ 2. In this instance, the ﬁve nodes v1 , v2 , v3 , v4 and v5 lie on the circle ∂D(v, g j−1 +ε) in the counterclockwise direction satisfying that u, v and v3 are on the same line and ∠vi vvi+1 = 74 ◦ for each 1 ≤ i ≤ 4 and ∠v5 vv1 = 64 ◦ , where ε > 0 is suﬃciently small and will be chosed later. These ﬁve nodes lie inside Aj (v) but very close to its inner circle. For each i = 6, 7, 8 and 9, the node vi lies on the circle ∂D(v, g j − ε) satisfying that vvi is the angle bisectors of ∠vi−5 vvi−4 . These four nodes lie inside Aj (v) but very close to its outer circle. The communication radius of vi

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is vvi for each 1 ≤ i ≤ 9. We further assume that j ≥ 2 is small enough such that uv ≥ 2g j−1 . The ﬁrst ﬁve nodes are independent as their mutual angle separations at v are all greater than 60 ◦ . Similarly, the last four nodes are also independent as their angle separations at v are also all greater than 60 ◦ . In addition, choose ε suﬃciently small such that each of the ﬁrst ﬁve nodes and each of the last four nodes are independent as their angle separation at v is strictly greater than 36 ◦ . Therefore, all the nine nodes are independent with each other for suﬃciently small ε. Since uv ≥ 2g j−1 and the ﬁve nodes v1 , v2 , v3 , v4 and v5 lie close to the inner circle of Aj (v), we choose ε suﬃciently small such that uvi > vvi = rvi for all 1 ≤ i ≤ 5. Note that uv is the angle bisector of ∠v5 vv1 . Then ∠uvv1 = 32 ◦ and ∠uvv6 = 69 ◦ . Thus, v6 is on the left-hand side of the vertical line v1 v5 since vv6 cos ∠uvv6 < g j cos 69 ◦ < g j−1 cos 32 ◦ < vv1 cos ∠uvv1 . Similarly, v9 is on the left-hand side of the vertical line v1 v5 . Therefore, uvi > vvi = rvi for all 6 ≤ i ≤ 9. Thus none of these nine neighbors of v is adjacent to the node u. Hence, we have |Ij (v)| = 9 in this example. Therefore, the claim |Ij (v)| ≤ 7 in Lemma 3.2 in [9] is incorrect. This error further propagates to the proof of Theorem 3.1 in [9].

v6

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Fig. 3. A packing of nine independent nodes adjacent to the node v but not adjacent to the node u in an annulus Aj (v) with j ≥ 2

Minimum Edge Interference in Wireless Sensor Networks Trac N. Nguyen, Nhat X. Lam, D.T. Huynh, and Jason Bolla University of Texas at Dallas, Richardson TX 75080, USA {nguyentn,nxl081000,huynh,jab018600}@utdallas.edu http://www.utdallas.edu

Abstract. The approach of using topology control to reduce interference in wireless sensor networks has attracted attention of many researchers. There are several deﬁnitions of interference in the literature. In a wireless sensor network, the interference at a node may be caused by an edge that is transmitting data, or it occurs because the node itself is within the transmission range of another. The interference load of an edge is the number of nodes that are in the disks deﬁned by the end nodes of this edge with a radius which is either the Euclidean distance or the power level of the end nodes. In this paper we show that the problem of assigning power level to a set of nodes in the plane to yield a connected geometric graph whose edges have bounded interference is NP-complete under both edge interference deﬁnitions. We also study the performance of a number of heuristics through simulation. Keywords: wireless sensor networks, interference, NP-completeness, geometric graphs, heuristic.

1

Introduction

One well known approach to minimize interference in Wireless sensor networks is to use topology control. This approach was ﬁrst discussed by a number of researchers including Burkhart et al. [4] and Rickenbach, et al. [11]. The authors in [4] deﬁned the notion of interference load of an edge in a network, and showed an interesting result that certain sparse networks may not have low interference. Following the work in [4], the authors in [11] introduced a notion of node interference that is caused by surrounding nodes whose transmission range includes the given node. They analysed the special case of the exponential 1-dimensional node chain which √is also called the highway model. They showed that this sparse network has Ω( n) node interference, where n is the number of nodes in the √ network. The authors described an algorithm that provides an O( 4 δ) approximation of the optimal connectivity preserving topology in the highway model where δ is the maximum node degree. Similarly, Moaveni-Nejad and Li [9], and Johansson and Carr-Motyckovaa [7] focused on the notion of interference that is based on edges of the network. [7] gave a distributed algorithm called Average Path Interference that tries to preserve the spanner property of the original graph while reducing the interference in the network at the same time. [9] showed G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 57–67, 2010. c Springer-Verlag Berlin Heidelberg 2010

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that the relative neighborhood graph and local spanning tree algorithms have a constant bounded average interferences ratio. The NP completeness of both types of node interference were discussed by Wu and Liao [14], Bilo and Proietti [2] and Buchin [3]. [14] extended the work of [4] with an NP-completeness proof for minimizing edge-based interference for general graphs along with a couple of heuristics, and [3] provided an NP-completeness proof for ﬁnding a spanning tree with minimum node interference for grid graphs. For the receiver-based interference model, in [2] the authors showed among other results that the problem of minimizing the maximum node interference is hard to approximate. On the √ other hand, [6] showed that for a set of n points in the plane a network with O( Δ) interference can be constructed using computational geometric tools (Δ is the maximum interference in the uniform-radius network). They left open the question whether this problem is NP-hard including the 1-dimensional case. [10] was able to provide an answer to some of the questions raised in [6] by showing that minimizing receiver-based node interference is NP-complete for the 2-dimensional case. However, for the sender-based model minimizing node interference is solvable in polynomial time as shown in [2]. In this paper, we are concerned with edge interference. We study the problem of assigning power to nodes in the plane to form a connected graph with bounded edge interference. Speciﬁcally, we prove that the problem of assigning power to a set of nodes in the plane to yield a connected geometric graph whose edges have bounded interference is NP-complete. Our result is signiﬁcant as it provides an NP completeness result for geometric graphs for some of the issues discussed in [4] and [7]. In fact, for this interference model there are no NP completeness results to date. Moreover, this result is surprising since edge interference is quite similar to sender-based node interference which was shown to be solvable in polynomial time by [2]. Note that in our work as well as most of the works in wireless sensor networks, two nodes are connected by an edge if they are within the transmission range of one another. This deﬁnition is not strictly followed in a number of papers including Benkert et al. [1] and Sharma et al. [12]. We also study the performance of a number of heuristics through simulation. The rest of this paper is organized as follows. Section II provides the definitions and explanations used in this paper. Section III is devoted to the NP completeness result, and Section IV discusses the performance of some heuristics. Section IV contains some concluding remarks.

2

Preliminaries

Consider a set V of transceivers (nodes) in the plane. Each node u is assigned a power level denoted by p(u). The signal transmitted by node u can only received by a node v if the distance between u and v, denoted by d(u, v), is ≤ p(u). We only consider the bidirectional case in which a communication edge exists between two nodes u and v, (u, v) if and only if both power levels p(u) ≥ d(u, v) and p(v) ≥ d(u, v). Thus, the set V of nodes in the plane together with the power levels assigned to the nodes deﬁne a geometric (also known as intersection) graph G = (V, E). A geometric graph is said to be planar if no edge crosses another.

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In this paper we are concerned with two deﬁnitions of edge interference described in [4] and [7]. Assuming that there is no obstacle blocking the broadcasting range of a node, the two deﬁnitions of the edge interference load are as follows. Let D(u, r) be the broadcasting disk of node u with radius r. The (Euclidean) distance-based edge interference (DEI) is deﬁned in [7] as follow: DEI(u, v) := |{w ∈ V |w ∈ / {u, v} and w ∈ D(u, d(u, v)) or w ∈ D(v, d(v, u))}| and for a geometric graph G = (V, E) DEI(G(V, E)) := max(u,v)∈E {DEI(u, v)

(1)

Similarly, the power level-based edge interference (PEI) is in [4] as follows: P EI(u, v) := |{w ∈ V )|w ∈ / {u, v} and w ∈ D(u, p(u)) or w ∈ D(v, p(v))}| and for a geometric graph G = (V, E) P EI(G = (V, E)) := max(u,v)∈E {P EI(u, v)}

(2)

Intuitively, the interference load of an edge is the total number of nodes that are in the area covered by the disks created by the two end nodes of that edge. The radius of a disk is equal the Euclidian length of the edge (DEI case) or the power level usage (PEI case).

3

The NP-Completeness of Minimum Edge Interference in Geometric Graphs

In this section, we ﬁrst prove the NP-completeness of the problem of assigning power levels to nodes in the plane to produce a connected geometric graph G with bounded edge interference load DEI(G) (MDEICG). As a corollary, the problem of assigning power to nodes to produce a geometric graph G with bounded P EI(G) (MPEICG) is also NP-complete. Problem MDEICG: Given a set of N nodes V = {v1 , v2 , ..., vN } in the plane, a set of M power levels P = {p1 , p2 , .., pM } at which a node can transmit, and a positive number R. Is there a power assignment to all nodes so that it induces a connected geometric graph G(V,E) such that DEI(G) is ≤ R? The problem MPEICG is deﬁned similarly. In the following we show that MDEICG is NP-complete. Theorem 1. MDEICG is NP-complete. Proof. It is straightforward to show that MDEICG is in NP. To prove its NPhardness, we construct a polynomial time reduction from the planar 3-SAT problem (P3SAT) which was proven NP-complete in D. Lichtenstein [8]. Consider an instance φ of P3SAT, and the planar instance graph G of φ, where G = (X ∪ C, E ∪ E ) with edge sets E = {{x, c}|x ∈ C ∨ ¬x ∈ C} and E = {{xi , xi+1 }|1 ≤ i ≤ n − 1}.

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(a) Gadgets used in representing a variable x in a P3SAT instance.

(b) A series of gadgets representing a variable x of a P3SAT instance. Fig. 1. Gadgets

To construct an instance < V , P, R > of MDEICG, we ﬁrst create two gadgets for each variable of φ. As shown in Figure 1(a), each gadget for variable x contains 4 nodes, which are 2 pairs of x and ¬x. These nodes are connected through straight and curved edges. The left gadget has 2 curved edges and 4 straight edges that form a square, whereas the right gadget has 3 straight edges and two curved edges. To have suﬃcient nodes for x so that we can connect x to some clause nodes, we replace a variable node x in G by a chain of these two gadgets in such a way that two gadgets at the ends of the chain have four straight edges, whereas every in-between gadget has three straight edges. This way, every node in the chain of gadgets has a degree of at least 3. Moreover, every second gadget has its nodes and edges rearranged as depicted in Figure 1(b). Note that consecutive gadgets are connected by two parallel straight edges. If the degree of node x is d, then the number of x s gadgets in this chain is 3 ∗ d. From each chain of gadgets representing a variable node in G, we only use the nodes with degree 3, to connect to a clause node. Furthermore, the variable link that connects all variables should enter each chain of gadgets via a degree 3 node at one end and exit via another degree 3 node at the other end. Thus, the new graph (also denoted G for convenience) obtained from the original graph has a maximum degree of 4 while the planarity of the graph is preserved. Next, we use Valiant’s result [13] to embed the graph G with maximum degree 4 into the Euclidean plane: A planar graph with maximum degree 4 can be embedded in the plane using O(|V |) area in such a way that its vertices are at integer coordinates and its edges are drawn so that they are made up of line segments of form x = i or y = j, for integers i and j. Moreover, this embedding process can easily be designed to satisfy the additional requirement that each line segment drawn to connect two original vertices of the graph G must be of length at least 6, and any two parallel line segments in the embedded graph are at least 6 units apart. Let us call the units of the embedded graph G original units. Each original unit is divided further into 12 smaller “pieces” of equal length. Let the original unit be δ. We deﬁne the radii ri , 1 ≤ i ≤ 8 as follows: ri := (i/12) ∗ δ. For the sake of convenience let us call the variable and clause nodes of G embedded in

Minimum Edge Interference in Wireless Sensor Networks

(a) Nodes added on the line segments replacing a straight edge (x,¬x) of G(V, E) at variable node x with degree 3.

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(b) Nodes added on the line segments replacing a straight edge (x,¬x) of G(V, E) on a variable link connecting y and x.

Fig. 2. Nodes added on the line segments in cases

the plane the variable and clause nodes, respectively. First, we have to make a change in the line segments representing a straight edge connecting x to ¬x in the embedded graph G as follows. 1. If these segments are strictly either horizontal or vertical segments, we create a U shape from these original segments. Starting from either node x or ¬x, we move 2 segments, say the third and the fourth segments, 2 units apart from their current location. We then connect both ends of these segments using additional 2-unit segments that form a U shape. 2. For the corners created by perpendicular unit segments, pick one corner and let this corner be the marked corner. Note that from the previous step, every line segments representing a straight edge in G connecting x to ¬x must have one marked corner. At the marked corner, replace the two perpendicular unit segments by a segment connecting two grid nodes (see Figure 4.) This segment is called a slanted segment. We now further modify each line segment by placing additional nodes to create the instance < V , P, R > of MDEICG as follows: 1. On every line segments, add one node at every grid point. Lets call these nodes grid nodes (e.g. nodes g s in Figure 2, 3 and 4.) 2. On the line segments representing the variable link connecting diﬀerent variable gadgets chains, add 2 nodes to each unit to divide it into 3 equal pieces of length r4 each (see Figure 2(b)). 3. On the line segments representing the connection from a variable node to a clause node, add the ﬁrst node (node o in Figure 3(b)) at distance r8 from the variable node, and successively add nodes at distance r4 from the previous one until the clause node is reached.

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(a) Nodes added on the line segments replacing a straight edge (x,¬x) of G(V, E) at variable node x with degree 4.

(b) Nodes added on the line segments replacing a straight edge (x,¬x) of G(V, E) on a connection to a clause node C.

Fig. 3. Node added on the line segments in cases

4. For a variable node with degree 3, add two nodes at distance r4 to the unit that does not correspond to any segment. (See nodes d and t in Figure 2(a).) 5. On the line segments representing a curved edge in G, add two nodes at distance r8 from the variable nodes. Then successively add nodes at distance r4 between these two new nodes (e.g. node i in Figures 2(a), 2(b), 3(a), 3(b) is at distance r8 from the variable node x, and the subsequent nodes are at distance r4 from the preceding ones.) 6. On the line segments representing a straight edge connecting a variable node x and its negated node ¬x of G, add new nodes on every unit line segment by the following steps: – On every unit segment add a node at distance r4 from the grid node. Let us call these nodes support nodes. (See nodes s’s in Fig. 2, 3 and 4) – Between the support node and the grid node on each unit segment, add one node (away from the segment), called auxiliary node, so that its distance to the support node or the grid node is exactly r8 . Moreover, these auxiliary nodes are added in such a way that they are not close to the other auxiliary nodes added on other unit segments (see nodes a s in Figures 2, 3 and 4.) – Between the support node and the variable node on every unit segment incident with a variable, add a node at distance r4 from the variable node and the support node. – In the middle of each slanted segment, add two nodes called middle nodes at distance r8 from the end (grid) nodes of the segment. Note that the distance between the two middle nodes is about 0.97. (See nodes k and w in Figure 4.) On both sides of the middle nodes, two more nodes are added such that one node is at distance r2 from the grid node and the other is at distance r2 from the closest middle node. (See Figure 4.) The

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Fig. 4. Nodes added on the line segments replacing a straight edge (x,¬x) of G(V, E). These line segments contain a slanted segment and segments that form a U shape.

distance between these two nodes is r4 . Between the two middle nodes and away from the line segment add an auxiliary node that is at distance r8 from each of the middle nodes. – Consider two perpendicular adjacent unit segments. On one of the unit segment add two nodes between the support node and the grid node so that the distance between these two nodes is r4 . These two nodes are at distance r2 to the grid node and the support node. On the remaining unit segments, add 2 pairs of nodes between the support node and the grid node. The distance of nodes in each pair is r1 . The distance between two adjacent pairs of nodes is r4 , and each of them is at distance r2 to the grid node or to the support node. These two pairs of nodes form a group of 4 nodes. (See Figures 2, 3 and 4.) Let P := {0, r1 , r2 , ..., r8 } and R := 8. The correctness of the above polynomialtime reduction follows from the following claim: Claim. The instance φ of P3SAT is satisﬁable if and only if the MDEICG instance < V , P, R > has a power assignment that yields a connected geometric graph G (V , E ) such that DEI(u, v) of each edge (u, v) ∈ E is ≤ R. Intuitively, some node on a straight edge connecting a variable node and its negated node must be assigned a high power level for the resulting graph to be connected. Moreover, if a variable node is assigned a high power level, then its negated node must be assigned a low power level due to the interference bound; or vice versa. The proof of the Claim is omitted due to space limitation. From the Claim, Theorem 1 follows.

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Then the NP completeness of MPEICG follows as a corollary. Theorem 2: MPEICG is NP-complete. Proof. The proof of Theorem 2 is similar to the proof of Theorem 1, and is omitted.

4 4.1

Heuristics and Their Performance Heuristics

Node interference approach. Our ﬁrst approach is built on the sender-based node interference problem, where the interference load of a node v is the number of nodes within the broadcasting disk of v and the problem is to minimize the maximum node interference. This problem has been shown to be solved in polynomial time in [2] through a simple greedy heuristic called OPT-MINMAXSIP. Based on this result, we propose the Node Power Level Search heuristic (NPLS) which ﬁrst computes the minimum sender-based node interference, and then reduce the power level for each node while preserving connectivity. Although this step does not necessarily decrease the maximum node interference of the graph, it may improve the interference load of other nodes, and hence the interference load of the edges. The pseudo code of this algorithm is reported in Figure 5. Input: a set V of nodes in the plane Output: A power assignment P that yields a connected geometric graph whose edge interference is minimized 1 Using the greedy technique, ﬁnd a power assignment P that yields a minimum node interference graph // Reduce power levels of nodes while maintaining connectivity 2 for each node v ∈ V 3 Using binary search ﬁnd the smallest power level for P [v] while P yields a connected graph 4 return the power assignment P Fig. 5. Node-Power-Level-Search

Minimum spanning tree approach. The basic idea of this approach is to ﬁnd a minimum spanning tree for a given complete graph. We use this approach in two diﬀerent algorithms, the distance-based minimum spanning tree (DMST) algorithm and the interference-based minimum spanning tree (IMST) algorithm. The main diﬀerence between these two algorithms is the deﬁnition of the edge weight. The former uses the Euclidean distance as edge weight while the latter uses the DEI edge interference load. In implementing these two algorithms, we use Kruskals algorithm [5] to ﬁnd the minimum spanning tree. This algorithm uses two well-known subroutines find-set and union. find-set(u) is to ﬁnd the root of tree-based connected component containing node u whereas union(u, v) is to connect two connected components of u and v. The details of these two subroutines can be found in [5].

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Figure 6 contains the pseudo code of the DMST algorithm. Observe that adding an edge may increase the power levels assigned to the two end nodes, thereby creating new edges (adjacent to the newly added edge). This is taken into account in lines 8-9 in Figure 6. Input: a set V of nodes in the plane Output: A power assignment P that yields a connected geometric graph whose edge interference is minimized 1 E = set of edges between all nodes in V 2 A=∅ 3 w(e) = length of e for all e ∈ E 4 Sort E in non-decreasing order according to weights w 5 for each edge (u, v) ∈ E 6 if find-set(u) = find-set(v) 7 add edge (u, v) to A and union(u, v) 8 B = edges created by adding (u, v) 9 A = A B and E = E \ B 10 Compute the power assignment P from A 11 return P Fig. 6. Distance-based-Minimum-Spanning-Tree

Regarding IMST algorithm, a possible improvement is to modify the deﬁnition of edge weight. As pointed out above, adding an edge may create some higher interference edges. So the weight of an unprocessed edge e is not its interference load, but the maximum interference load of those edges, which would be created if e were chosen to be added to A, including e itself. At every step, after A is updated, we recalculate the weights of unprocessed edges and sort them in non-decreasing order. 4.2

Experimental Results

In our experiments, we only consider geometric graphs. We randomly generate 100 nodes on an area of size of 10000x10000. For each graph, we run all three algorithms. We use the percentage diﬀerence as a metric to evaluate their performance. We compare IMST against the other two, namely DMST and NPLS. The results for the DEI as well as the PEI models in the table below demonstrate that the IMST algorithm performs signiﬁcantly better. With regard to the DEI model, DMST and NPLS are always worse than IMST. However, for the PEI model, the DMST algorithm in some cases performs better as the negative value in Table I shows. By analysing the data concerning these cases, we are able to generate a simple example where DMST works better than IMST. In this case, IMST gives a graph whose maximum edge interference is 5 with the maximum interference edge (5, 6) (Figure 7(a)), while the maximum interference found by DMST is 4 (Figure 7(b)). Remark. Note that we have not provided a formal analysis of the running time of the above heuristics as the techniques used are well known. For eﬃcient implementation of the algorithms using computational-geometric techniques, the

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Min Max Average Better Worse Equal

Distance based DMST NPLS 0.00% 0.00% 200.00% 83.33% 23.57% 28.41% 0 0 136 163 64 37

Power Level based DMST NPLS -36.36% 14.29% 200.00% 114.29% 19.85% 61.32% 9 0 125 200 66 0

y 15

y 15

2

2 10

10

5

5 6

5 4

0

1

3

0

6

5 0 10

(a) IMST algorithm

4

0 20 x

0

1

3

0 10

20 x

(b) DMST algorithm

Fig. 7. An Example Showing The Better Result of DMST

reader is referred to [1]. We also note that the improvement described in Lines 9-12 in the IMST algorithm does not show any concrete result in our simulation as the connected geometric graphs appear to have minimum edge interference even without this improvement. It would be interesting to formally investigate how well IMST performs.

5

Conclusions

In this paper we have shown that the problems of minimizing DEI or PEI interference are both NP-complete for (planar) connected geometric graphs. This is the ﬁrst NP completeness result concerning the edge interference model for geometric graphs. We also compare the performance of some heuristics through simulation. These results provide new insight on some of the issues discussed in [4] and [7].

References 1. Benkert, M., Gudmundsson, J., Haverkort, H., Wolﬀ, A.: Introduction to Algoˇ rithms. In: Wiedermann, J., Tel, G., Pokorn´ y, J., Bielikov´ a, M., Stuller, J. (eds.) SOFSEM 2006. LNCS, vol. 3831, pp. 166–176. Springer, Heidelberg (2006) 2. Bil` o, D., Proietti, G.: On the Complexity of Minimizing Interference in Ad-Hoc and Sensor Networks. Theorectical Computer Science 402, 43–55 (2008)

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3. Buchin, K.: Minimizing the Maximum Interference is Hard (2008), http://arxiv.org/abs/0802.2134 4. Burkhart, M., von Rickenbach, P., Wattenhofer, R., Zollinger, A.: Does Topology Control Reduce Interference? In: MOBIHOC 2004, pp. 9–19 (2004) 5. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Constructing InterferenceMinimal Networks. Section 23.2: The Algorithms of Kruskal and Prim, pp. 567–574. McGraw-Hill, New York (2001) ISBN 0-262-03293-7 6. Halld´ orsson, M.M., Tokuyama, T.: Minimizing Interference of a Wireless Ad-Hoc Network in a Plane. In: Nikoletseas, S.E., Rolim, J.D.P. (eds.) ALGOSENSORS 2006. LNCS, vol. 4240, pp. 71–82. Springer, Heidelberg (2006) 7. Johansson, T., Carr-Motyckova´ a, L.: Reducing Interference in Ad Hoc Networks Through Topology Control. In: DIALM-POMC 2005, pp. 17–23 (2005) 8. Lichtenstein, D.: Planar Formulae And Their Uses. SIAM J. of Compt. 11(23) (1982) 9. Moaveni-Nejad, K., Li, X.-Y.: Low-Interference Topology Control for Wireless Ad Hoc Networks. Ad Hoc and Wireless Sensor Networks 1, 41–64 (2005) 10. Nguyen, T.N., Huynh, D.T.: Minimum Interference Planar Geometric Topology in Wireless Sensor Networks. In: Liu, B., Bestavros, A., Du, D.-Z., Wang, J. (eds.) WASA 2009. LNCS, vol. 5682, pp. 149–158. Springer, Heidelberg (2009) 11. Rickenbach, P.V., Schmid, S., Wattenhofer, R., Zollinger, A.: A Roburst Interference Model for Wireless Ad-Hoc Networks. In: Proc. 19th IEEE Int. Par. and Dist. (2005) 12. Sharma, A., Thakral, N., Udgata, S., Pujari, A.: Heuristics for Minimizing Interference in Sensor Networks. In: Garg, V., Wattenhofer, R., Kothapalli, K. (eds.) ICDCN 2009. LNCS, vol. 5408, pp. 49–54. Springer, Heidelberg (2009) 13. Valiant, L.: Universality Considerations in VLSI Circuits. IEEE Trans. on Compupters C-30, 135–140 (1981) 14. Wu, K.-D., Liao, W.: On Constructing Low Interference Topology in Multihop Wireless Networks. Int. J. of Sensor Networks 2, 321–330 (2007)

Maximum Weighted Independent Set of Links under Physical Interference Model Xiaohua Xu, Shaojie Tang, and Peng-Jun Wan Illinois Institute of Technology, Chicago IL 60616, USA

Abstract. Interference-aware scheduling for wireless communications is crucial to improve the network throughput. In this paper, we study the problem of Maximum Weighted Independent Set of Links (MWISL) under the physical interference model in wireless networks. Given a set of communication links distributed in a two-dimensional Euclidean plane, assume each link is associated with a positive weight which represents the benefit of transmitting along the link, the objective is to seek an independent set of links subject to the physical interference constraints with maximum weighted sum. To the best of our knowledge, no algorithm for MWISL under physical interference model has been proposed. We focus on MWISL in the oblivious power assignment setting.

1 Introduction Link scheduling in wireless networks plays a critical role for wireless networking performances, especially when the network has stringent quality of service restrictions. One challenge for link scheduling lies in interferences among concurrent transmissions. Unlike the wired networks, the signal interference casts significant effect on the fundamental limit on the data throughput that any scheduling protocols (centralized or distributed) can achieve. It is well-known that a number of scheduling problems (e.g., maximum throughput scheduling) become NP-hard to solve when considering wireless interference, while their counter-parts are solvable in polynomial time for wired networks. Thus, the scheduling protocols for wireless networks (even the benchmark performances obtained by centralized scheduling approaches) often rely on heuristics that approximately optimize the throughput. We address a fundamental problem for scheduling in wireless communications: Maximum Weighted Independent Set of Links (MWISL): given an input set of links, a subset of links is an independent set of links iff they can transmit concurrently. Assume each link is associated with a positive weight (representing the benefit of transmitting along the link), the objective is to seek an independent set of links with maximum weighted sum under given interference model. We focus on MWISL under physical interference model, where a signal is received successfully if the Signal to Interferenceplus-Noise Ratio is above a threshold depending on hardware and coding method. In this practical model, the definition of a successful transmission accounts also interference generated by transmitters located far away. Thus, traditional methods in graphbased interference models (e.g., protocol interference model, RTS/CTS model et al.)

This work was supported in part by NSF of USA under grant CNS-0831831.

G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 68–74, 2010. c Springer-Verlag Berlin Heidelberg 2010

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cannot be directly applied here. The MWISL problem has several variations: such as different power assignments and different weight distributions. Even for the special case where every link has unit weight, MWISL is proved to be NP-hard [6, 7]. Our main contribution is the algorithm design for MWISL problem under the oblivious power assignment setting where each link l (or its corresponding sender) is assigned a transmission power c · lβ , here c is a constant and 0 < β ≤ κ is a constant. We can prove that our solution satisfies the interference constraints and achieves constant approximation ratio of the optimum when β = κ. Our main idea is like this: by using partition and shifting strategies, we find multiple sets of well-separated links and then select the one with the largest weight. The rest of the paper is organized as follows. Section 2 formulates our problems. Section 3 presents our algorithm design for the MWISL problem. Section 4 outlines the related work. Finally, Section 5 concludes the paper.

2 Network Model All the networking nodes V lie in plane and have a maximum transmission power P . The Euclidean distance between any pair of nodes is denoted by uv. Let r be the distance between a closest pair of nodes in V . The path-loss over a distance l is ηl−κ , where κ is path-loss exponent (a constant greater than 2 and 5 depending on the wireless environment), and η is the reference loss factor. Since the path-loss factor over the distance r is less than one, we have η < rκ . In an oblivious power assignment, a β node u transmits to another node v always at the power c uv for some constants c > 0 and 0 < β ≤ κ. This assumption implicitly imposes an upper bound on the distance between a pair of nodes which directly communicate with each other: For u to be able to directly communicate with v, we must have c uvβ ≤ P and hence 1/β uv ≤ (P/c) . Let ξ be the noise power, and σ be the signal to interference and noise ratio (SINR) threshold for successful reception. Then, in the absence of interference, the transmission by a node u can be successfully received by another node v if β −κ cη ≥ σ which is equivalent to uvκ−β ≤ σξ . Note that when and only if cuv ·ηuv ξ κ−β

cη uv = σξ , link uv can only transmit alone since any other link will conflict with uv. Thus we can disregard these links in A and assume that κ−β

uv

<

cη σξ

Therefore, the set A of communication links consists of all pairs (u, v) of distinct nodes cη satisfying that uvβ ≤ P/c and uvκ−β < σξ . Let R be the maximum length of the links in A. Then, Rκ = Rβ · Rκ−β < Hence, R < r

Pη P κ P cη · = < r . c σξ σξ σξ

P σξ

1/κ .

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A set I of links in A is said to be independent if and only if all links in I can transmit successfully at the same time under the oblivious power assignment, i.e., the SINR of each link in I is above σ. We denote by I the collection of independent sets of links in A. Given a link weight function d ∈ RA + , the problem Maximum Weighted Independent Set of Links (MWISL) seeks a set I ∈ I with maximum total weight d (I) = a∈I d (a).

3 Algorithm Design In this section, we present our algorithm design for MWISL in the oblivious power assignment setting under physical interference model.

K

Fig. 1. Grid partition of the plane

√ We employ a grid partition of the plane (Fig. 1). Let = R/ 2. The vertical lines x = i · for i ∈ Z and horizontal lines y = j · for j ∈ Z partition the planes into half-open and half-closed grides of side (here Z represents the integer set): {[i, (i + 1) ) × [j, (j + 1) ) : i, j ∈ Z} . For all i, j ∈ Z, we denote Aij to be the set of links in A whose senders lie in the grid [i, (i + 1) ) × [j, (j + 1) ). We first give a sufficient condition for a set I of links to be independent. Let −1/κ √ √ −1 −1 K = 2 (4τ ) σ −1 − ξ(cη) Rκ−β + 2 Generally, K’s value depends on R, we can see that when β = κ, −1/κ √ √ K = 2 (4τ )−1 σ −1 − ξ(cη)−1 + 2 which is a constant independent of R. Lemma 1. Consider any two nonnegative integers k1 and k2 which are at most K. Suppose I is a set of links satisfying that for each i, j ∈ Z, |I ∩ Aij | ≤ 1 if i mod (K + 1) = k1 and j mod (K + 1) = k2 and |I ∩ Aij | = 0 otherwise. Then, I is independent.

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Proof. Consider any link a = (u, v). The wanted signal strength is β

−κ

c a · η a

β−κ

= cη a

≥ cηRβ−κ .

Consider any link a = (u , v ) in I other than a. We have u u ≥ K. Therefore, √ u v ≥ u u − uv ≥ K − R = K/ 2 − 1 R. The total interference to a from all other links in I is at most √ −κ cRβ · η x2 + y 2 · K/ 2 − 1 R (x,y)∈Z2 \{(0,0)}

−κ √ = cηRβ−κ K/ 2 − 1

−κ x2 + y 2

(x,y)∈Z2 \{(0,0)}

≤ 4cηR ≤ 4cηR

β−κ

β−κ

−κ √ K/ 2 − 1 −κ √ K/ 2 − 1

√ −κ = 4τ cηRβ−κ K/ 2 − 1 , where

∞

i

−κ

+

∞ ∞

x2 + y 2

x=1 y=1

i=1 κ

κ(1 + 2− 2 ) π2−κ/2 + κ−1 2(κ − 2)

−κ

κ

τ=

π2−κ/2 κ(1 + 2− 2 ) + . κ−1 2(κ − 2)

Thus the SINR at the receiver of the link is at least cηRβ−κ √ −κ ≥ σ ξ + 4τ cηRβ−κ K/ 2 − 1 since

−1/κ √ K/ 2 − 1 ≥ (4τ )−1 σ −1 − ξ(cη)−1 Rκ−β

Next, we give a necessary condition for a set I of links to be independent. Let

κ 2 P + 1 . ω= σ2 ξ Lemma 2. For any I ∈ I and any i, j ∈ Z, |I ∩ Aij | ≤ ω. Proof. Let Iij = I ∩ Aij . Assume a = (u, v) be the shortest link in Iij , consider any link a = (u , v ) in Iij other than a, the distance between the sender u and v satisfies √ u v ≤ u u + uv ≤ 2 + uv ≤ 2R,

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The SINR at a from all other links in Iij is at most β

c a · η a

a ∈Iij \{a}

Since

β

−κ

c a · η u v

r −κ (|Iij |−1)(2R)−κ

−κ

≤

−κ

a

a ∈Iij \{a}

≥ σ, we have |Iij | ≤

2κ Rκ σ rκ

u v−κ

+1<

≤

2κ P σ σξ

r−κ (|Iij | − 1) (2R)−κ

+1=

2κ P σ2 ξ

+ 1.

Input: A set of links L = {l1 , l2 , · · · , ln } dmax ← 0 ; for k1 = 0, · · · , K and k2 = 0, · · · , K do for i, j ∈ Z and the grid gi,j contains links from L do if i ≡ k1 mod (K + 1) and j ≡ k2 mod (K + 1) then select one link with the maximal weight whose sender lies within gi,j All the selected links form a set Ik1 k2 ; if d(Ik1 k2 ) > dmax then S ← Ik1 k2 ; dmax ← d(Ik1 k2 ); return A schedule S containing a subset of links in L.

Algorithm 1. Scheduling for MWISL Our partition-based scheduling method for input L is shown in Algorithm 1. The correctness of the algorithm follows from Lemma 1. Next, we derive its approximation bound. Theorem 1. The approximation ratio of our algorithm for MWISL is at most (K + 1)2 ω. Proof. Let I ∗ be a maximum-weighted independent set of links. For any pair of nonnegative integers k1 and k2 , let Ik∗1 k2 denote the set of links in I ∗ which lie in the grids [i, (i + 1) ) × [j, (j + 1) ) for all i, j ∈ Z satisfying that i mod (K + 1) = k1 and j mod (K + 1) = k2 . By Lemma 2, the set Ik∗1 k2 contains at most ω links from each grid [i, (i + 1) ) × [j, (j + 1) ) with i mod (K + 1) = k1 and j mod (K + 1) = k2 . Hence, d Ik∗1 k2 ≤ ωd (Ik1 k2 ) ≤ ω max d (Ik1 k2 ) . 0≤k1 ,k2 ≤K

Therefore, d (I ∗ ) =

0≤k1 ,k2 ≤K d

So, the theorem holds.

∗ Ik1 k2 ≤ (K + 1)2 ω

max

0≤k1 ,k2 ≤K

d (Ik1 k2 ) .

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4 Literature Review For general graph-based interference model, the maximum throughput link scheduling problem is NP-complete [5]. Both probabilistic scheduling protocols [5, 10, 11] and distributed link scheduling protocols [13, 14] are proposed to maximize the throughput. For physical interference model, the problem of joint scheduling and power control has been well studied in [3, 4]. In [12], a power-assignment algorithm which schedules a strongly connected set of links in poly-logarithmic time is presented. [6] shows that the scheduling problem without power control under physical interference model, where nodes are arbitrarily distributed is NP-complete. A greedy scheduling algorithm with approximation ratio of O(n1−2/(Ψ (α)+ ) (log n)2 ), where Ψ (α) is a constant that depends on the path-loss exponent α, is proposed in [1]. Notice that this result can only hold when the nodes are distributed uniformly at random in a square of unit area. In [6], an algorithm with a factor O(g(L)) approximation guarantee in arbitrary topologies, where g(L) = log ϑ(L) is the diversity of the network, is proposed. In [2], an algorithm with approximation guarantee of O(log Δ) was proposed, where Δ is the ratio between the maximum and the minimum distances between nodes. Obviously, it can be arbitrarily larger than ϑ(L). Recently, Goussevskaia et al. [7] proposed a method for MWISL in the unweighted case which was claimed to have constant approximation bound. Unfortunately, as observed in Xu and Tang [16], their method [7] works correctly in absence of the background noise. Wan et al. [15] resolved this issue by developing the first correct constantapproximation algorithm. Most Recently, Halldorsson et al. [9] presented a robustness result, showing that constant parameter and model changes will modify the min-length link scheduling result only by a constant. They [8] also studied the scheduling problem under power control.

5 Conclusions We studied MWISL problem under the physical interference model with oblivious power assignment. Some interesting questions are left for future research. The first is to extend our algorithm to deal with a more general path loss model. The second is to develop constant approximation algorithms for MWISL under uniform power and adjustable power assignment settings.

References 1. Brar, G., Blough, D., Santi, P.: Computationally efficient scheduling with the physical interference model for throughput improvement in wireless mesh networks. In: ACM MobiCom (2006) 2. Chafekar, D., Kumar, V., Marathe, M., Parthasarathy, S., Srinivasan, A.: Approximation Algorithms for Computing Capacity of Wireless Networks with SINR Constraints. In: IEEE INFOCOM, pp. 1166–1174 (2008) 3. Cruz, R., Santhanam, A.: Optimal routing, link scheduling and power control in multihop wireless networks. In: IEEE INFOCOM, vol. 1 (2003)

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4. ElBatt, T., Ephremides, A.: Joint scheduling and power control for wireless ad hoc networks. IEEE Transactions on Wireless Communications 1, 74–85 (2004) 5. Sharma, G., Joo, C., Shroff, N.: Distributed scheduling schemes for throughput guarantees in wireless networks. In: Allerton 2006 (2006) 6. Goussevskaia, O., Oswald, Y., Wattenhofer, R.: Complexity in geometric SINR. In: Proceedings of ACM MobiHoc, pp. 100–109 (2007) 7. Goussevskaia, O., Wattenhofer, R., Halld´orsson, M.M., Welzl, E.: Capacity of Arbitrary Wireless Networks. In: IEEE INFOCOM 2009 (2009) 8. Halld´orsson, M.: Wireless scheduling with power control. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 361–372. Springer, Heidelberg (2009) 9. Halldorsson, M., Wattenhofer, R.: Wireless Communication is in APX. In: Automata, Languages and Programming, pp. 525–536 10. Joo, C., Shroff, N.: Performance of random access scheduling schemes in multi-hop wireless networks. In: IEEE INFOCOM (2007) 11. Lin, X., Rasool, S.: Constant-time distributed scheduling policies for ad hoc wireless networks. In: IEEE CDC (2006) 12. Moscibroda, T., Wattenhofer, R.: The Complexity of Connectivity in Wireless Networks. In: IEEE INFOCOM (2006) 13. Penttinen, A., Koutsopoulos, I., Tassiulas, L.: Low-complexity distributed fair scheduling for wireless multi-hop networks. In: WiOPT 2005: First Workshop on Resource Allocation in Wireless Networks, RAWNET (2005) 14. Sanghavi, S., Bui, L., Srikant, R.: Distributed link scheduling with constant overhead. In: ACM SIGMETRICS, pp. 313–324. ACM Press, New York (2007) 15. Wan, P., Jia, X., Yao, F.: Maximum Independent Set of Links under Physical Interference Model. In: Liu, B., Bestavros, A., Du, D.-Z., Wang, J. (eds.) WASA 2009. LNCS, vol. 5682, pp. 169–178. Springer, Heidelberg (2009) 16. Xu, X., Tang, S.: A constant approximation algorithm for link scheduling in arbitrary networks under physical interference model. In: Proceedings of the 2nd ACM International Workshop on Foundations of Wireless Ad Hoc and Sensor Networking and Computing, pp. 13–20. ACM, New York (2009)

A QoS-Guaranteed Energy-Efficient Packet Scheduling Algorithm for WiMax Mobile Devices Hung-Cheng Shih and Kuochen Wang Department of Computer Science, National Chiao Tung University {hcshih,kwang}@cs.nctu.edu.tw

Abstract. A mobile device may run energy consuming multimedia applications, such as video streaming. Thus, power management is an important issue in mobile devices. In this paper, we propose a QoS-guaranteed energy-efficient packet scheduling algorithm for IEEE 802.16-2009 (WiMax) network interfaces. The integration of packet scheduling and sleep mode operation can make IEEE 802.16-2009-enabled mobile devices more energy efficient. Simulation results show that the proposed packet scheduling algorithm has 14.37% less energy consumption than the naïve algorithm, and it does not sacrifice average packet delay. The contributions of our proposed algorithm are that our algorithm is suitable for wireless environments where connections come and go rapidly and the time complexity of our algorithm is low. Keywords: Energy-efficient, IEEE 802.16-2009, packet scheduling, sleep window, WiMax.

1 Introduction Power management is an important issue in mobile devices (called subscriber stations (SSs) in the IEEE 802.16-2009), which are powered by batteries. Because the capacity of a battery is limited, we have to reduce the energy consumption in every way, such as decreasing the working time of a mobile device. In a mobile device with the capability of wireless communications, the energy consumed by active transmissions is high because the transmitter and the receiver have to be turned on during the transmission. If the network interface is not active, the mobile device can turn off the transmitter and/or the receiver to save energy. In order to use the energy efficiently, the transmission time has to be shortened as much as possible, so that the transmitter and the receiver can be shutdown most of the time. The IEEE 802.16-2009 [3] is to provide a high speed, wireless last mile to benefit the rural area lack of wire-line infrastructure [1][2]. It can serve as a backbone of IEEE 802.11 networks so that the cost of deploying the wireless networks can be decreased [1][2]. Users can connect their notebooks or PDAs directly to the IEEE 802.16 network without using the IEEE 802.11 network to get higher transmission rates and better QoS [2]. The communication range of the IEEE 802.16-2009 standard is up to 31 miles and the transmission rate is up to 100 Mbps [2]. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 75–79, 2010. © Springer-Verlag Berlin Heidelberg 2010

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To reduce the energy consumption of SSs, the sleep mode was introduced in the IEEE 802.16-2009 to achieve energy saving. The sleep mode is a state for an SS to be temporarily absent from the serving base station (BS) air interface [3]. Note that in the IEEE 802.16-2009, normal transmission periods, which are called normal mode, and sleeping periods, which are called sleep mode, are interleaved with each other.

2 Existing Approaches The characteristics of wireless networks are high error rate, high channel variation, and high transmission delay, and we must consider these characteristics when we design a packet scheduling algorithm [4]. In [5], the authors proposed a Deficit Fair Priority Queue (DFPQ) algorithm, which uses two-tier architecture. The first tier ensures that low priority data streams will have the opportunity to transmit packets. The second tier uses a multiple queue mechanism and applies a different packet scheduling algorithm according to each queue’s characteristic. The drawback of this algorithm is that the transmission of low priority data streams may interrupt the transmission of high priority data streams. In [6], the authors proposed an algorithm called Preemptive DFPQ to fix the problem mentioned above. However, these two algorithms did not consider the wireless network channel condition. In [7], the authors proposed an algorithm that considers network traffic patterns and SS mobility to maximize energy efficiency. The authors suggested several calculation methods of the length of a sleep window to avoid the problem of serious transmission delay caused by a long sleep window. The disadvantage of this algorithm is that the total sleep time will be shortened. In [8], the authors proposed another way to adjust the parameters of sleep mode for the traffic patterns of constant bit rate (CBR) and FTP. However, they set the round trip time (RTT) to a constant in their experiment, which is not realistic in real network environments. In [9], the authors proposed an algorithm to determine the lengths of sleep window and listening window based on the QoS constraints of real-time communications. Packets are scheduled by the EDF algorithm. The disadvantages of this algorithm are that it uses linear integer programming to find an optimum solution, which results in high time and space complexities. In [10], the authors proposed a packet scheduling algorithm not only considering the delay constraint, but also the delay jitter constraint. Packets are also scheduled by the EDF algorithm. The disadvantage of this algorithm is that it is applied to PSC type 3, which will be deactivated when the sleep period is expired. As a result, the calculation of PSC parameters and the negotiation of PSC creation must be performed repeatedly, which are not energy-efficient.

3 Design Approach In the IEEE 802.16-2009, we must combine packet scheduling and power saving modes altogether to achieve low power consumption, while maintaining the QoS requirement. Therefore, we propose a QoS-guaranteed and energy-aware packet scheduling algorithm. The proposed algorithm is divided into two steps: power saving class generation and packet scheduling, which involve simple computations so as to have low time complexity.

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3.1 Power Saving Class Generation Because the connections in PSC type 2 have strict QoS constraints, we describe the proposed power saving class generation of PSC type 2 first. In PSC type 2, we define the length of listen windows tl as follows: C

tl = ∑ i =1

pi R j × TF

(1)

where pi is the average packet length of connection i, Rj is the raw data rate under burst profile j, TF is the frame duration and C is the number of connections. The length of sleep window ts is the minimum delay constraint in all connections. In PSC type 1, the suitable data delivery services are NRT-VR and BE services. In our design, the connections which belong to PSC type 1 can use the remaining bandwidth of the PSC type 2 listen window after PSC type 2 connections have been served. PSC type 3 was designed for connections that carry management operations as well as multicast connections. According to the specification of PSC type 3, when the periodic ranging is active, the maximum sleep window length tmax must be set to the value indicated by the Next Periodic Ranging TLV, which is carried in the RNG-RSP or MOB_SLP-RSP messages [3]. The start frame number of sleep window nsf must be set to the frame number next to the transmission of RNG-RSP or MOB_SLP-RSP messages. 3.2 Packet Scheduling Because the connections using UGS scheduling services have strict QoS requirements, the UGS connections have the highest priority. The connections of the ertPS and rtPS scheduling services have the second and third priority, respectively. We use the EDF algorithm in UGS, ertPS, and rtPS scheduling services. Packets that miss their transmission deadline will be discarded. The nrtPS and BE scheduling services are not delay sensitive. Therefore, the packet scheduling algorithm for nrtPS connections is WFQ, and the packet scheduling algorithm for BE connections is RR.

4 Experimental Results We evaluate the proposed algorithm using the ns-2 simulator and NIST WiMAX modules for ns-2. Table 1 shows the experiment parameters we used, which are defined in the IEEE 802.16-2009 [3]. We compared average packet delay under various numbers of connections. In UGS and ertPS connections, we used G.711, G.723.1, G.726, and G.729a as the network flow model. In rtPS, we used H.264/MPEG-4 AVC as the network flow model. As to the nrtPS and BE connections, we used the Poison process as the network flow model. Fig. 1 shows that the average packet delay of our algorithm is almost identical (0.000677% higher) to that of the naïve algorithm. Fig. 2 illustrates the relationship between energy consumption and number of connections. When the number of connections increases, the total number of packets transmitted increases; thus it causes the increase of energy consumption. Simulation results show that the energy consumption of our algorithm is 14.37% less than that of the naïve algorithm.

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Description

BW

Nominal channel bandwidth

Value 10MHz

NFFT

Number of subcarriers

256

Nd

Data subcarriers

192

G

Guard fraction

1/4

TF

Frame duration

5 ms

Average packet delay (msec)

5.69650 5.69640

5.69633494 5.696296025

5.69630 5.69620 5.69610 5.69600 Naïve

Proposed Algorithm

Fig. 1. Average packet delay under different numbers of connections Naïve

Proposed Algorithm

Energy Consumption (joule)

5 4.5 4 3.5 3 2.5 2 1

2

3

4

5

6

7

8

9

10

Number of Connections

Fig. 2. Energy consumption under various numbers of connections

5 Conclusion As mobile computing is getting more popular, the demand of efficient power source usage will become even greater. In this paper, we have presented a QoS-guaranteed, energy-efficient packet scheduling algorithm. Simulation results show that our proposed algorithm has 14.37% less energy consumption than the naïve algorithm. In addition, our algorithm does not sacrifice the average packet delay. Currently, we are working on comparing with some representative related work. The contributions of our proposed algorithm are that our algorithm is suitable for wireless environments where connections come and go rapidly and the time complexity of our algorithm is low.

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Acknowledgments This work was supported in part by the National Science Council, Taiwan, under Grants NSC96-2628-E-009-140-MY3 and NSC98-22218-E-009-008.

References 1. Vaughan-Nichols, S.J.: Achieving wireless broadband with WiMax. IEEE Computer 37(6), 10–13 (2004) 2. Abichar, Z., Peng, Y., Chang, J.M.: WiMax: the emergence of wireless broadband. IEEE IT Prof. 8(4), 44–48 (2006) 3. IEEE Standard for Local and Metropolitan Area Networks Part 16: Air Interface for Fixed Broadband Wireless Access Systems, IEEE Std 802.16-2009 (2009) 4. Tsai, T.Y., Tsai, Z.: Design of a Packet Scheduling Scheme for Downlink Channel in IEEE 802.16 BWA Systems. In: IEEE Wireless Communications and Networking Conference, pp. 1453–1458 (2008) 5. Chen, J., Jiao, W., Wang, H.: A service flow management strategy for IEEE 802.16 broadband wireless access systems in TDD mode. In: IEEE International Conference on Communications, vol. 5, pp. 3422–3426 (2005) 6. Safa, H., Artail, H., Karam, M., Soudah, R., Khayat, S.: New scheduling architecture for IEEE 802.16 wireless metropolitan area networks. In: IEEE/ACS International Conference on Computer Systems and Applications, pp. 201–210 (2007) 7. Lee, N.H., Bahk, S.: MAC sleep mode control considering downlink traffic pattern and mobility. In: IEEE 61st Vehicular Technology Conference, vol. 3, pp. 2076–2080 (2005) 8. Jang, J., Han, K., Choi, S.: Adaptive power saving strategies for IEEE 802.16e mobile broadband wireless access. In: Asia-Pacific Conference on Communications, pp. 1–5 (2006) 9. Tsao, S.L., Chen, Y.L.: Energy-efficient packet scheduling algorithms for real-time communications in a mobile WiMAX system. Computer Communications 31(10), 2350–2359 (2008) 10. Liao, W.H., Yen, W.M.: Power-saving scheduling with a QoS guarantee in a mobile WiMAX system. Journal of Network and Computer Applications 32(6), 1144–1152 (2009)

Minimum Energy Cost k -barrier Coverage in Wireless Sensor Networks Huiqiang Yang, Deying Li , Qinghua Zhu, Wenping Chen, and Yi Hong Key Laboratory of Data Engineering and Knowledge Engineering, MOE School of Information, Renmin University of China, P.R. China [email protected]

Abstract. Barrier coverage problem is one of important issues in wireless sensor networks. In this paper we study the minimum energy cost k-barrier coverage problem in wireless sensor network in which each sensor has l + 1 sensing power levels. First, we transform the minimum energy cost k-barrier coverage problem into a minimum cost ﬂow problem with side constraints. Then we use Lagrangian relaxation technique to solve this minimum cost ﬂow problem. Moreover, we propose two eﬃcient heuristics for the minimum energy cost k-barrier coverage problem. Simulations evaluate that our algorithms are eﬃcient. Keywords: k-barrier coverage, sensing power level, minimum energy cost, sensor networks.

1

Introduction

Wireless sensor networks [1] provide an eﬃcient way to guard the boundaries of critical infrastructures or assets. By deploying a number of wireless sensors along the boundaries, intruders could be detected once they enter into the areas covered by the sensors, where the union of the covered areas of sensors forms a barrier. Fig.1 shows a typical example of barrier coverage. Every path crossing the belt barrier region will be detected by at least one sensor. Compared with full coverage which requires every point in the deployment region to be covered, barrier coverage requires much fewer sensors. If the width of the deployment region is three times the sensing range, full coverage requires more than twice the density of barrier coverage. Saving in sensors grows linearly with width [2]. In recent years, with the requirement of protecting sensitive military facilities and national borders increasing, barrier coverage attracted more and more interest of researchers. In some literatures about barrier coverage, barriers are constructed as rectangular regions, which have been pre-selected as moats around the protected area. The barrier can also be annulus [2]. In practice, a certain degree of redundancy is required in barrier coverage because some sensors may fail due to draining of energy, malfunction or malicious damage and so on. k barrier coverage which every path crossing barrier region can be detected by at

Corresponding author.

G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 80–89, 2010. c Springer-Verlag Berlin Heidelberg 2010

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Fig. 1. An example of barrier coverage

least k sensors is proposed to address this situation. The authors [2] proposed eﬃcient algorithms to quickly determine whether a region is k -barrier covered. In [8], the authors proposed LBCP algorithm to achieve local barrier coverage by limiting the cross path. A distributed algorithm in [9] is proposed to construct strong sensor barriers without any constraint on crossing paths. In all the above works, the sensing ranges of sensors are not adjustable. Since sensors are battery powered and it is diﬃcult to replace the batteries, sensors may be designed to adjust sensing range adaptively to save energy consumption. In this paper, we study the problem of minimizing the energy consumption in k -barrier coverage. Each sensor has several sensing power levels and each sensing power level corresponds to a sensing range. We need to assign a sensing power level to each sensor and the result sensor network can form a k -barrier coverage. Our goal is ﬁnding such an assignment that the total sensing power is minimized. To solve the problem, we ﬁrst transform our problem into minimum cost network ﬂow problem with side constraints. Second, we use the Lagrangian relaxation technique [3] to get a lower bound of the problem. Then, we design two heuristic algorithms with diﬀerent ways of constructing auxiliary graph to solve our problem. The rest of this paper is organized as follows. We discuss the related work in Section 2. Then, we give the network models and the problem statements in Section 3. In section 4, we ﬁrst transform the problem to the minimum cost ﬂow problem with side constraints, then use the Lagrangian Relaxation method to get a lower bound for our problem. In Section 5, we propose two eﬃcient heuristic algorithms to solve our problem. Section 6 presents the performance evaluation results. Finally, Section 7 concludes the paper.

2

Related Work

The traditional coverage problems, including point coverage [4] which requires every target point to be covered by at least one or k sensors, and area coverage [5] which requires every point in the area to be covered by at least one or k sensors, have been studied intensively. The concept of barrier coverage was ﬁrst introduced in the context of control schemes for multi-robot systems [6]. Kumar et al. [2] proposed eﬃcient algorithms to quickly determine whether a region is k -barrier covered. Morever, they established an optimal deployment pattern to achieve k -barrier coverage when sensors were deployed deterministically. Diﬀerent

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from omni-sensing model in [2], Kuo-Feng Ssu et al. [7] studied the k -barrier coverage problem in directional sensing model and developed a distributed algorithm to construct k -barrier coverage in randomly deployed directional sensor ﬁelds. Due to the observation that movements were likely to follow a shorter path when crossing a belt region, [8] introduced the concept of local barrier coverage which guaranteed to detect all movements whose trajectory were conﬁned to a slice of the belt region of deployment and proposed LBCP algorithm to achieve local barrier coverage. Although local barrier coverage does not always guarantee global barrier coverage, the authors showed that for thin belt regions, local barrier coverage almost always provides global barrier coverage. In [9], a distributed algorithm were proposed to construct strong sensor barriers on long boundary areas of irregular shape without any constraint on crossing paths. The above works were based on the assumption that sensors were uniformly deployed in a large area at random. Saipulla, et al. [10] studied the barrier coverage when sensors were deployed along a line and established a tight lower-bound for the existence of barrier coverage in line-based deployments. All these works assumed that the sensing power of sensor were ﬁxed. In this paper, based on multiple sensing power levels model, we study the problem of minimizing the total energy cost of k -barrier coverage in a sensor network.

3

Network Model and Problem Description

We assume that a set S of sensors are deployed randomly in a two-dimensional rectangular area where sensors do not move after they are deployed. We assume that each sensor knows the coordinates (x, y) of its own location. This may be detected through GPS or be obtained through a localization mechanism. Suppose each sensor node has l + 1 sensing power levels, 0 ≤ p1 ≤ p2 · · · ≤ pl , i.e., each sensor has l + 1 diﬀerent corresponding sensing ranges {0, r1 , r2 , · · · , rl }. Every active sensor is assigned with one sensing power level in {p1 , p2 , · · · , pl }. For a given power assignment of the sensors, i.e., each sensor si is assigned with a sensing power level P L(si ),PL(si ) ∈ {0, p1 , p2 , · · · , pl }, we deﬁne the total P L(si ). energy cost of the network as si ∈S

We give some deﬁnitions as follows: Definition 1. Crossing path (or Incursive path): A path is said to be a crossing (or incursive) path in a belt region if it crosses the complete width of the region from one side to the other side. An example is illustrated in Fig.1. The red sketch path is an incursive path while the two blue paths are not crossing paths. Definition 2. A crossing path P is said to be k-covered if P intersects with at least k active sensor’s sensing disks. Definition 3. k-barrier coverage: A belt region is said to be k-barrier covered by a sensor network deployed over it if and only if all the crossing paths through the belt region are k-covered by this sensor network.

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Definition 4. Minimum energy k-barrier coverage problem: Given a sensor network over the objective belt region and each sensor has l + 1 sensing power levels, find a sensing power level assignment such that the given belt region is k-barrier covered and the total energy cost is minimized. The goal of the paper is to ﬁnd a sensing power level assignment for each sensor, i.e., the set of sensing power {P L(s1 ), P L(s2 ), · · · , P L(sn )} such that the sensor network can form a k-barrier coverage and the total energy cost is minimum among all the assignments.

4

Lagrangian Method for Minimum Energy Cost k-barrier Coverage Problem

Given a set of sensors S = {s1 , s2 , · · · , sn }, each sensor node si has l + 1 sensing power levels, 0 ≤ p1 ≤ p2 ≤ · · · ≤ pl . We consider the minimum energy cost k-barrier coverage problem in a long narrow region referred to as a belt, where sensors are deployed randomly over it. Before solving the minimum energy cost k-barrier coverage problem, we ﬁrst restate the method for determine whether a belt region is k-barrier covered [2]. Corresponding to a sensor network deployed in a belt region, we derive a coverage graph CG = (V, E), where V is the set of all sensor nodes plus two virtual vertexes s and t (see Figure 2). The set of edges E is derived as follows: each pair of sensors whose sensing disks overlap is connected by an edge. Additionally, the sensors whose sensing disks intersect with the left boundary are connected to node s and the sensors whose sensing disks intersect with the right boundary are connected to node t. An example of constructed coverage graph is shown in Figure 2.

Fig. 2. A simple example of coverage graph

Lemma 1. [2] An open belt region is k-barrier covered if and only if s and t are k-connected in the corresponding coverage graph, CG. In order to solve the minimum energy cost k-barrier coverage problem, we construct an auxiliary weighted directed graph Ga = (Va , Ea , Ca , Ua ) as follows. In Ga , Va denotes the set of nodes, Ea denotes the set of edges, Ca denotes the set of edges’ cost and Ua denotes the set of edges’ capacity.

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t h t h t h First, we construct l node pairs {qi,1 , qi,1 }, {qi,2 , qi,2 }, · · · , {qi,l , qi,l } for each t h t sensor si in S. For each node pair {qi,k , qi,k }, we add a directional edge from qi,k h h t h to qi,k with cost pk . We add a directed edge (qi,k , qj,m ) with cost 0 from qi,k to t qj,m if and only if si ’s sensing range intersects with sj ’s sensing range while sensor si working at sensing power level pk and sensor sj working at sensing power level pm . In addition, we add a new virtual node s at the left side of the belt region and a new virtual node t at right side of the belt region. Then, there is a directed edge t with cost 0 if and only if si ’s sensing range intersects with the left from s to qi,k border of the belt region when si working at sensing power level pk . Last, there is h to t with cost 0 if and only if si ’s sensing range intersects a directed edge from qi,k with the right border of the belt region when si working at sensing power level pk . Each edge in Ea is associated with capacity 1. To formulate our problem rigorously, we use a binary variable xi,j to represent if there is an edge from i to j in Ga = (Va , Ea , Ca , Ua ). For the simplicity of the mathematical representation of the problem, we use e(si ) to denote the set of t h t h t h , qi,1 ), (qi,2 , qi,2 ), · · · , (qi,l , qi,l )} for each sensor si in S. Let X denote edges {(qi,1 the row vector of variables corresponding to each edge (i.e., X = {xi,j |(i, j) ∈ Ea }). Let C denote the row vector of the cost of edges in the same order with X (i.e., C = {ci,j |(i, j) ∈ Ea }). Let X be the transpose of vector X. Our problem can be formulated by an integer programming problem as follows:

min C · X subject to

(i,v)∈Ea

xi,v −

(1)

xu,i = 0, for each node i ∈ Va − {s, t}

(2)

xs,v = k

(3)

xu,t = k

(4)

xu,v = 1, for each edge set e(si )

(5)

xi,j = 0, 1

(6)

(u,i)∈Ea

(s,v)∈Ea

(u,t)∈Ea

(u,v)∈e(si )

From the formulation, it is easy to know that if there are not constraints (5), the problem is a classical minimum cost ﬂow problem. Adding constraints (5), the problem becomes the minimum cost network ﬂow problem with side constraints (5). From Lemma 1 and the structure of the auxiliary graph, we have the following result. Lemma 2. The minimum energy cost k-barrier coverage problem is equivalently transformed to the minimum cost k-flow problem with side constraints. We can use the Lagrangian relaxation algorithm in [3] to obtain a lower bound of our problem. Lagrangian relaxation has been used in various approaches to the solution of constrained capacitated network ﬂow problems [11,12].

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Two Heuristic Algorithms for Minimum Energy Cost k-barrier Coverage Problem

In this section, we propose two eﬃcient heuristic algorithms (referred as Heuristic-1 and Heuristic-2) for the minimum energy cost k-barrier coverage problem. Every sensor si has its coordinates si (x, y) relative to the bottom-left corner of the belt region which is the original point. We use si .x to represent x-coordinate x of si . The main idea of two heuristic algorithms is that:(1) we construct an auxiliary graph according to CG; (2) use the minimum cost ﬂow algorithm on the auxiliary graph to ﬁnd k node-disjoint paths from s to t; (3) according to the construction of graph, we can adjust the sensing power levels of sensors in each path to make 1-barrier coverage. Thus we get k barrier coverage in our original sensor network. In order to make sure that a sensor node si is used only once in all paths , we split each sensor node in CG into two nodes qit and qih . This transformation replaces each original undirected edge (si , sj ) in CG by a directed edge (qih , qjt ). The main diﬀerence between Heuristic-1 and Heuristic-2 is the way of assigning cost for each edge. In Heuristic-1, we use the distance between si and sj as the cost of edges while in Heuristic-2 we use the maximum sensing power levels as the cost of edges. 5.1

Heuristic-1

In Heuristic-1, an auxiliary weighted directed graph Gb = (Vb , Eb , Cb , Ub ) where Vb denotes the set of nodes, Eb denotes the set of edges, Cb denotes the set of edges’ cost and Ub denotes the set of edges’ capacity, is constructed as follows: we construct one node pair {qit , qih } for each sensor si originally in S. For each node pair {qit , qih }, we add a directed edge from qit to qih with cost 0. We add a directed edge (qih , qjt ) with cost of dist(si , sj ) from qih to qjt if and only if the Euclidian distance between si and sj (i.e., dist(si , sj )) is less than 2 ∗ rl , where rl is the largest sensing range. In addition, we also add a new virtual node s at the left side of the belt region and a new virtual vertex t at right side of the belt region. Then, there is a directed edge from s to qit if and only if si ’s sensing range intersects with the left border of the belt region when works at its maximum sensing power level pl . The cost of this edge is si .x. There is a directed edge from qih to t if and only if si ’s sensing range intersects with the right border of the belt region when works at its maximum sensing power level pl . The cost of this edge is LEN − si .x, where LEN is the length of the belt region. Each edge in Eb is associated with capacity 1. Then we use the minimum cost ﬂow algorithm on Gb to ﬁnd k node-disjoint paths from s to t. And adjust the sensing power level of each sensor node in each path to make this path form 1-barrier coverage. According to Lemma 1, we know that the found paths form a k-barrier coverage in the sensor network. Note that we use R(pi ) to denote the sensing range corresponding to the sensing power level pi .

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Algorithm Heuristic-1 is formally represented as following: Algorithm 1. Heuristic-1 1: Construct an auxiliary graph Gb = (Vb , Eb , Cb , Ub ) as we have just illustrated above. 2: Using minimum cost ﬂow algorithm in [3] to ﬁnd k-minimum cost ﬂow from s to t. 3: For each path found Pi = {s, qit1 , qih1 , · · · , qitm , qihm , t}, assign a power level for each sensor node as follows: P L(si1 ) = min{pk |rk ≥ max(si1 .x, dist(si1 , si2 ) − rl )} P L(sij ) = min{pk |rk ≥ max(dist(sij−1 , sij ) − R(P Lsij−1 ), dist(sij , sij+1 ) − rl )}, for 2 ≤ j ≤ m − 1 P L(sim ) = min{pk |rk ≥ max(dist(sim−1 , sim ) − R(P Lsim−1 ), LEN − sim .x)}

5.2

Heuristic-2

Now we will introduce another Heuristic algorithm for our problem. As in Heuristic-1, we ﬁrst construct an auxiliary graph Gc = (Vc , Ec , Cc , Uc ), where Vc denotes the set of nodes, Ec denotes the set of edges, Cc denotes the set of edges’ cost and Uc denotes the set of edges’ capacity. We construct one node pair {qit , qih } for each sensor si originally in S. For each node pair {qit , qih }, we add a directed edge (qit , qih ) from qit to qih with cost pl . We add a directed edge ( qih , qjt ) with cost 0 from qih to qjt if and only if the Euclidian distance between si and sj (i.e., dist(si , sj )) is less than 2 ∗ rl , where rl is the largest sensing range. We also add a new node s at the left side of the belt region and a new node t at right side of the belt region. There is a directed edge from s to node qit with cost 0 if and only if the sensor si ’s sensing range intersects with the left border of the belt region when si working at the maximum sensing power level pl . And there is a directed edge from qih to t with cost 0 if and only if si ’s sensing range intersects with the right border of the belt region when si working at the maximum sensing power level pl . Each edge in Ec is associated with capacity 1. Then we use the minimum cost ﬂow algorithm on Gc to ﬁnd k node-disjoint paths from s to t. And adjust the sensing power level of each sensor node in each path to make this path feasible. Heuristic-2 is formally represented as follows: Algorithm 2. Heuristic-2 1: Construct an auxiliary graph Gc = (Vc , Ec , Cc , Uc ) as we have just illustrated. 2: Using minimum cost ﬂow algorithm in [3] to ﬁnd k-minimum cost ﬂow from s to t. 3: For each found path Pi = {s, qit1 , qih1 , · · · , qitm , qihm , t}, assign a power level for each sensor node as follows: P L(si1 ) = min{pk |rk ≥ max(si1 .x, dist(si1 , si2 ) − rl )} P L(sij ) = min{pk |rk ≥ max(dist(sij−1 , sij ) − R(P Lsij−1 ), dist(sij , sij+1 ) − rl )}, for 2 ≤ j ≤ m − 1 P L(sim ) = min{pk |rk ≥ max(dist(sim−1 , sim ) − R(P Lsim−1 ), LEN − sim .x)}

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Both in Heuristic-1 and Heuristic-2, the step 1 takes time complexity O(n). In step 2, using the minimum cost ﬂow algorithm takes time O(n3.5 ). Step 3 takes time O(n · k). Because k is always less than n, step 3 takes time at most O(n2 ). Thus, the whole algorithm takes time O(n3.5 ).

6

Performance Evaluation

We develop a custom simulator and use it to evaluate the performance of our proposed algorithms. In the simulation, we randomly deploy sensors in a rectangular barrier with 100 units long and 5 units wide. Each sensor has 4 diﬀerent sensing power levels {0, 16.0, 36.0, 64.0}. Without loss of generality, the sensing ranges are the square root of the power levels in the simulation, i.e., each sensor has 4 diﬀerent sensing ranges {0, 4.0, 6.0, 8.0}. In Lagrangian Relaxation algorithm, we set the = 0.0001. Due to time cost for converging to a feasible solution is very high, we terminate the algorithm after 100 iterations and get a lower bound of an instance. In experiment, we will consider the eﬀect of factors such as belt length, sensor density and number of required barrier coverage. The performance of the barrier construction schemes was evaluated by the total energy cost of the formed k-barrier coverage. 6.1

Total Energy Cost vs. Number of Sensors

5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

Heuristic-1 Heuristic-2 Lagrangian

100

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Total Energy Cost

Total Energy Cost

To study the eﬀect of number of sensors on total energy cost, we set up the networks with the number of sensors increasing from 100 to 220 with an increment of 20. The length of belt region is set to 100. Fig.3 presents the performance of the algorithms with respect to the number of sensors. We can see that Heuristic2 has better performance than Heuristic-1. The total energy cost consumed by Heuristic-1 has at most two times of the lower bound of the optimal solution. This showed our heuristic algorithms are eﬃcient.

140

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Number of sensors

(a) k = 5

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(b) k = 8

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(a) k = 5

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Fig. 4. Total energy cost vs. Length of Belt region

6.2

Total Energy Cost vs. Lenght of Belt Region

To study the eﬀect of the length of belt region, we vary the length of belt region from 100 to 200 with an increment of 20. The number of sensors is set to 250. As shown in the Fig.4, the total energy cost increases when the length increases. This is because every path has to cover longer length. As shown in Fig.4, Heuristic-2 has better performance than Heuristic-1. The total energy cost consumed by Heuristic-1 has at most two times of the lower bound of the optimal solution. Above simulation results show that our proposed heuristic are eﬃcient.

7

Conlusions

In this paper, we study the minimum energy cost of k-barrier coverage problem and formulate the problem into a minimum cost ﬂow problem with side constraints. We use the classical Lagrangian algorithm to ﬁnd a lower bound of the total energy cost. Then we propose two heuristic algorithms for this problem. At last, we implement all three algorithms and compare their performances. The two heuristic algorithms consume the total energy cost at most two times of the lower bound of the optimal solution. These experiments show the proposed algorithms’ eﬃcient.

Acknowledgment This paper was supported in part by Renmin University of China under Grants 10XNJ032 and 10XNG037 and 863 high-tech project under Grant 2008AA01Z120.

References 1. Akyildiz, I., Su, W., Sankarasubramaniam, Y., Cayirci, E.: A survey on sensor networks. IEEE Communications Magazine, 102–114 (2002) 2. Kumar, S., Lai, T.H., Arora, A.: Barrier Coverage with Wireless Sensors. In: Proc. ACM MobiCom (2005)

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3. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliﬀs (1993) 4. Cardei, M., Thai, M.T., Li, Y., Wu, W.: Energy-eﬃcient target coverage in wireless sensor networks. In: IEEE INFOCOM, pp. 1976–1984 (2005) 5. Zhou, Z., Das, S., Gupta, H.: Connected K-coverage problem in sensor networks. In: ICCCN (2004) 6. Gage, D.: Command control for many-robot systems. In: The Nineteenth Annual AUVS Technical Symposium (1992) 7. Ssu, K.-F., Wang, W.-T., Wu, F.-K., Wu, T.-T.: K-barrier coverage with a directional sensing model. International Journal on Smart Sensing and Intelligent Systems, 75–83 (2009) 8. Chen, A., Kumar, S., Lai, T.-H.: Designing localized algorithms for barrier coverage. IEEE Transactions on Mobile Computing, 491–504 (2010) 9. Liu, B., Dousse, O., Wang, J., Saipulla, A.: Strong Barrier Coverage of Wireless Sensor Networks. In: Proceeding of ACM Mobicom (2008) 10. Saipulla, A., Westphal, C., Liu, B., Wang, J.: Barrier Coverage of Line-Based Deployed Wireless Sensor Networks. In: IEEE INFOCOM, pp. 127–135 (2009) 11. Fisher, M.L.: The Lagrangian relaxation method for solving integer programming problems. Management Science, 1–18 (1981) 12. Held, M., Wolfe, P., Crowder, H.D.: Validation of subgradient optimization. Mathematical Programming, 62–88 (1974)

On the Performance of Distributed N-Cooperation Power Allocation via Differential Game in Cognitive Radio System Shunxi Gao1, Long Zhang2, Suqin Fan1, Wei Huang2, Qiwu Wu2, and Yu Deng3 1

Mechanical and Electrical Services Department, Handan Polytechnic College, Handan 056001, China 2 School of Information Engineering, University of Science and Technology Beijing, Beijing 100083, China 3 Communications Research Group, Department of Electronics, University of York, Heslington YO10 5DD, UK [email protected]

Abstract. In this paper, we study the distributed cooperative power allocation problem in a single-cell cognitive radio system. We build up a dynamic power allocation model via differential game using two formulated cost functions. Moreover, we propose a distributed cooperative power allocation algorithm based on differential game (DCPADG) under the condition that the average power limits and QoS constraints of secondary users (SUs) are guaranteed. Numerical results demonstrate that the proposed DCPADG algorithm can effectively reduce the transmit power and improve the throughput of SUs. Keywords: Cognitive radio; cooperative power allocation; differential game.

1 Introduction Recently, with the increasing interest in cognitive radio (CR) [1], there has been significant attention in power allocation research for CR system. As a framework for modeling, game theory has gained more attention as an effective tool to analyze the distributed power allocation in CR system. In [2], L. Zhang, et al. proposed a distributed noncooperative power control algorithm based on differential game (DNPCDG), which achieves the distributed dynamic power control under the condition that the average power threshold and quality of service (QoS) of secondary users (SUs) are guaranteed. However, the DNPCDG algorithm applied the idea of noncooperative game to solve the problem of distributed power allocation. In this paper, we take into account the problem of cooperative power allocation in CR system. We build up a dynamic power allocation model via differential game using two formulated cost functions. Moreover, we present a distributed cooperative power allocation algorithm based on differential game (DCPADG) under the same condition with the DNPCDG algorithm. Numerical results demonstrate that the proposed DCPADG algorithm can effectively reduce the transmit power and improve the throughput of SUs by comparison with the DNPCDG algorithm. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 90–94, 2010. © Springer-Verlag Berlin Heidelberg 2010

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2 System Model Consider a cellular primary user (PU) radio system consisting of a finite number of cells and base stations. Assume that n SUs are randomly distributed in a single cell over the time interval [t0,T] and SUs are either fixed or slowly moving in that cell. The SUs constitute a spread spectrum CR system centered around the fixed CR base station over the total available authorized spectrum in an ad hoc mode to share the corresponding spectrum holes (SH). Let N = {1,2,", n} denote the set of SUs. At time instant s ∈ [t0 , T ] , the signal-to-interference ratio (SIR) of SU i, for i ∈ N , is equal to ⎛ γi ( s ) = (W R ) ⋅ ⎜⎜ hi pi ( s) ⎝

⎞ h j p j ( s ) + v 2 ⎟⎟ , where pi(s) is the transmit power of SU i, hi j ∈N \ i ⎠

∑

is the channel gain from SU i to the base station of CR system, v2 is the constant background noise power of the base station of CR system, W is the system bandwidth, and R is the transmission rate of SUs. Let pi and d r denote the maximum transmit power of SU i and the cell radius of CR system, respectively. Therefore, the transmit power of SU i should satisfy pi ( s) ∈ [0, pi ] . Assume that pi is a monotone increasing function of Euclidean distance di ( 0 ≤ d i ≤ d r ) from SU i to the base station of CR system, i.e., pi = μ d i5 , where μ > 0 is the power regulation factor. Moreover, assume that di can be acquired through sensing the surrounding environment by SU i. The channel gain hi is given by hi = A diθ , where A > 0 is the constant factor and θ ∈ [3,6] is the path loss factor. To guarantee a QoS requirement, the SIR of SU i should exceed the desired target SIR γitar to recover the transmitted data correctly, i.e., γi ( s ) ≥ γitar . Furthermore, to protect PUs from the harmful interference, it is clearly required that the total transmit power of SUs should not exceed the constraint of the average power limit pth , i.e., ∑ pi ( s ) ≤ pth . Assume that the transmission paths in CR system are valid i∈N

in AWGN channels. According to [3], the throughput of SU i at time instant s ∈ [t0 , T ] , for i ∈ N , can be written as ti ( s ) = log 2 (1 + k ⋅ γ i ( s )) , where k > 0 is the constant factor.

3 Distributed N-Cooperative Power Allocation Definition 1. The N-cooperation is defined as a scenario that all of SUs in a single cell within CR system agree to cooperate, i.e., full cooperation. For N-cooperation power allocation, assume that SUs agree to constitute grand coalition N through full cooperation for their common interests. We introduce two cost functions of SU i to formulate the payoff function, i.e., the power reduction (PR) cost function Ci ( pi ( s )) and the power radiation damage (PRD) cost function Di ( x( s )) . The PR cost function of SU i at time instant s ∈ [t0 , T ] is defined as

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Ci ( pi ( s )) =

λ 2

⋅ ( pi ( s ) − pi )

2

(1)

where λ > 0 is the power regulation factor. To describe the negative effect caused by interference from SUs, we define the PRD cost function of SU i as Di ( x( s )) = π ⋅ x( s)

(2)

where π > 0 is power radiation damage factor and x( s ) is the stock of the accumulated power radiation at time instant s ∈ [t0 , T ] . Combining with differential game theory [4], x( s ) in (2) can be regarded as state variable in differential game. At time instant s ∈ [t0 , T ] , the instant payoff function of SU i can be written as the sum of two cost functions, i.e., gi =

λ 2

⋅ ( pi ( s ) − pi ) + π ⋅ x( s ) 2

(3)

where pi(s) can be regarded as the control variable of SU i in differential game. Therefore, the payoff function of SU i can be given by T 2 ⎡λ ⎤ min ∫ exp(− rs ) ⎢ ⋅ ( pi − pi ) + π ⋅ x ⎥ ds t0 pi ⎣2 ⎦

(4)

where r ∈ [0,1] is the discount rate. According to [4], x( s ) is assumed to satisfy the following differential equation, i.e., x ( s ) = ∑ pi ( s) − δ ⋅ x( s ), i∈ N

x (t0 ) = x0

(5)

where δ > 0 is the power radiation absorption factor. For the convenience of derivation, we relax the time interval of the game and discuss the infinite-horizon differential game [4], i.e., T → ∞ and t0=0 in (4). Let { p1Ν , p2Ν ,", pnΝ } be the transmit power of n SUs under the condition of grand coalition N and assume that there exists continuously differentiable function W ( N , x) , which satisfies Bellman equation in [4], i.e., ⎧ ⎡λ ⎞⎫ 2 ⎤ ∂W ( N , x ) ⎛ rW ( N , x) = min ⎨ ∑ ⎢ ⋅ ( pi − pi ) + π ⋅ x ⎥ + ⎜ ∑ pi − δ ⋅ x ⎟ ⎬ pi ∂x ⎦ ⎝ i∈N ⎠⎭ ⎩i∈N ⎣ 2

(6)

Performing the indicated minimization in (6) yields piN = pi −

1 ∂W ( N , x) ⋅ ∂x

λ

(7)

Substituting piN in (7) into (6), we obtain ∂W ( N , x) ⎡ ⎛ 1 ∂W ( N , x) ⎞ ⎤ ⎪⎫ 2 ⎪⎧ λ rW ( N , x) = min ⎨ ⋅ ( pi − pi ) + π ⋅ x + ⎟ − δ ⋅ x⎥ ⎬ ⎢ ∑ ⎜ pi − λ ⋅ pi ⎪ 2 ∂ x ∂ x ⎠ ⎣ i∈ N ⎝ ⎦ ⎭⎪ ⎩

(8)

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Solving the differential equations in equation (8), we obtain W ( N , x) =

⎫⎪ nπ ⎧⎪ ⎡ n 2π ⎤ ⎨ ⎢ ∑ pi − ⎥ + rx ⎬ , r (r + δ ) ⎪⎩ ⎣i∈N 2λ (r + δ ) ⎦ ⎪⎭

piN = pi −

nπ

λ (r + δ )

(9)

Compared with the transmit power of SU i in DNPCDG algorithm [2], the transmit power of SU i of the N-cooperation power allocation is theoretically reduced. The DCPADG algorithm is given as follows. Step 1. SUs obtain SH by sensing spectrum. Step 2. Initialization at time instant t0=0: parameters of CR system and dynamic power allocation model, pth , pi (t0 ) ( pi (0) ∈ [0, pi ] , ∑ pi (0) ≤ pth ), and γitar . i∈N

Step 3. Update pi over time within time interval [t0 , T ] . Step 4. Regulate pi ( s) at time instant s ∈ [t0 , T ] through (10). If

∑ pi (s) > pth , go to

i∈N

Step 2. If γi ( s ) < γitar , go to Step 2. If PUs use the SH, go to Step 1. Or go to Step 3. Table 1. Parameters used in simulations Symbol Average power limit Pth Background noise power v2 System bandwidth W Transmission rate R Initialized transmit power Desired target SIR Path gain factor A Path loss factor θ Power regulation factor μ Constant factor for throughput of single SU k Power reduction factor λ Power radiation damage factor π Discount rate r Power radiation absorption factor δ

Default Value 2.5W 2×10-15mW 106Hz 104bits/s 2.22×10-16mW 8 0.097 4 2×10-14 0.2 1.25×10-6 5×10-3 0.05 0.03

4 Numerical Results and Conclusions In the simulations, we evaluate the performance of DCPADG algorithm by comparing it with the DNPCDG algorithm [2] and the conventional SIR-balancing algorithm [5]. The iterative function of power regulation of SU i in SIR-balancing algorithm [5] is given by pi(l +1) = ( γitar γi(l ) ) ⋅ pi(l ) , l = 0,1, 2," The feedback Nash equilibrium of the transmit power of SU i in DNPCDG algorithm [2] is given by pi∗ = pi − π λ (r + δ ) . We consider a single-cell CR system with n = 10 randomly distributed SUs and dr=500m. The distance from SUs to the base station of CR system is set to the same values as that in [2]. The other parameters used in simulations are given in Table 1. Fig. 2 compares the transmit power of the DCPADG algorithm, the DNPCDG algorithm and the SIR-balancing algorithm. We observe from Fig. 2 that the transmit power of the proposed DCPADG algorithm is lower than those of the DNPCDG

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Fig. 1. Comparison of transmit power of the proposed DCPADG algorithm, the DNPCDG algorithm and the SIR-balancing algorithm.

Fig. 2. Comparison of throughput of the proposed DCPADG algorithm, the DNPCDG algorithm and the SIR-balancing algorithm.

algorithm and the SIR-balancing algorithm in the range of distance from SU to base station between 200m and 500m, which indicates that the DCPADG algorithm is more suitable for the SUs with the farther distance to the base station of CR system. Fig. 3 shows that the comparison of throughput of the DCPADG algorithm, the DNPCDG algorithm and the SIR-balancing algorithm. It can be easily seen that the throughput of the DCPADG algorithm, the DNPCDG algorithm and the SIRbalancing algorithm is shown as a gradual increasing trend with the growth of the distance from SU to base station. It is obvious that the throughput of the DCPADG algorithm is higher than those of the DNPCDG algorithm and the SIR-balancing algorithm in the range of distance from SU to base station between 45m and 430m. In this paper, we study the distributed cooperative power allocation problem in a single-cell CR system and further propose the DCPADG algorithm. Numerical results demonstrate that the proposed algorithm can effectively reduce the transmit power and improve the throughput of SUs. Acknowledgments. This work is funded in part by the NSFC under Grant No. 60903004.

References 1. Mitola III, J., Maguire, G.: Cognitive radio: making software radios more personal. IEEE Personal Communications 6(4), 13–18 (1999) 2. Zhang, L., Zhou, X., Wang, J., Huang, W., Ma, Z.: Power Control Algorithm Based on Differential Game for CR System. Journal of Electronics & Information Technology 32(1), 141–145 (2010) 3. Qiu, X., Chawla, K.: On the performance of adaptive modulation in cellular systems. IEEE Transactions on Communications 47(6), 884–895 (1999) 4. Yeung, D.W.K., Petrosyan, L.A.: Cooperative Stochastic Differential Games. Springer, Heidelberg (2005) 5. Koskie, S., Gajic, Z.: A Nash game algorithm for SIR-based power control in 3G wireless CDMA networks. IEEE/ACM Transactions on Networking 13(5), 1017–1026 (2005)

Energy-Efficient Restricted Greedy Routing for Three Dimensional Random Wireless Networks Minsu Huang1, Fan Li2, , and Yu Wang1, 1

University of North Carolina at Charlotte, USA {mhuang4,yu.wang}@uncc.edu 2 Beijing Institute of Technology, China [email protected]

Abstract. In this paper, we investigate how to design energy-efficient localized routing in a large-scale three-dimensional (3D) wireless network. Several 3D localized routing protocols were proposed to seek either energy efficiency or delivery guarantee in 3D wireless networks. However, recent results [1, 2] showed that there is no deterministic localized routing algorithm that guarantees either delivery of packets or energy efficiency of its routes in 3D networks. In this paper, we focus on design of a simple localized routing method which can provide energy efficiency with high probability in a randomly deployed 3D network. In particular, we extend our previous routing method designed for 2D networks [3] to 3D networks. The proposed 3D routing method is a simple variation of 3D greedy routing and can guarantee energy efficiency of its paths with high probability in random 3D networks. We also study its asymptotic critical transmission radius to ensure the packet delivery with high probability in random 3D networks. Simulation results confirm our theoretical results.

1 Introduction Recently, three-dimensional (3D) wireless network has received significant attention [4, 5, 6, 7, 8], due to its wide range of potential applications (such as underwater sensor networks [9]). However, the design of networking protocols for 3D wireless networks is surprisingly more difficult than that for 2D networks. In this paper, we focus on one particular problem in 3D networks: energy efficient localized geographic routing. In localized geographic routing, the forwarding decision is made by the intermediate node based solely on its local information. Without the route discovery phase and maintenance of routing tables, it enjoys the advantages of lower overhead and higher scalability than other traditional routing protocols. This makes it very suitable for 3D networks. Greedy routing is one of the most popular localized routing methods, in which a packet is greedily forwarded to the closest node to the destination in order to minimize the average hop count. Greedy routing can be easily extended to 3D case. Actually, several under-water routing protocols [6, 10] are just variations of 3D greedy routing.

The work of F. Li is supported in part by the National Natural Science Foundation of China under Grant No. 60903151, the Beijing Key Discipline Program, and funds provided by Beijing Institute of Technology. The work of M. Huang and Y. Wang is supported in part by the US National Science Foundation (NSF) under Grant No. CNS-0721666 and No. CNS-0915331. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 95–104, 2010. c Springer-Verlag Berlin Heidelberg 2010

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However, to guarantee packet delivery or energy efficiency of 3D routing is not straightforward and very challenging. Simple greedy routing may fail to reach the destination when falls into a local minimum (a node without any “better” neighbors). Face routing [11] can be used on planar topology to recovery from local minimum and guarantee the delivery in 2D networks, but there is no planar topology concept any more in 3D networks. In fact, Durocher et al. [1] recently proved that there is no deterministic localized routing algorithm for 3D networks that guarantees the delivery of packets. On the other hand, even a localized routing method can find the route to deliver the packet, it may not guarantee the energy efficiency of the path, i.e., the total power consumed compared with the optimal could be very large in the worst case. Several energy-aware localized 2D routing protocols [12, 13, 14] already took the energy concern into consideration, but none of them can theoretically guarantee the energy-efficiency of their routes. This is true for all existing 3D localized routing methods too. Recently, Flury and Wattenhofer [2] showed an example of a 3D network (Figure 1 of [2]) where the path found by any localized routing protocol to connect two nodes s and t has energy consumption asymptotically at least Θ(d3 ) in the worst case. Here d is the optimal energy consumption to connect s and t. Therefore, in this paper, we focus on the design of a simple localized routing method which can provide energy efficiency with high probability in a randomly deployed 3D network. In particular, we extend our previous routing method designed for 2D networks [3] to 3D networks. The proposed 3D routing method is a simple variation of 3D greedy routing and can guarantee energy efficiency of its path with high probability if it finds one in random 3D networks. To ensure that our routing method can find paths with high probability, we also study its asymptotic critical transmission radius (CTR) in random 3D networks. We prove that for a 3D network formed by nodes, that are generated by a Poisson point process of density n over a convex compact region of unit volume, the CTR for our proposed 3D routing is asymptotic almost sure (a.a.s.) at most ln n 2 for any β > β0 and at least 3 3β4πn for any β < β0 . Here, β0 = 1−cosα where α is an parameter used by our routing method. The rest of the paper is organized as follows. Section 2 presents our network model and a formal definition of CTR. Section 3 gives our proposed 3D localized routing protocol and derived bounds on its CTR. Section 4 shows our simulation results and Section 5 summarizes the paper. 3

3β ln n 4πn

2 Preliminaries Network Model and Assumptions: We consider a set V of n wireless devices (called nodes hereafter) uniformly distributed in a compact and convex 3D region D with unitvolume in R3 . By proper scaling, we assume the nodes are represented by a Poisson point process Pn of density n over a unit-volume cube D. Each node knows its position information and has a uniform transmission radius r (or rn ). Then the communication network is modeled by a unit disk graph G(V, r), where two nodes u and v are connected if and only if their Euclidean distance is at most r. Hereafter, we use u − v to denote the Euclidean distance between u and v. For a link uv ∈ G(V, r), we use uv

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to denote its length. We further assume that the energy needed to support the transmission of a unit amount of data over a link uv is e(uv), where e(x) is a non-decreasing function on x. |A| is shorthand for the volume of a measurable set A ⊂ R3 . An event is said to be asymptotic almost sure if it occurs with a probability converges to one as n → ∞. To avoid trivialities, we assume n to be sufficiently large if necessary. Critical Transmission Radius for Greedy-based Routing: In any greedy-based routing, the packet may be dropped by an intermediate node when it could not find any of its neighbors that is “better” than itself. One way to ensure that the routing is successful for every source-destination pair is during the topology control phase each node is set with a sufficiently large transmission radius such that every intermediate node will always find a better neighbor. Critical transmission radius for routing algorithm is first studied by [15]. A routing method A is successful over a network G if the routing method A can find a path for any pair of source and destination nodes. Then we can define the critical transmission radius (CTR) of A as follows: Definition 1. Given a routing method A and a set of wireless nodes V , the critical transmission radius, denoted as ρA (V ), for successful routing of A over V is the minimum transmission radius r such that method A over the network G(V, r) is successful. The subscript A will be omitted from ρA (V ) if it is clear from the context. In [15], Wan et al. derived the CTR of 2D greedy routing in random networks. Recently, Wang et al. [8] also derived the CTR of 3D greedy routing in random 3D networks. In this paper, we will use similar techniques from [15, 8] to study the CTR of our 3D routing.

3 Energy-Efficient Restricted 3D Greedy Routing In this section, we present details of our energy-efficient localized 3D routing method and theoretical analysis on its critical transmission radius in 3D random networks. 3.1 Routing Method Our energy-efficient localized 3D routing method is a variation of classical 3D greedy routing and an extension of a localized routing method [3] we designed for 2D networks. In 3D greedy routing, current node u selects its next hop neighbor based purely on its distance to the destination, i.e., it sends the packet to its neighbor who is closest to the destination. However, such choice might not be the most energy-efficient link locally, and the overall route might not be globally energy-efficient too. Therefore, our routing method use two concepts energy mileage and restricted region to refine the choices of forwarding nodes in 3D greedy routing. Energy Mileage: Given a energy model e(x), energy mileage is the ratio between x . the transmission distance and the energy consumption of such transmission, i.e., e(x) r0 x Let r0 be the value such that e(r0 ) = maxx e(x) . We call r0 as the maximum energy x mileage distance1 under energy model e(x). We assume that the energy mileage e(x) is an increasing function when x < r0 and a decreasing function when x > r0 . 1

Here, we assume that d( e(x) )/dx is monotone increasing, thus, r0 is unique. x

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This assumption is true for most of commonly used energy models. For example, if 2 e(uv) = uv + c is the energy √ used by sending message from u to v, the maximum energy mileage distance r0 = c. Our 3D localized routing greedily selects the neighbor who can maximize the energy mileage as the forwarding node. Restricted Region: Instead of selecting the forwarding node from all neighbors of current node u (a unit ball in 3D as shown in Figure 1(a)), our 3D routing method prefers the forwarding node v inside a smaller restricted region. The region is defined inside a 3D cone with an angle parameter α < π/3, such that angle ∠vut ≤ α, as shown in Figure 1(b). The use of α (restricting the forwarding direction) is to bound the total distance of the routing path. Then the restricted region is a region inside this 3D cone and near the maximum energy mileage distance r0 , such that every node v inside this area satisfies η1 r0 ≤ uv ≤ η2 r0 , as shown in Figure 1(b). Here, η1 and η2 are two constant parameters. This can help us to prove the energy-efficiency of the route. Notice that both these ideas are not completely new. Restricted region with an angle has been used in some localized routing methods, such as nearest/farthest neighbor routing [16], while concepts similar to energy mileage have been used in some energyaware localized routing methods [12,14]. However, combining both of these techniques to guarantee energy efficiency is first done in our previous work [3] for 2D networks. In this paper, we further adapt them into 3D routing. Our energy-efficient localized 3D routing protocol is given in Algorithm 1. There are four parameters used by our method. Three adjustable parameters 0 < α < π3 and η1 < 1 < η2 define the restricted region, while r0 is the best energy mileage distance based on the energy model. For example, the following setting √ of these parameters can be used for energy model e(x) = x2 + c: α = π4 , r0 = c, η1 = 1/2 and η2 = 2. Hereafter, we denote the routing algorithm, energy-efficient restricted greedy, as ERGrd if no greedy routing (Grd) is used when no node v satisfies that ∠vut ≤ α. If Grd is applied afterward, then the routing protocol is denoted ERGrd+Grd. Notice that if Grd fails to find a forwarding node, randomized scheme [2] could also be applied.

r

t

u

r0 u

t

α

v

(a) all possible forwarding neighbors (b) best energy mileage forwarding v u

α

t

u

α

t

v

(c) greedy forwarding inside 3D cone

(d) classical greedy forwarding

Fig. 1. Illustrations of our 3D routing: (a) energy-efficient forwarding in the restricted region, (b) greedy forwarding in the 3D cone, (c) greedy forwarding when the 3D cone is empty

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Algorithm 1. Energy-Efficient Restricted 3D Greedy Routing 1: while node u receives a packet with destination t do 2: if t is a neighbor of u then 3: Node u forwards the packet to t directly. 4: else if there are neighbors inside the restricted region and r0 < r then uv is 5: Node u forwards the packet to the neighbor v such that its energy mileage e(uv) maximum among all neighbors w inside the restricted region, as shown in Figure 1(b). 6: else if there are neighbors inside the 3D cone then 7: Node u finds the node v inside the 3D cone (Figure 1(c)) with the minimum t − v. 8: else 9: Greedy routing (Figure 1(d)) is applied, or the packet is simply dropped. 10: end if 11: end while

The path efficiency of ERGrd is given by the following two theorems. The detail proofs of these two theorems are exactly the same with the proofs of Theorems 1-3 in [3] for 2D network, thus are ignored here. Theorem 1. When ERGrd routing indeed finds a path PERGrd (s, t) from the source s to the target t, the total Euclidean length of the found path is at most δt − s where δ = 1−2 1sin α , thus, a constant factor of the optimum. 2

Theorem 2. When ERGrd routing indeed finds a path PERGrd (s, t) from the source s to the target t, the total energy consumption of the found path is within a constant factor σ of the optimum. When r0 ≥ r, σ depends on α; otherwise, depends on η1 , η2 and α. 3.2 Critical Transmission Radius of 3D ERGrd Routing Notice that ERGrd routing may fail, as all other greedy-based methods do, when an intermediate node cannot find a better neighbor to forward the packet. We now study the critical transmission radius for ERGrd routing in random wireless networks. Given a set of nodes V distributed in a region D, the critical transmission radius ρ(V ) for successful routing by 3D ERGrd is max

min

u,v w: ∠wuv≤α

w − u.

(1)

By setting the r = ρ(V ), ERGrd can always find a forwarding node inside the 3D cone region, thus can guarantee its packet delivery. In this section, we prove a similar result as in [15, 3, 8] for our restricted 3D greedy routing method, 3D ERGrd. 2 Theorem 3. Let β0 = 1−cosα and n 43 πrn3 = β ln n for some β > 0. Then, 1. If β > β0 , then ρ (Pn ) ≤ rn is a.a.s.. 2. If β < β0 , then ρ (Pn ) > rn is a.a.s.. 4π/3 2 = 1−cosα is the ratio between the volume of a unit ball Here, β0 = 2π(1−cosα)/3 and the volume of a 3D cone (the forwarding region) inside the ball. Next, we present the detailed proofs for two parts of this theorem. To simplify the argument, we ignore boundary effects by assuming that there are nodes outside D with the same distribution. So, if necessary, packets can be routed through those nodes outside D.

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Upper Bound of Theorem 3: To prove the upper bound, it is sufficient to show that when each node has a transmission radius rn satisfying the above condition and β > β0 , for every pair of nodes u and v there is always a node w ∈ Pn such that ∠wuv ≤ α and w − u ≤ rn , i.e., any intermediate node u can find a “better” neighbor w towards the destination node v. Given a point distribution Pn , let S(Pn , rn ) be the minimum number of such neighboring nodes w that can be chosen by any intermediate node u for any possible destination v. As proved in [15], it suffices to prove that the cardinality |S(Pn , r)| > 0. Instead, we now prove a stronger result shown in the following lemma. Lemma 1. Suppose that n 43 πrn3 = β ln n for β > β0 . Then for any constant some β1 β1 ∈ (β0 , β), it is a.a.s. that S (Pn , rn ) > L β0 ln n. Here L(x) = xφ−1 (1/x) for x > 0 and φ(x) = 1 + x ln x − x for x > 0.

Proof. Given a node u, the region that node u can choose its neighbor to forward data is a 3D cone with angle 2α, as shown in Figure 1(c). Let Y denote this 3D cone and d denote its diameter (i.e., the largest distance between any two points inside √ it). Clearly d = rn when α ≤ π6 , and d = 2 sin α · rn when π6 ≤ α < π3 . Thus, d < 3rn . Assume that the space is partitioned into 3D grids (equal-size cubes) of side length η, which we call it η-tessellation of space. Here, we consider an εd-tessellation, where 4 ε = 27β (1 − ββ2 ). A polycube is defined as the set of cubes that intersect with a 0 convex and compact region, e.g., Y. Notice that when the grid-partition shifts, we will have different polycubes for the fixed region Y. A polycube in a η-tessellation is said to have a span s if it can be contained in a cube of side-length s · η. We are only interested in polycube that has span at most 1ε and volume at least a certain fraction of 4π 3 3 rn . Assume that, given Y, there are In different such polycubes that are completely β2 ln n 3 contained inside, with span at most 1ε and volume at least ββ02β 4π 3 rn = ( β0 ) n . For ith such polycubes, let Xi denote the number of nodes of V contained inside. Then Xi is a Poisson RV with rate at least ββ20 ln n. Since the number of cubes in D is 1 3 O εd = O lnnn , by Lemma 3 of [8], In = O lnnn . By Lemma 6 of [15], it is In mini=1 Xi β2 β1 a.a.s. that ≥ L > L . ln n β0 β0 n To prove the lemma, it is sufficient to show that S(Pn , r) ≥ minIi=1 Xi . In other words, we only need to show that for any Y, it contains a polycube P has span at r3 . For a 3D cone Y, let P denote the polycube inmost 1ε and volume at least ββ02β 4π 3 n duced by Y−√3εd . Here Y−x denotes the region of Y whose points are of distance at x from the boundary of Y. Then, P ⊆ Y, and the span of P is at most least √ d−2 3εd 4 + 1 < 1ε . By Lemma 2 in [8] and the fact that |Y| = 43 πrn3 β10 > 9√ πd3 β10 , εd 3 √ √ 0 we have |P | ≥ Y−√3εd ≥ |Y| − πd2 3εd = |Y| − 3επd3 > |Y| − 27β 4 ε |Y| = β2 β2 4 3 1 0 |Y| 1 − 27β 4 ε = β |Y| = β0 3 πrn β . This completes the proof of the lemma. ln n Lower Bound of Theorem 3: We now show that, if rn = 3 3β4πn for any β < β0 , a.a.s., there are two nodes u and v such that we cannot find a node w for forwarding by node u, i.e., there does not exist node w inside the 3D cone. Again we partition the space using equal-size cubes (called cells) with side-length ηrn for a constant 0 < η to

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(1+δ)rn

δrn+ rn

u u

w rn

v

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(a) selection of u

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Fig. 2. Illustrations of the proof of lower bound: (a) a cubic cell and the region where we select a node u; (b) the event that node u cannot find a forwarding node w to reach a node v

be specified later. Thus the number of cells, denoted by In here, that are fully contained inside the compact and convex region D with unit volume, is Θ( η31r3 ) = Θ( lnnn ). Let n Eu,v denote the event that no forwarding node w (in the 3D cone) exists for node u to reach node v. Then to prove our claim, it is equivalent to prove that the probability of at least one of the event Eu,v happens a.a.s., i.e., 1 − Pr(none of event Eu,v happens). Since the events Eu,v are not independent for all pairs u and v, we will only consider a special subset of events that are independent. Consider any cell produced by the 3D grid partition that are contained inside D. For each cell, we draw a shaded cube with side-length (η − 2(1 + δ))rn and it is of distance (1 + δ)r to the boundary of the cell, as shown in Figure 2 (a). We only consider the case when node u is located in this shaded cube. We also restrict the node v to satisfy that rn < u − v ≤ (1 + δ)rn , i.e., in the torus region in Figure 2 (b). Clearly, node v will also be inside this cell, and the shaded 3D cone where the possible forwarding node could locate is also inside this cell. Thus, events Eu1 ,v1 and Eu2 ,v2 are independent if u1 and u2 are selected as above from different cells. For each cell i, we compute the probability that event Eui ,vi happens, where ui is selected from the shaded cube of cell i and vi is selected such that rn < vi − ui ≤ (1 + δ)rn . Recall that for any region A, the probability that it is empty of any nodes is e−n|A| . 3 3 Clearly, the probability that node ui exists is 1 − e−n(η−2−2δ) rn since the shared cube 3 3 4 has volume (η −2−2δ)3 rn3 ; the probability that node vi exists is 1−e−n 3 π((1+δ) −1)rn 4 since the torus has volume 3 π((1+δ)3 −1)rn3 . Given node ui and vi , the probability that 3 2 event Eui ,vi happens is e−n 3 π(1−cosα)rn = e−β/β0 ln n = n−β/β0 . Consequently, event 3 3 Eu,v happens for some node pairs ui and vi is Pr(Eui ,vi ) ≥ (1 − e−n(η−2−2δ) rn )(1 − 3 3 3 3 4 e−n 3 π((1+δ) −1)rn )n−β/β0 = (1 − n−β(η−2−2δ) 3/4π )(1 − n−β((1+δ) −1) )n−β/β0 . Thus, the probability that ERGrd routing fails to find a path for some source/destination pairs is Pr(at least one of events Eu,v happens) ≥ Pr(at least one of Eui ,vi happens)= 1 − Pr(none of Eui ,vi happens) = 1 − (1 − Pr(Eui ,vi ))In = 1 − eIn ·ln (1−Pr(Eui ,vi )) ≥ 3 1 − e−In ·Pr(Eui ,vi ) . Notice that In · Pr(Eui ,vi ) = Θ( lnnn )(1 − n−β(η−2−2δ) 3/4π )(1 − 3

1−β/β0

n−β((1+δ) −1) )n−β/β0 n ln n , which goes to ∞ as n → ∞ when β < β0 , η − 2 − 2δ > 0, and δ > 0. This can be easily satisfied, e.g., δ = 1, η = 5. Thus, limn→∞ 1 − e−In ·Pr(Eui ,vi ) = 1. This completes the proof.

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4 Simulation In this section, we study performance of our ERGrd routing in random 3D networks via extensive simulation. Critical Transmission Radius of ERGrd Routing: To confirm our theoretical analysis, we conduct simulations to study the real values of CTR of ERGrd routing method in random 3D networks. We randomly generate 1000 networks with n nodes in a 100 × 100 × 100 cubic region, where n is from 50 to 500. For each network V , we compute the CTR ρ(V ) of ERGrd by using Equation (1). Figure 3 show the probability distribution function of ρ(V ) when α = π/6 and α = π/4. It is clear that the CTR satisfies a transition phenomena, i.e., there is a radius r0 such that ERGrd can successfully deliver all packets when r > r0 and can not deliver some packets when r < r0 . Notice that the transition becomes faster when the number of nodes increases. This confirms our theoretical analysis on the existence of CTR. In addition, from the figures, we can find that larger node density always leads to smaller value of CTR. The practical value of ρ(V ) is larger than the theoretical bound in our analysis, since the theoretical bound is standing for n → ∞. Compared the two cases with α = π/6 and π/4, larger CTR is required if smaller restricted region (i.e. smaller α) is applied. Probability of r>ρ(V)

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Fig. 3. PDF curves of 3D ERGrd routing when α = π/6 (30◦ ) and α = π/4 (45◦ )

Network Performance of ERGrd Routing: We implement the classic 3D greedy routing (Grd) and variations of our proposed restricted greedy routing (specifically, RGrd-30, ERGrd-45, ERGrd+Grd-30, and ERGrd+Grd-45, where 30 or 45 is the degree value of α.) in our simulator. We assume that the energy consumption of a link uv is 2 e(uv) = uv + c, where c = r2 /4. The values of η1 and η2 are 1/2 and 2. By setting various transmission radii, we generate 100 connected random networks with 100 wireless nodes again in a 100 × 100 × 100 cubic region. We randomly select 100 source-destination pairs for each network and test five greedy-based 3D routing. All results presented hereafter are average values over all routes and networks. Figure 4 illustrates the average delivery ratios of the five routing methods. Clearly, the delivery ratio increases when r increases. After r is larger than a certain value, it

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Fig. 4. Average delivery ratios of 3D ERGrd routing methods in random networks

always guarantees the delivery. Notice that ERGrd methods without Grd backup have lower delivery ratio under the same circumstance, since they have smaller region to select the next hop node. With Grd backup, the delivery ratios of ERGrd+Grd methods are almost the same with those of Grd (simple 3D greedy). Figure 5(a) and (b) illustrate the average length stretch factors and energy stretch factors of all routing methods, respectively. Here, the length/energy stretch factor of a path from node s to node t is the ratio between the total length/energy of this path and the total length/energy of the optimal path connecting s and t. Smaller stretch factor of a routing method shows better path efficiency. For the length stretch factor, the ERGrd with α = π/6 has the best length efficiency. It is surprising that with α = π/4 the length of ERGrd path could be longer than simple greedy. However, when considering the energy efficiency, all ERGrd methods can achieve better path efficiency than simple greedy method. Notice that smaller restricted region leads to better path efficiency, however it also has lower delivery ratio. Therefore, it is a trade-off between path efficiency and packet delivery. It is also clear that when the network is dense (with large transmission radius), ERGrd and ERGrd+Grd are almost the same, since ERGrd can always find nodes inside the 3D cone. 1.2 Grd ERGrd−30 ERGrd−45 ERGrd+Grd−30 ERGrd+Grd−45

1.08 1.07 1.06 1.05 1.04 1.03 1.02 1.01 1

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5 Conclusion In this paper, we propose a simple localized routing protocol ERGrd for 3D networks. The proposed method achieves the energy efficiency by limiting its choice inside a restricted region and picking the node with best energy mileage. We then theoretically derive the critical transmission radius of our proposed 3D routing for random 3D networks. This provides the insight about how large the transmission radius should be set for our method to guarantee the delivery of packets between any two nodes. We also conduct extensive simulations to confirm our theoretical results.

References 1. Durocher, S., Kirkpatrick, D., Narayanan, L.: On routing with guaranteed delivery in three-dimensional ad hoc wireless networks. In: Rao, S., Chatterjee, M., Jayanti, P., Murthy, C.S.R., Saha, S.K. (eds.) ICDCN 2008. LNCS, vol. 4904, pp. 546–557. Springer, Heidelberg (2008) 2. Flury, R., Wattenhofer, R.: Randomized 3D geographic routing. In: IEEE Infocom (2008) 3. Wang, Y., Song, W.-Z., Wang, W., Li, X.-Y., Dahlberg, T.: LEARN: localized energy aware restricted neighborhood routing for ad hoc networks. In: Proc. of IEEE SECON (2006) 4. Alam, S.M.N., Haas, Z.J.: Coverage and connectivity in three-dimensional networks. In: Proc. of ACM Mobicom (2006) 5. Kao, G., Fevens, T., Opatrny, J.: Position-based routing on 3D geometric graphs in mobile ad hoc networks. In: Proc. of CCCG 2005 (2005) 6. Pompili, D., Melodia, T.: Three-dimensional routing in underwater acoustic sensor networks. In: Proceedings of ACM PE-WASUN 2005, Montreal, Canada (October 2005) 7. Ravelomanana, V.: Extremal properties of three-dimensional sensor networks with applications. IEEE Transactions on Mobile Computing 3(3), 246–257 (2004) 8. Wang, Y., Yi, C.-W., Li, F.: Delivery guarantee of greedy routing in three dimensional wireless networks. In: Li, Y., Huynh, D.T., Das, S.K., Du, D.-Z. (eds.) WASA 2008. LNCS, vol. 5258, pp. 4–16. Springer, Heidelberg (2008) 9. Akyildiz, I.F., Pompili, D., Melodia, T.: Underwater acoustic sensor networks: research challenges. Ad Hoc Networks 3(3), 257–279 (2005) 10. Xie, P., Cui, J.-H., Lao, L.: VBF: Vector-based forwarding protocol for underwater sensor networks. In: Boavida, F., Plagemann, T., Stiller, B., Westphal, C., Monteiro, E. (eds.) NETWORKING 2006. LNCS, vol. 3976, pp. 1216–1221. Springer, Heidelberg (2006) 11. Bose, P., Morin, P., Stojmenovic, I., Urrutia, J.: Routing with guaranteed delivery in ad hoc wireless networks. ACM/Kluwer Wireless Networks 7(6) (2001) 12. Li, C.-P., Hsu, W.-J., Krishnamachari, B., Helmy, A.: A local metric for geographic routing with power control in wireless networks. In: Proc. of IEEE SECON (2005) 13. Melodia, T., Pompili, D., Akyildiz, I.F.: Optimal local topology knowledge for energy efficient geographical routing in sensor networks. In: Proc. of IEEE INFOCOM (2004) 14. Seada, K., Zuniga, M., Helmy, A., Krishnamachari, B.: Energy-efficient forwarding strategies for geographic routing in lossy wireless sensor networks. In: Proc. ACM Sensys (2004) 15. Wan, P.-J., Yi, C.-W., Yao, F., Jia, X.: Asymptotic critical transmission radius for greedy forward routing in wireless ad hoc networks. In: Proc. of ACM Mobihoc (2006) 16. Li, X.-Y., Calinescu, G., Wan, P.-J., Wang, Y.: Localized Delaunay triangulation with applications in wireless ad hoc networks. IEEE Trans. on Parallel and Distributed Systems 14(10), 1035–1047 (2003)

Adaptive Energy and Location Aware Routing in Wireless Sensor Network Hong Fu1,*, Xiaoming Wang1, and Yingshu Li2 1

Department of Computer Science, Shaanxi Normal University, Xi’an, China, 710062 [email protected], [email protected] 2 Department of Computer Science, Georgia State University, Atlanta, USA, 30303 [email protected]

Abstract. This paper proposes a novel routing algorithm called AELAR for wireless sensor networks to meet QoS requirements: energy conservation and data delay reduction. Firstly, we propose a novel method of dividing routing request zone and construct a select equation which can enlarge the energy awareness as network time goes on. We make the routing request zone and factors in select equation changed automatically. Simulation results show that AELAR outperforms traditional algorithms in the performance of network lifetime, utilization and consumption balancing of energy and data delay. Keywords: Wireless sensor networks, Energy efficient routing, Distance level.

1 Introduction Sensor nodes in Wireless sensor networks (WSNs) have resource constraints which include limited energy, processing capacity, storage, communication range and bandwidth [1]. These unique features have raised special problems that must be solved while designing routing protocols. For instance, in some applications, routing algorithms must meet QoS requirements such as energy conservation and data delay reduction. Some routing algorithms (e.g., [2], [3], [4]) limit the routing search to request zones to reduce the cost of flooding. Geographical routings (e.g., [5], [6]) use greedy forwarding mechanisms to forward packets. GPSR [7] is such an efficient, classic algorithm. But GPSR is based on a single metric. GEAR [8] uses energy aware metric, together with geographical information, to make routing decisions. However, the energy balancing strategy in GEAR increases the average path length. With the progress on the semiconductor technology, wireless sensors’ capabilities of computation, storage may not be limitations in future. However, how to consume energy efficiently is still one of the most challenging problems in WSNs’ researches [9]. The main goals of the research presented in this paper are to: *

This work was supported by NSFC (60773224, 60970054), the Key project of Chinese Ministry of Education under Grant (107106) and the Scientific Research Foundation of State Education Ministry for the Returned Overseas Chinese Scholars.

G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 105–109, 2010. © Springer-Verlag Berlin Heidelberg 2010

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Prolong the network lifetime of WSN to deliver more packets; Balance the energy consumption of nodes and achieve energy conservation; Find the routing path as short as possible to reduce data delay.

The rest of the paper is organized as follows: Section 2 describes the network system. Section 3 presents our proposed algorithm. Section 4 demonstrates the effectiveness of AELAR via simulations. Finally, section 5 gives concluding remarks.

2 The Network System Model To simple the question, we assume that all sensors are homogeneous. They have the same ability of communication and the radius of communication is r. They know their neighbors’ and their own location information by location services, e.g., Ad hoc Positioning System with Angle of Arrival [10]. We also assume that there is only one sink in the WSN. Besides, we give the follow definition for our research: Distance level. For a given sensor node ni, let did is the distance from node ni to the sink, the distance level of node ni, dl, is defined as

d l = ⎣d id / m ⎦ .

(1)

Where, m is the distance level dividing standard. It can be determined by special applications. The distance level can indirectly indicate the distance from the sink.

3 Adaptive Energy and Location Aware Routing In this section, we present a novel method of dividing routing request zone and construct a select equation which can be adjusted according to nodes’ distance levels. 3.1 Ellipse-Shape Routing Request Zone and Density Self-adaptation We employ the virtual coordinates [11] to identify the routing request zone. In Fig. 1(a), given the node I, letting O be the origin, the virtual coordinates of a neighboring node (e.g. J), (xj’, yj’), can be calculated by the following equations: ⎧⎪ x j ' = cos(θ ) × ( x j − xi ) + sin(θ ) × ( y j − yi ) ; y − yi . θ = arctan( j ) ⎨ x j − xi ⎪⎩ y j ' = cos(θ ) × ( y j − y i ) − sin(θ ) × ( x j − xi )

(2)

In AELAR, we define the routing request zone as the overlap area of an elliptic region and a disk. In fig.1 (b), the shallow area shows the proposed routing request zones. Letting dij be the distance form node I to node J, the node I can use (3) to learn whether a neighboring node locates in its routing request zone. ( x' − a)2 ( y')2 + ≤1 2 a b2

and

d ij ≤ r .

(3)

From Fig.1 (b), we can find that ellipse parameter a and b (especially, b) have great impact on the size of routing request zone. To simplify the discussion, we assign the

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(a) Virtual coordinate of node J

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(b) Different value of b for different nodes

Fig. 1. Virtual coordinate and ellipse-shape routing request zone of a node

value of a as transmission radius r. However, we must take seriously the value of b because it has the most important effect on the size of routing request zone. According to geometric knowledge, the value of y’ of the neighboring nodes which are closer from the sink than the node I is limited by the inequality: y'≤

(4)

2 rd id − r 2 .

Where, did is the distance from the node I to the sink. We hope that the nodes which have long distance from the sink can forward data packets as soon as possible and the nodes near the sink have more options to choose the next hop, so that the nodes near the sink can better balance the energy consumption. To make the node I can adjust the value of b according to its location, we make node I determine the value of b as: (5)

b = δ 2 r d id − r 2 .

Where, δ is determined by the distance level and the transmission radius of each node. Only the neighboring nodes that locate in the routing request zone are considered to be next-hop candidates in AELAR. 3.2 The Select Equation

To select an appropriate node as the next hop for forwarding data packets, AELAR strikes a balance between residual energy and distance metrics. We use the natural logarithm function for residual energy level to enlarge the residual energy difference among neighboring nodes. Letting er be the residual energy of a neighboring node J and ei be the initial energy of node I. Then, the node J’s select factor fj is defined as

{

f j = ln (er / ei ) (did / d jd ) α

β

}.

(6)

In above select equation, α and β are the weight factors of energy consumption ratio and distance ratio. In AELAR, we make the nodes adjust their weight factors automatically according to their distance levels. In WSNs, the nodes closer to the sink usually carry out more forward tasks. We hope these nodes consider more distance

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factor so that the total energy consumption is lower. AELAR selects the neighboring node that not only locates in the routing request zone but also has the greatest value of fj as the next hop. The energy effect on the above equation will drastically increase if the residual energy of the node is too low. So, nodes with too low residual energy will be difficult to be chosen as the next hop in the route discovery, which is helpful to balance the energy expenditure.

4 Simulation and Analysis We investigated the AELAR’s performance by comparing it with traditional GPSR and energy-aware GEAR. We also optimized our algorithm (AELAR-O) with the method in [12] to confirm that the paths discovered by AELAR are almost as short as possible. Fig.2(a) shows that number of packets delivered by AELAR is more than GPSR and GEAR during the network lifetime. From Fig.2(b) and Fig.2(c), we can learn that AELAR successfully achieves energy efficiency and has lower energy deviation because the nodes’ energy differences is exaggerated by logarithmic function. Fruthermore, owing to the use of the adaptive mechanism, we learn that AELAR almost has the same average path hop as GPSR from Fig.2(d), which is beneficial to the data delay reduction while considering energy conservation.

(a) Number of packets delivered to the sink

(c) The energy standard deviation

(b) Energy utilization radio of WSN

(d) Average hops of searched paths

Fig. 2. Simulation results for evaluating the performance of AELAR

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5 Conclusions This paper has proposed a new routing algorithm, AELAR, which is location-based, energy-aware, and scalable to discover the paths between the source node and the sink. We first present a novel method of dividing routing request zone and then construct a select equation for a node to choose its next hop. Both the routing request zone and the select equation can be adjusted by the node according to its own location. This adaptive mechanism makes sure that the paths found by AELAR are as short as possible while considering the energy conservation. The simulation results show that AELAR performs better in terms of network lifetime, energy utilization, energy consumption balancing and data delay reduction than traditional routing algorithms.

References 1. Yicka, J., Mukherjeea, B., Ghosal, D.: Wireless sensor network survey. Computer Networks 52(12), 2292–2330 (2008) 2. Woo, K., Yu, C., Lee, D., Youn, H.Y., Ben, L.: Non-blocking, localized routing algorithm for balanced energy consumption in mobile ad hoc networks. In: Proceedings of Ninth International Symposium on Modeling, Analysis and Simulation of Computer and Telecommunication Systems, Cincinnati, OH, USA, pp. 117–124 (2001) 3. Ko, Y.B., Vaidya, N.H.: Location-aided routing in mobile ad hoc networks. Wireless Networks 6(4), 307–321 (2000) 4. Shih, T.F., Yen, H.C.: Location-aware routing protocol with dynamic adaptation of request zone for mobile ad hoc networks. Wireless Networks 14(3), 321–333 (2008) 5. Chen, M., Leung, V.C.M., Mao, S., Xiao, Y., Chlamtac, I.: Hybrid geographic routing for flexible energy-delay tradeoff. IEEE Transactions on Vehicular Technology 58(9), 4976–4988 (2009) 6. Kranakis, E., Singh, H., Urrutia, J.: Compass routing on geometric networks. In: Proceedings 11th CCCG Vancouver, BC, Canada, August 1999, pp. 51–54 (1999) 7. Karp, B., Kung, H.T.: GPSR: greedy perimeter stateless routing for wireless networks. In: Proceedings of ACM Annual Conference on Mobile Computing and Networking, Boston, MA, USA, pp. 243–254 (August 2000) 8. Yu, Y., Govindan, R., Estrin, D.: Geographical and energy aware routing: a recursive data dissemination protocol for wireless sensor networks. In: UCLA Computer Science Department Technical Report, pp. 1–23 (2001) 9. Li, Y., Ai, C., Cai, Z., Raheem, B.: Sensor Scheduling for p-Percent Coverage in Wireless Sensor Networks. Journal of Cluster Computing (May 2009) 10. Kieb, W., Fubler, H., Widmer, J.: Hierarchical location service for mobile ad hoc networks. Mobile Computing and Communications Review 8(4), 47–58 (2003) 11. Chen, M.L., Victor, C.M., Mao, S., Yuan, Y.: Directional geographical routing for real-time video communications in wireless sensor networks. Computer Communications Concurrent Multipath Transport 30(17), 3368–3383 (2007) 12. Shu, L., Zhang, Y., Yang, L.T., Wang, Y., Hauswirth, M.: Geographic routing in wireless multimedia sensor networks. In: Proceedings of the 2nd International Conference on Future Generation Communication and Networking (FGCN 2008), Hainan, China (December 2008)

Utilizing Temporal Highway for Data Collection in Asynchronous Duty-Cycling Sensor Networks Tao Chen, Deke Guo, Honghui Chen, and Xueshan Luo Key Lab of C4 ISR, National University of Defense Technology, 410073 Changsha, China

Abstract. Duty cycling technique is widely used in sensor networks to save energy. Due to coverage and other requirements, sensor nodes may have asynchronous duty schedules, which bring additional challenges for data collection. Existing approaches suﬀer from diﬀerent limitations such as requirement of time synchronization. This paper proposed a crosslayer design, which utilizes multihop information to ﬁnd shortcuts and reduces the delay in data collection. Keywords: sensor networks, data collection, duty cycle.

1

Introduction

Wireless sensor network (WSN) is a key enabling issue for many applications, such as environment monitoring, battleﬁeld surveillance and so on. While sensor nodes are expected to operate for a long periods of time in these applications, they are usually deployed in unattended area and equipped with limited battery. Idle listening is considered to be one of the most signiﬁcant sources of energy consumption in sensor networks, which reduces the lifetime of network. Duty cycling mechanisms are proposed to reduce such energy consumption. Each node turns its radio oﬀ (sleep state) for most of the time, and turns the radio on (active state) periodically for a short period of time. When node is active, it is able to transmit or receive packets. But almost all the proposed duty-cycling protocols suﬀer from the shortcomings of synchronization protocols. To address such challenge, we propose a cross-layer design for data collection in asynchronous duty-cycling sensor networks called ”T-Highway”. Our design uses a hybrid method of two strategies in routing. The basic strategy is hop based shortest path tree (SPT) routing. The other strategy tries to utilize a sequence of active nodes which constitute a pipeline. We call the pipeline ”highway” because packet could be forwarded through it immediately without waiting for next hop to wake up. The highway is temporal, because it disappears when some node in the sequence go to sleep. Node ﬁrst tries to use the second strategy to route data packet. If no highway is available, node switches to the ﬁrst strategy and store the packet in local buﬀer, waits for the next hop to wake up. We use Fig. 1 to illustrate the beneﬁt of highway. Grey nodes are active, while the white ones are asleep. Solid lines denote links in shortest path tree. When source node generates a packet, it G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 110–114, 2010. c Springer-Verlag Berlin Heidelberg 2010

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Fig. 1. An example of highway

chooses node B as receiver instead of node A, because the packet could soon be passed through the highway marked with solid arrows to node D which is much nearer to sink. The rest of this paper is organized as follows. We brieﬂy survey related works in Section 2. The design of T-Highway is presented in Section 3. Section 4 evaluates the performance of T-Highway through simulations. Section 5 concludes this work.

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Related Works

Duty cycling is a widely used technique to reduce energy wastage in wireless sensor networks. The existing literatures about duty cycling could be grouped into two categories: well-designed-schedule based and independent schedule based. The well-designed schedule based approaches assign a duty schedule to each node according to the given duty ratio and global or regional topology information. Duty schedules of the whole network form a certain pattern. For example, in GreenOrbs [1], all nodes wake up and go to sleep simultaneously. Some more general MAC protocols with similar approach are also proposed, such as S-MAC [2] and RMAC [6]. Those protocols synchronize neighboring nodes to align their active or sleeping period, which separate the network into several components where all nodes in the same component share the same schedule. Cao et al. proposed a protocol for data collection [3], which synchronizes duty cycles of nodes into a streamlined sequence to pipe the data eﬃciently. Multi-parent scheme [7] is a similar approach, which assign multiple parents with diﬀerent wakeup schedules to each node. All aforementioned approaches need global time synchronization, which introduces extra overhead. Moreover, those approaches suffer from clock drift, which are common problems in working system deployment. The independent schedule based approaches allow each node operate on its own duty schedule, and try to improve the eﬃciency by other techniques. Dynamic programming is used in DSF [4] to achieve high performance with the consideration of both low-duty-cycle and unreliable links. But they still suﬀer from the overhead and clock drift problem of local time synchronization. A typical example of time-synchronization-free approaches is B-MAC [5] which employing low power listening. However, preamble transmission may occupy the medium for a

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relative long time, which could prevent all neighboring nodes from transmitting and add to the delay of data delivery. RI-MAC [8] is proposed to minimize the medium occupation while still decoupling duty schedules of nodes as B-MAC does. It applies the idea of receiver-initiated transmission to duty cycle MAC protocols. But no data collection protocol based on RI-MAC has been proposed.

3 3.1

T-Highway Design Overview

The workﬂow of T-Highway consists of four components as illustrated in Fig. 2. In node initialization, each node generates a ﬁxed duty schedule, which makes the node periodically change the mode between sleep and active. The schedule can also be assigned after deployment by using dissemination protocols [9]. In network initialization, all nodes keep awake and try to ﬁnd all the neighbors. Then the sink node initializes a ﬂooding for building shortest path tree. Node could change to use the assigned duty schedule after it overhears the broadcast of all its neighbors. Highway maintenance is the key component which is used to ﬁnd and maintain highways. We will explain the detail in Subsection 3.2. Routing component is used to choose the appropriate next hop. Node may switch between the SPT and highway based routing.

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Beacon message is used for highway maintenance. It can be denoted as a threetuple Beacon =< Src, d, l >. Src is the address of the source node which broadcasts the beacon. d is the hop distance between the endpoint of highway and the sink. It is used for nodes who overhear the beacon to make routing decisions or maintain highways. l is the estimated lifetime of the aforementioned highway. As the lifetime passes, the highway expired and is no longer used for and routing. Since no synchronization and global knowledge is available, node cannot predict the wakeup of neighbors. So it is impossible to select an optimal next hop with least delay to sink. In T-Highway, a receiver initialized MAC such as RIMAC is used to help gathering information, which in turns to select a next hop with greedy heuristic intuition. Whenever a node is wake up, it broadcast a beacon message if the medium is not busy; otherwise a backoﬀ mechanism is employed to avoid collisions. Each node maintains a highway list in memory. When node wakes up, it initializes a new highway list, and adds itself as the ﬁrst highway into the list. The lifetime of the ﬁrst highway is the length of duty

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period of the node. Then it builds a beacon message for the ﬁrst highway and broadcast it to its neighbors which are awake. When node receives a beacon, it also extracts the highway information, and scans the local highway list to check whether the new highway could provide further progress or longer lifetime than all the highways with further progress in the list. If the new highway has advantage in either aspect, it is added to the list. Only the ﬁrst node in the new highway sequence need to be stored rather than the whole sequence for space eﬃciency. Highways which are worse in both aspects than any other one are removed from the list, because they will never be used. After the scan process, if new highway is added, node sends a beacon. Node clears the highway list only when it goes to sleep. Considering the temporal character of highway, the lifetime update should be in time to avoid selecting the wrong next hop according to an expired highway. Each time when a new highway is added to the list, a local timer is initialized with the length of the associated lifetime. Those timers are used to update lifetime of highways in the list when receives a beacon, because it could record how long has since the highway was established. When a timer ﬁres, the associated highway expires and is removed from the list. As described above, the lifetime of highway is always less than the working period time(no more than a few seconds in most cases) of a sensor node. Even if clock drift happens, it cannot be accumulated before it can signiﬁcant inﬂuence the performance. So we can say our design is free of the shortcomings of synchronization protocols.

4

Simulation Results

The performance of T-Highway is compared to SPT based ﬁxed routing. A receiver initialized MAC protocol similar to RI-MAC is used in simulations. We use a random topology with 1000 nodes distributed in a 600m × 600m square. In the baseline scenario, the communication range is set to 40m. In each duty cycle, node keeps active for 1 second, then sleeps for 9 seconds(the duty ratio is 10%). Fig. 3(a) shows the end-to-end delay distribution of both approaches in baseline scenario simulation. T-Highway result has more packets whose delay is between 30s and 60s, but fewer packets whose delay is between 60 s and 100s. It means T-Highway could reduce the delay caused by waiting. Fig 3(b)

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shows the CDF curves of delay distribution in a baseline simulation result. Solid line is always above the dashed line, which means T-Highway can achieve less end-to-end delay in data collection than SPT. To further investigate how much T-Highway can gain over SPT with diﬀerent duty ratios, we keep the length of a duty cycle unchanged and change the length of active time so as to change the duty ratio from 1% to 100%, and run the simulations under diﬀerent settings. The results of average reduced delay per packet are plotted in Fig 3(c). It shows that T-Highway overwhelms SPT in a wide range of duty ratio.

5

Conclusion

In this paper, we have presented an approach for data collection in asynchronous duty-cycling wireless sensor networks. It releases the requirement for time synchronization, and reduces the delay for data collection. Simulation results show that the proposed design could reduce the end-to-end delay.

Acknowledgement This work is supported in part by the NSF China under grant No. 60903206, the Military Pre-research Foundation under grant No. 9140A06050610KG0117, 9140A06020709KG01, and Military Pre-research Project 51306040101.

References 1. Mo, L., He, Y., Liu, Y., et al.: Canopy Closure Estimates with GreenOrbs: Sustainable Sensing in the Forest. In: 7th ACM Conference on Embedded Networked Sensor Systems (2009) 2. Ye, W., Heidemann, J.S., Estrin, D.: An Energy-Eﬃcient MAC Protocol for Wireless Sensor Network. In: 21st INFOCOM (2002) 3. Cao, Q., Abdelzaher, T., He, T., Stankovic, J.: Towards optimal sleep scheduling in sensor networks for rare-event detection. In: 4th International Symposium on Information Processing in Sensor Networks (2005) 4. Gu, Y., He, T.: Data Forwarding in Extremely Low Duty-cycle Sensor Networks with Unreliable Communication Links. In: 5th International Conference on Embedded Networked Sensor Systems (2007) 5. Polastre, J., Hill, J., Culler, D.: Versatile Low Power Media Access for Wireless Sensor Networks. In: 2nd International Conference on Embedded Networked Sensor Systems (2004) 6. Du, S., Saha, A.K., Johnson, D.B.: RMAC: A Routing-enhanced Duty-cycle MAC Protocol for Wireless Sensor Networks. In: 26th INFOCOM (2007) 7. Keshavarzian, A., Lee, H., Venkatraman, L.: Wakeup Scheduling in Wireless Sensor Networks. In: 7th ACM International Symposium on Mobile Ad hoc Networking and Computing (2006) 8. Sun, Y., Gurewitz, O., Johnson, D.B.: RI-MAC: A Receiver-Initiated Asynchronous Duty Cycle MAC Protocol for Dynamic Traﬃc Loads in Wireless Sensor Networks. In: 6th ACM Conference on Embedded Networked Sensor Systems (2008) 9. Chen, T., Guo, D., Liu, X., Chen, H., Luo, X., Liu, J.: BDP: A Bloom Filters Based Dissemination Protocol in Wireless Sensor Networks. In: 6th IEEE International Conference on Mobile Ad Hoc and Sensor Systems (2009)

The Impact of Reader to Tag Collision on RFID Tag Identification Yiyang Zhao1, Weijun Hong2, S.C. Cheung1, and Shufang Li2 1

Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Hong Kong, China SAR {zhaoyy,scc}@ust.hk 2 Department of Information and Communiction Engineering, Beijing University of Posts and Telecommunications, Beijing, China {hongwj,lisf}@bupt.edu.cn

Abstract. RFID has been applied in many areas. Since the dense deployment of readers and tags, the collision problem of RFID system becomes a critical issue. In this paper, we introduce a novel probabilistic model to estimate the impact of reader to tag collision based on the information from physical layer. According to the theoretical analysis, our model ensures that more than one tag can be identified by both readers when they communicate with tags. To obtain the bit information, we employ the NI RFID emulator and LabView software. The simulation results show that the probability of miss-identification is lower than 17%. Keywords: RFID, Collision, Probabilistic model, Physical Layer.

1 Introduction With the contactless capabilities, Radio Frequency Identification (RFID) systems were widely used in many areas, from traditional logistics management to internet of things. Tracking and real time localization [1,2] are also important applications of RFID systems and other wireless systems [3], where the dense deployment of readers and tags is required to guarantee the quality of services. Since RFID systems employ the radio frequency as the communication media [4], the collision phenomenon is an inherent feature of those systems. To deal with the collision problem, people proposed a lot of MAC layer protocols. For those RFID systems working on Low Frequency band (LF, 125kHz~134kHz) and High Frequency band (HF, typically 13.56MHz), such as Access Control System (ACS), only one tag in the interrogation area can be identified by reader typically, due to the relative short interrogation range. However, the really attractive properties of RFID technology are long operating ranges, high data rate and high level of identify capability for large number of objects. The RFID systems working on UHF and microwave frequency band provide more powerful capability of identification than previous systems. Therefore the collision problem becomes one of the very important issues of RFID research. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 115–119, 2010. © Springer-Verlag Berlin Heidelberg 2010

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Illustrated as Figure 1, the effective ranges for different operations are discriminative. Normally, the write operation requires more energy. By contrary, the identification range is farther. Since the signal is attenuate within a distance, the interference range is farthest. In this paper, we only consider the identification range and the interference range. The collision problems in RFID systems can be classified as three situations: tag collision problem, reader collision problem and reader to tag collision problem. When a reader tries to identify all tags in its identification range and tags may communicate with the reader in same time slot, the tag collision occurs. Aloha-based approaches and tree-based approaches are proposed to avoid tag collision. As Fig. 1 shown, the interference range is largest. Hence, the reader collision happens when interference ranges of two readers are intersected. To deal with reader to reader collision, the frequency assignment methods are applied. Differing with previous collision, the reader to tag collision means that tags may not reply the readers if they locate in identification zones of two and more readers. Similar with ‘hidden terminal’ in traditional wireless networks, readers involved in the reader to tag collision will not able to recognize tags. Our probability model will help to increase the identification rate. Our major contributions are described as followings. Based on the information of the physical layer, we proposed the probabilistic identification model for multiple tags in reader to tag collision situation, which is more close to the practical scenario. Not only we deliver the theoretical analysis about the probability model, but also our simulation results show that our new model more effective than the traditional deterministic model. We verify our model by the NI emulator. Applying existing protocols, we evaluate the performance of RFID systems. In terms of successful reading rate and system throughput, the performance of the RFID system gets remarkable improvement. The remainder of the paper is organized as follows. In Section 2 we explore the capability of RFID tags to communicate with two readers based on the standard. In Section 3 we introduce the critical multiple tag inventory process and propose our mathematical model. In Section 4, we provide the analysis. Related work will be introduced in Section 5. Finally, we conclude our work in Section 6.

Interference Range Identification Range Read Range Write Range

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2 Communication Capability of an RFID Tag Similar with EPCglobal C1G2 standard, if a tag provides the independent memory called ‘session’ which can store corresponding state during communicating with different readers, more than one reader can identify common tags in a time-interleave way. According to our novel design, tags will provide 4 sessions (denoted S0, S1, S2, and S3) and maintain an independent inventoried flag for each session to indicate the state that it has been identified or not. Each of the four inventoried flags has two values, which are denoted A and B separately. Different from the standard, four independent parts called ‘Reply Slot No.’ are inserted in tag memories. Those parts will record slot numbers of tag replies in corresponding sessions. Shown in Figure 2, at the beginning of each inventory round, a reader chooses to inventory either A or B flag in one of the four sessions. After interpreting the length of inventory round, each tag produces a pseudo number which is the slot it intends to reply, and record it in ‘Replying Slot No.’. Tags participating in an inventory round in one session shall neither use nor modify the inventoried flag and ‘Replying Slot No.’ for a different session. Based on above scheme, we can assume that each tag can store some commands from readers.

3 Probabilistic Model Considering the RFID communication protocols, we will find that there is a chance to use a probabilistic model to describe the reader to tag collision. Based on the different length of commands, the probability for recognizing tags can be obtained. Since there are various scenarios, we will discuss them according to different conditions. Generally, in dense reader scenarios, there would be more than one tag in the intersection area of interference ranges of two readers, as shown in Fig. 3. Two readers, A and B, will communicate to tags in their identification ranges. However, they do not interfere each other due to the limited interference range. Two or more tags, called common tag population, may locate in the intersection area of two readers, like the gray area in Fig. 3. Since reader A and B do not sense each other, they will communicate with two tags in the gray area simultaneously. In traditional way, two tags are refused to identify. By contrary, according to our scheme two tags have chance to be recognized stochastically. In some previous work [5], the single tag within the collision area was considered. However, only the single tag situation was discussed. In the genuine scenario, the model does not work. Hence, our model has the common adaptability. To understand bit level behavior of RFID system, we use NI RFID simulator and LabView software. Fig. 4 illustrates the signal of captured by NI simulator. When the collision happens, the bit information still can help us to do the recovery. Suppose the total number of tags inside the intersection is m, there are four possible outcomes. P1 denotes that the all information from two readers can be recovered. P2 and P3 denote the probability that one of readers established communication with tags. P4 indicates the probability that no useful information was resumed from readers. The equation (1) represents the probabilities. P݅ = ሼpA ሺmሻ = 1 or 0, pB ሺmሻ = 1 or 0ȁm > 1ሽ, ݅ = 1 to 4

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Fig. 3. Reader to Tag collision with tags

Fig. 4. Captured signal by NI emulator

We assume that the total number of bits captured by NI emulator is N, which is the number of tag response slots for each reader. Suppose the length of reader commands is L, the probabilities can be obtained by following equations. P1 =

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According to the model, we can predict the readability of readers in reader to tag collision situation.

4 Analysis As discussion in previous sections, our novel model can ensure that tags in intersection area have particular probability to be identified. In this section, we analysis the effect of our model based on the simulation results. Fig. 5 shows that the possibility that two readers can both identify all the tags inside the intersection area of their interrogation range increases with the increasing of the length of inventory round of readers. However, as the scale of common tag population increase, the probability decreases fast. The probability that both readers miss all tags in common tag population is showed in Fig. 6. It is always lower than 17% even there is only one tag in the intersection area. It decreases quickly to approximate

Fig. 5. The PDF of all identified tags by both readers.

Fig. 6. The PDF of identified tags by one of two readers

Fig. 7. The PDF of unidentified tags

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to zero as the number of tags or the length of inventory round increases. As shown in Fig. 7, the probability that both readers can only identify a part of tags in the common tag population decreases quickly as the length of inventory round of reader increases, on the contrary it increases slowly as the number of tags increases.

5 Related Work A lot of protocols are introduced to solve the tag to tag collision problems, such as tree-based and aloha-based protocols. However, previous protocols only deal with the multiple tags and one reader model. For reader to tag collision with multiple tags, those algorithms do not work well. Colorwave [6] employes a distributed TDMA based method to address the reader collision issues. If there is a collision occurs, the relative readers will choose other time slots to retransmit. Unfortunately, Colorwave only solve the problem in an ideal situation. Researchers also proposed several scheduling-based algorithms, such as Channel Monitoring Algorithm and Enhanced Pulse Protocol Algorithm. Since those algorithms mentioned above focus on the multiple readers and employ the deterministic model, they do not adapt the reader to tag collision scenarios.

6 Conclusion and Future Work We have presented a novel model to deal with the reader to tag collision problem in RFID systems. Based on the information obtained by NI RFID emulator and LabView software, we can employ our probabilistic model to help readers to identify more tags, when the reader to tag collision occurs. The simulation results indicate that our model increases the identification rate. In the future, we plan to setup a test bed to implement our model.

Acknowledgement This research was supported in part by the Hong Kong ITC grant ITP/022/08LP, China NSFC Grants 60933011, Technology Major Project of China under Grant No. 2009ZX03006-001 and National Basic Research Program of China (973 program) under Grant No. 2006CB303000.

References 1. Ni, L.M., Liu, Y., Lau, Y.C., Patil, A.: LANDMARC: Indoor Location Sensing Using Active RFID. In: Proceeding of PerCom (2003)

2. Liu, Y., Chen, L., Pei, J., Chen, Q., Zhao, Y.: Mining Frequent Trajectory Patterns for Activity Monitoring Using Radio Frequency Tag Arrays. In: Proceeding of PerCom (2007)

3. Li, M., Liu, Y.: Rendered path: range-free localization in anisotropic sensor networks with holes. In: Proceeding of MobiCom 2007, Canada (2007)

4. Lu, L., Liu, Y., Hu, L., Han, J., Ni, L.M.: A Dynamic Key-Updating Private Authentication Protocol for RFID Systems. In: Proceeding of PerCom (2007)

5. Hong, W., Liu, Y., Li, S., Wang, J.: Tag Identifiable Probability Study While Reader Collision Occurs. In: Proceeding of the fifth ICWCNMC, Beijing, pp. 3688–3691 (2009)

6. Waldrop, J., Engels, D.W., Sarma, S.E.: Colorwave: An Anticollision Algorithm for the Reader Collision Problem. In: IEEE WCNC, pp. 1206–1210 (March 2003)

A Desynchronization Tolerant RFID Private Authentication Protocol Qingsong Yao1 , Yong Qi1 , Ying Chen2 , and Xiao Zhong2 1

School of EIE, Xi’an Jiaotong University, Xi’an, China 2 IBM China Research Lab, Beijing 100094, China [email protected], [email protected], {yingch,zhongx}@cn.ibm.com

Abstract. Previous designed synchronization approaches advocate an O(1) search complexity. Although it is eﬃcient, such an approach is vulnerable to Desynchronization Attacks, in which the secret information will become incrementally diﬀerent between the tag and reader. Either adversary can utilize this to distinguish tags or the legitimate tag and reader cannot authenticate with each other. Even worse, synchronization approaches suﬀer from replay attacks. To address these problems, we propose a DESynchronization Tolerant RFID private authentication protocol, DEST, which forces a tag to keep its behaviors undistinguishable. DEST provides desynchronization tolerance, replay attack resistance, and forward secrecy. The analysis results show that DEST eﬀectively enhances the privacy protection for RFID private authentication, and provides the same eﬃciency, O(1), as traditional synchronization approaches. Keywords: RFID, authentication, privacy, desynchronization, tolerant.

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Introduction and Related Work

The identifying message in RFID systems may leak private information of their owners, for example, the owner’s ID. To address this problem, a number of privacy preserving authentication (PPA) protocols [1, 2] are proposed to prevent information leaking. In a PPA scheme, each tag shares a distinct symmetric key (or keys) with backend database. The RFID reader can securely access the backend database that stores all the keys of tags. During the authentication, the RFID reader sends an authentication request. The tag replies a message that is encrypted using the shared secret key. Upon the message relayed by the RFID reader, the backend database searches among all tags’ records to ﬁnd a key that can correctly generate the identical cipher. If there is a match, the RFID reader thereby authenticates the tag. For the ease of discussion, we term ’reader’ as the combination of RFID reader and backend database in the following. We also denote T as the tag and R as the reader. Early designed protocols, such as [3], usually store keys linearly in the backend database. The searching eﬃciency of these approaches is usually O (N ) due to the linear structure, where N is the number of tags in the system. O (N ) complexity is ineﬃcient enough in huge RFID systems. For fast authentication, a special G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 120–124, 2010. c Springer-Verlag Berlin Heidelberg 2010

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kind of protocols based on synchronization messages and limited pre-computed records were proposed. In [4] Ohkubo et al. limit the possible output number from each tag to a predeﬁned value. When the tag is overrunning, it sends random responses or no response. In [5], Henrici et al. use a counter for each tag to record the successful authentication times. The reader authenticates the tag by checking if the counters are equal. But the counters sent in plain text can help tracking by just sending queries to a tag and observe the regular counters. In [6], Juels propose a method to loop through a window to authenticate tags. These protocols have a O (1) search complexity, but suﬀer from desynchronization attack [3]. The adversary can launch such attacks by just sending a large volume of authentication queries to a tag. The tag will become incrementally desynchronized with its information stored in the reader. However, the counter in the tag can only increase a limited amount due to some predeﬁned threshold. When the diﬀerence exceeds the threshold, the tag sends constant messages, or alternatively sends random messages or no message which cannot be accepted by a legitimate reader. All the cases will help the adversary to recognize the tag and thus track it. Even worse, responses are a kind of pre-computed value, thus a replay of historical response from a tag may fool a legitimate reader. An adversary may replay response from a cheap commodity to pay little for an expensive one. Unfortunately, how to defend against the these attacks while remaining the high authentication eﬃciency of synchronization approaches remains open. Although in [7], an approach is proposed for O (1) search complexity and good security properties, it may incur extra storage cost on reader side. In this paper, we propose a DESynchronization Tolerant PPA protocol, DEST. DEST enables the tolerance of desynchronization attacks and can defend against replay attack and provide forward secrecy. Utilizing DEST, the storage and computing complexity of a tag is O (1). DEST provides O (1) authentication eﬃciency for tags that are not badly attacked. Diﬀerent from the previous synchronization approaches, DEST can cope with the authentication requests from those desynchronized tags. The rest of this paper is organized as follows. We present the DEST design in Section 2. In Section 3 we analyze the security and eﬃciency of DEST. At last, we conclude the paper.

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The basic idea of DEST is to provide function for overrunning tags, so that the attacker cannot ﬁnd out if a tag is overrunning. DEST comprises of three components, System initialization, Authentication, and Key-updating. In DEST, each tag Tj is embedded with a one-way cryptographic hash function h(·), a pseudo random number generator (PRNG), and a counter counterj . The length of PRNG output is equal to the length of the output of h(·). The counter keeps the information that how many times the tag has been queried. Each

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Tj and R shares a secret key Kj . On the other side, R has the similar oneway cryptographic hash function h(·) and pseudo random number generator (PRNG). R also keeps a distinct counter counterj for each tag Tj . R maintains a threshold to limit the number of possible outputs pre-stored for each tag. The counters in both sides are used to synchronize a tag and the reader. The values of the counters are initialized to 0. In the initialization component, R pre-computes a hash table Table. In the Table, each tag Tj has correlated records (IDj , h(Kj , counterj,i ), Kj ), 1≤j ≤N and 0≤i≤threshold, where the IDj is ID correlated to the tag Tj and counterj,i are the counters kept for Tj . As i is a cursor, we denote it by counterj brieﬂy for reader in the following. R holds a temporal key K and counter counter for the latest successful authentication. The Authentication process comprises four phases. 1) R sends a query to Tj , comprising a request and a random number r1 from PRNG. 2) Upon the query, Tj generates a random number r2 by PRNG. Then a message component I is generated as h(Kj , counterj ). A message component M is also generated as h(0, Kj , r1 , r2 ), corresponding to this authentication session. Then r2 , I and M are sent to R as a response, and counterj of Tj is increased by 1. 3) When R receives the response from Tj , it determines whether it is legitimate. R launches a search in Table, to ﬁnd a h(Kj , counteri ) which equals to I. If exists such a Kj , R can determine I is generated by Tj related to Kj . Then R checks whether the response is a replayed one by checking M. If M equals to h(0, Kj , r1 , r2 ), Tj is accepted as a legitimate tag. Otherwise Tj is rejected as an illegal one. If no matching Kj is found in Table, Tj should be either overrunning or illegal. Then R computes h(0, Kj , r1 , r2 ) for each Kj in the system and compares the hash values to I. If there is a Kj that matches, Tj related to Kj should be the tag under authentication. R then accepts Tj as a legitimate one. Otherwise, Tj is rejected as an illegal one. For a legitimate tag, R computes a message component O as h(1, Kj , r1 , r2 ) if matched by a pre-computed record, else O is generated as h(2, Kj , r1 , r2 ) and counterj is set to 0, meaning matched by exhaustive search. For an illegal tag, O is generated from PRNG. Then, R sends O to Tj as an authentication message for R. If R determines that Tj is legitimate, it records current key K and counter counter. Then it launches Key-updating. The key-updating of DEST has two goals, updating secret key on both sides to provide forward secrecy, and updating the pre-computed Table in the reader. The new key is generated as h(3, Kj , r1 , r2 ). The second goal is done by computing a serial of threshold values starting from h(Kj , counter + 1) to h(Kj , counter + threshold ). When the key of a tag is updated, the pre-stored hash values are updated correspondingly. 4) The tag Tj receives O and checks if O is the value of h(1, Kj , r1 , r2 ) or h(2, Kj , r1 , r2 ). If it is, R is authenticated as a legitimate reader. Then Key-updating process at tag side is launched.

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First, we give the security analysis. We show that DEST fulﬁlls the general security requirements, Confidentiality, Untraceability, Cloning resistance, Forward secrecy, and Compromising resistance, proposed in [7]. We also consider the desynchronization tolerance. For the limited space, we don’t give the deﬁnitions. Confidentiality: Messages on the air only include hash values and random numbers. No private information is leaked in the DEST’s messages. Cloning attack: In DEST, the response from a tag has a component M. The value of M is computed by enrolling two conversation relative nonces from both the tag and reader as session tokens, thus an adversary cannot predict their values. Therefore, replaying or cloning cannot pass the veriﬁcation of R. Forward secrecy: DEST poses a Key-updating process after each successful mutual authentication, in which the key is renewed by computing the hash value of old key and session tokens. The attackers cannot deduce previous keys from current one according to the preimage resistance property of cryptographic hash function. Therefore, DEST can provide forward secrecy. Tracking: In DEST, a tag’s response includes r2 , I, and M. In all four phases of the authentication, messages are sent in the same format. Thus, attackers cannot distinguish a tag through those messages no matter whether the tag is overrunning or not. Among them, r2 is a random number. M is a hash value using two random nonces r1 , r2 , and a secret key Kj as the input. Without knowing Kj , counterj , and r2 in advance, the adversary cannot link a tag to its responses. Therefore, the adversary cannot distinguish a tag from others by comparing the hash values. Compromising resistance. In DEST, tags share no keys. Thus compromising of some tags will help nothing for an adversary to attack other tags. Desynchronization Tolerance: We carefully design the Key-updating process of DEST to avoid desynchronization. In DEST, the Key-updating process takes place only after successful mutual authentications. Between successful mutual authentications of tag and reader, only the counterj embedded in tag is updated upon the authentication query. And the value of counter is never sent out in plain text. In this way, an adversary cannot track a tag by observing its responses. An adversary may interrupt the updating on the tag side after a legitimate reader authenticates the tag. In this case, the state of tag and the record in the reader are temporally desynchronized. In DEST, we keep a copy of old key K and old counter for each tag in the reader to record the information of last successful authentication. Then a desynchronized tag can still pass the authentication of the reader, which is impossible in existing synchronization approaches. Second, we give the performance analysis. Both the storage and computing complexity of DEST are O (1) for the tags. The storage cost of DEST protocol for the reader is O (threshold * N ). Considering the threshold is a predeﬁned parameter, it is O (N ). Pre-computing can be done oﬄine, which does not aﬀect authentication eﬃciency. We care more about the authentication eﬃciency at reader side. For a non-overrunning tag, the reader can ﬁnd its hash value in

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Table via an O (1) lookup. For an overrunning tag, the reader needs to compute M with each known key. This leads to O (N ) searching complexity. In large-scale RFID systems, the ratio of overrunning tags should be very small. We explain this in the following. We deﬁne rc as the ratio of the number of overrunning tags to the number of all tags in the system. Then the cost of the adversary is rc * threshold * N. Considering rc as a parameter of the adversary’s goal, the cost means threshold times of continuous queries on separate rc * N tags. That needs the adversary to continuously send O (N ) queries to rc * N tags, and force the tags to update their counters. Considering N is a huge number, this can be prohibitively hard. DEST protocol provides functionality for overrunning tags, and turns them to normal ones by updating. This increases the diﬃculty of the adversary.

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Conclusions

In this paper, we present a desynchronization tolerant RFID private authentication protocol, DEST. DEST can eﬀectively tolerate desynchronization attack while maintaining the high authentication eﬃciency O (1) at most of time. It can also satisfy general security requirements. Acknowledgments. This work was supported in part by China 863 Program under grants No. 2009AA01Z116 and 2009AA011903, NSFC under grant No. 60933003 and 60873262, Shaanxi ISTC under grant No. 2008KW-02, and IBM Joint Project.

References 1. Bibliography on Security and Privacy in RFID Systems, http://www.avoine.net/rfid 2. Juels, A.: RFID Security and Privacy: a Research Survey. IEEE Journal on Selected Areas in Communication 24(2), 381–394 (2006) 3. Ohkubo, M., Suzuki, K., Kinoshita, S.: Cryptographic Approach to Privacy-friendly Tags. In: Proceedings of RFID Privacy Workshop. MIT, Cambridge (2003) 4. Ohkubo, M., Suzuki, K., Kinoshita, S.: Eﬃcient Hash-Chain based RFID Privacy Protection Scheme. In: Proceedings of UbiComp, Workshop Privacy (2004) 5. Henrici, D., Mller, P.: Hash-based Enhancement of Location Privacy for RadioFrequency Identiﬁcation Devices Using Varying Identiﬁers. In: Proceedings of IEEE PerCom Workshops (2004) 6. Juels, A.: Minimalist Cryptography for Low-Cost RFID Tags. In: Blundo, C., Cimato, S. (eds.) SCN 2004. LNCS, vol. 3352, pp. 149–164. Springer, Heidelberg (2005) 7. Yao, Q., Qi, Y., Han, J., Zhao, J., Li, X., Liu, Y.: Randomizing RFID Private Authentication. In: Proceedings of IEEE PerCom (2009)

Study of Joint Routing and Wireless Charging Strategies in Sensor Networks Zi Li, Yang Peng, Wensheng Zhang, and Daji Qiao Iowa State University, Ames, IA 50011 {zili,yangpeng,wzhang,daji}@iastate.edu

Abstract. In recent years, wireless charging (a.k.a. wireless energy transferring) [3] has been recognized as a promising alternative to address the energy constraint challenge in wireless sensor networks. Comparing to the conventional energy conservation or harvesting approaches, wireless charging can replenish energy in a more controllable manner and does not require accurate location of or physical alignment to sensor nodes. In spite of these advantages, there has been little research on how much potential performance improvement may be achieved by applying the wireless charging approach to sensor networks and how to fully leverage its potential. In this paper, as one of the first efforts to study these issues, we (1) formulate the problem of maximizing the sensor network lifetime via codetermining routing and charging (ML-JRC), (2) prove the NP-hardness nature of the problem and derive an upper bound of the maximum sensor network lifetime that is achievable with ML-JRC, and (3) present a set of heuristics to determine the wireless charging strategies under various routing schemes, and demonstrate their effectiveness via in-depth simulation.

1 Introduction The limited energy supply has constrained the lifetime of a sensor network, and this has been a long-lasting and fundamental challenge in sensor networks that are designed for long-term operation. Over the years, various schemes have been proposed to address this challenge, such as energy conservation [4], environmental energy harvesting [2, 7], incremental sensor deployment, and battery replacement [10, 11]. Unfortunately, all these schemes have their limitations. Energy conservation schemes can only slow down energy consumption but cannot compensate energy depletion. Harvesting the environmental energy is subject to their availability which often is uncontrollable by people. Incremental sensor deployment may not be environmentally friendly because deserted sensor nodes can pollute the environment. As the wireless charging technology is being commercialized [9], it has become a promising alternative to address the energy constraint challenge in wireless sensor networks. Different from energy harvesting technologies, the wireless charging technology, together with more and more mature and inexpensive mobile robots, will make the controllable energy replenishment possible, with which energy can be replenished proactively to meet application requirements rather than passively from the energy resources available in the environment. Comparing with sensor node or battery replacement approaches, the wireless charging technology allows a mobile charger (MC) to

This work is supported partly by the NSF under Grant CNS-0831874.

G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 125–135, 2010. c Springer-Verlag Berlin Heidelberg 2010

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transfer energy to sensor nodes wirelessly when they are apart by up to a few feet, and therefore does not require accurate localization of sensor nodes or strict physical alignment between the MC and sensor nodes. In spite of the potential advantages of the wireless charging technology, little research has been conducted on how much sensor network lifetime may be prolonged with the technology and how to fully leverage its potential. Since communication is a major source of energy consumption in sensor networks and routing strategy has a significant impact on the communication efficiency, we are particularly interested in the problem of co-determining wireless charging and routing strategies to maximize the network lifetime. In general, this is a complicated problem due mainly to the mutual dependency between wireless charging and routing strategies. For example, depending on the routing schemes, such as the routing metric to be used and whether data is aggregated along the route, the MC may vary the sequence of the sensor nodes to be charged and the amount of energy allocated to each sensor. On the other hand, the location and moving speed of the MC may affect sensor nodes’ routing decisions as it could be more beneficial in terms of energy efficiency by routing data through longer paths but closer to the MC. In this paper, as the starting point to study this complicated problem, we investigate a basic version of the problem by making a few simplifying assumptions. Particularly, we assume no aggregation during data forwarding and zero delay or energy cost for MC movement. We refer to this version of the problem as the ML-JRC (Maximizing Lifetime via Joint Routing and Charging) problem. The rest of the paper is organized as follows. In Section 2, we formulate the MLJRC problem, prove its NP-hardness nature and derive an upper bound of the maximum sensor network lifetime achievable with ML-JRC. In Section 3, we present a set of heuristics to determine the wireless charging strategy under various routing schemes. Section 4 evaluates the proposed heuristics against the theoretical upper bound via simulation and Section 5 concludes the paper.

2 Problem Formulation and Analysis 2.1 System Models Network Model. We consider a wireless sensor network deployed for long-term, continuous monitoring of a certain field [6, 12]. A subset of sensors (denoted as src) in the network monitor the environment continuously and generate sensory data at the same constant rate (r). All sensors collaborate in forwarding sensory data to the base station hop by hop. There is only one base station in the network. We assume that data is not aggregated along the forwarding path. Each sensor is initially provisioned with a battery with full energy (Es ) and equipped with two antennas, one for wireless communication and the other for wireless charging. Previous studies have shown that among all components of a sensor node, radio consumes the most significant amount of energy. Therefore, we only consider the communication energy cost and neglect others (i.e., sensing cost and computational cost). Based on [1], we model the energy consumption of transmitting one data packet between nodes i, j as eij = L ∗ γ ∗ dnij , where L is the packet length (in bits), γ and n are the constants for a specific wireless system (usually 2 ≤ n ≤ 4), and dij is the distance between nodes i, j. In this paper, we set γ = 0.001 and n = 3.

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Wireless Charging Model. The wireless charging-related assumptions are made based on the wireless charging devices provided by Powercast [9]. A mobile charger (MC) equipped with an energy transmitting antenna can move around in the field. Charging is conducted wirelessly without requiring that the MC and sensors touch each other or are aligned in position. The wireless band used for charging is different from that for communication; for example, the Powercast chargers transfer energy in the 903 − 927MHz band while most sensor nodes use the 2.4GHz band for communication. When the MC operates, its power consumption is 3W (Λc ). The effective amount of power that can be captured by a sensor (denoted as λc ) varies with its distance to the MC. The relation is illustrated in Figure 1, which is obtained through our field test with the Powercast wireless energy transmitter and receiver. The antenna gain is 1.15 for both the transmitter and receiver. As shown in the figure, at the distance of 10cm, the receiver λc can receive about 30mW power, meaning that the charging efficiency η = Λ is about c 1%. As the distance increases, the charging efficiency decreases drastically. Initially, the MC is equipped with a battery with energy Ec for charging. We assume that the MC has the full knowledge about the network, including the geographic locations of all sensor nodes and their current energy levels. Moreover, we assume zero delay or energy cost for the MC movement. 2.2 Problem Definition and NP-Hardness Nature The problem to be studied in this paper is that, given the above system models, how to schedule the routing of sensory data packets from the source nodes to the base station and the charging activities of the MC so that the network lifetime may be maximized. Here, the network lifetime is defined as the time elapsed before the network fails to route a data packet from any source node to the base station (due to that either the source node has depleted its energy or all the routes from the source node to the base station have been broken). We refer to this problem as the ML-JRC (Maximizing Lifetime via Joint Routing and Charging) problem, which is NP-hard as to be proved below. The decision version of the ML-JRC problem can be formally expressed as follows: Given an ML-JRC instance G = (V ∪ {bs}, E), src, eij , r, Es , Ec , Λc , η, can the network survive for N time slots (where the length of a slot is given)? Our proof is motivated by [8] and based on the Disjoint Connection Path (DCP) problem which is well-known to be NP-complete. The DCP problem tries to determine that, given a graph G = (V , E ) (either directed or undirected) and a set of k disjoint source

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and destination vertex pairs (si , ti ), i = 1, 2, . . . , k, whether there are k vertex-disjoint paths each of which connects an (si , ti ) pair. Theorem 1. The decision version of the ML-JRC problem is NP-hard. Proof. (sketch) Given a DCP instance G , (si , ti ) where i = 1, 2, . . . , k, we can construct an ML-JRC instance with the following procedure: – Add one bs node, one source node src0 , and links (src0 , si ) and (ti , bs) (where i = 1, 2, . . . , k) to G to construct G; – Set esrc0 ,si = 0, and eij = 1 for all other links in G; – r is set to one packet per time slot, Es is set to 1, Ec , Λc and η are set to 0; – N is set to k. The above construction procedure can be completed in polynomial time. Next, we need to prove that a given DCP instance and its corresponding ML-JRC instance are equivalent. Suppose there exist k vertex-disjoint paths Pi where each Pi connects the pair (si , ti ), i = 1, 2, . . . , k in the DCP problem. Then, in the ML-JRC problem, we have k disjoint paths from src0 to bs. As a result, the system can survive N = k time slots since in each time slot, the src0 could use one of the k disjoint paths to deliver a data packet to bs. Conversely, if in the ML-JRC problem, the system has a lifetime of N time slots, there must exist N disjoint paths (denoted as Pi ) from src0 to bs. This is because each node in the network only has the energy to transmit one data packet. Therefore, no node can be part of multiple paths to deliver multiple packets. Let Pi be the path obtained from Pi by removing the nodes src0 , bs and the links (src0 , si ), (ti , bs) i = 1, 2, . . . , N . Then, all Pi paths are vertex disjoint, meaning that there exist k vertex-disjoint paths in the corresponding DCP instance. Since DCP is known to be NP-complete, we have proved that ML-JRC is NP-hard. 2.3 Upper Bound of Lifetime Achievable with ML-JRC In this section, we utilize the linear programming (LP) technique to derive an upper bound of the maximum network lifetime that can be achieved in the ML-JRC problem. Formally, it is formulated as follows: max p/r subject to: p+ j∈Ni

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Here, p stands for the total number of data packets generated by each source node during the network lifetime. Since the data generation rate r is the same over all source nodes, they should produce the same number of packets during the network lifetime. Therefore, p/r defines the lifetime. ΔEi is the total amount of energy used by the MC to charge node i. fij is the total number of packets transmitted from node i to node j. Constraints (1), (2) and (3) reflect the flow conservation requirements. Constraint (4) reflects that the energy used for transmission should be smaller than Es – the initial battery energy at a sensor node – plus the energy charged from the MC. Constraint (5) states that the total energy used for charging cannot exceed Ec – the initial battery capacity of the MC. The output fij , ΔEi is the joint routing and charging decision. More specifically, it specifies the number of data packets transmitted over link (i, j) and the amount of energy distributed to node i by the MC so that the network lifetime can be maximized.

3 Heuristic Solutions As the ML-JRC problem is NP-hard, finding an optimal solution may have high computational complexity as the network scale increases. Hence, we propose low-complexity heuristic solutions. Instead of solving the complicated joint problem directly, the proposed heuristic solutions adopt the following philosophy: each node in the network runs a routing algorithm which requires only little, locally available information, and the MC runs a charging scheduling algorithm based on the global network knowledge. The general framework of the proposed heuristic solutions is as follows. Time is divided into slots and the length of each time slot is δt . At the beginning of each time slot, each sensor node exchanges the latest routing cost to the base station with its neighbor nodes and, based on the received information, selects the least-cost route to deliver its data towards the base station. Meanwhile, based on its knowledge about the routes selected by all source nodes, the MC plans its activity to spend δt Λc amount of energy to charge sensor nodes during the slot. As Λc is the power consumed by the MC for charging operation, δt Λc gives the maximum energy available for charging in a slot. In the following, we describe the routing metrics used by sensor nodes to define least-cost routes and the heuristic algorithms for the MC to decide its charging activity. 3.1 Routing Metric As there have been lots of research on designing routing algorithms to prolong the network lifetime, we adopt a representative one and focus on designing charging algorithms with this particular routing algorithm. Specifically, the routing metric we adopt is wi,t Cij,t = eij ∗ u1− Es [5], where u is a system parameter and wi,t is the residual energy of node i at the beginning of time slot t. This metric is a combination of the minimum wi,t energy (eij ) and max-min residual energy (u1− Es ) metrics. When u = 1, it is reduced to the minimum energy metric; when u > 1, the route is selected by trading off the communication cost and nodal residual energy. The larger is u, the more balanced is the energy distribution among sensor nodes after the route is used. 3.2 Heuristic Charging Algorithms An effective charging algorithm should charge first the nodes whose lifetime constrains the network lifetime the most, and should utilize the network routing information to

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adjust the charging decisions adaptively. With these considerations, we propose three heuristics to prolong the network lifetime. Least Residual Energy First (LRE). Balancing energy consumption among all nodes is a well-known technique to prolong the network lifetime. We propose a simple LRE heuristic that applies this technique. Specifically, at the beginning of time slot t, the MC sorts all nodes into a list denoted as l = (l0 , . . . , ln ) in the ascending order of nodal residual energy, and performs the following max-min calculation: lm

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In the above, Δei,t is the energy to be charged to node i in time slot t. (7) restricts the total energy allocated in a time slot, (8) specifies the m nodes selected to be charged to the same energy level, and (9) constrains the maximum energy level of each node to be below the battery capacity. All of the δt Λc amount of energy shall be used up in the slot unless all nodes already have their batteries fully charged to the capacity. Least Estimated Lifetime First assuming Fixed Routes (LEL). Similar to LRE, the intuition behind LEL is also to charge first the nodes whose lifetime has been the bottleneck of the network lifetime. However, different from LRE, LEL considers not only nodes’ current residual energy but also their future energy consumption rates. To estimate nodes’ future energy consumption rates, routes adopted by nodes should be predicted. In LEL, it is simply assumed that the routes used currently are the same as those to be used in the future. Therefore, the lifetime of each node is estimated as follows: lifetimei,t = wi,t /workloadi,t , workloadi,t = α ∗ workloadi,t−1 + (1 − α) ∗ (wi,t − wi,t−1 ). The list l used in LRE is sorted by the estimated lifetime and (8) is modified to wl0 ,t +Δel0 ,t ∗η workloadl0 ,t

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In Section 2.3, the solution to the LP formulation, fi,j , ΔEi , contains both flow information and energy allocated to each node during the network lifetime. Yet, the exact amount of energy replenished to each node at each time slot was not given. This formulation, however, inspires us to design the AEA heuristic which tells the MC the exact amount of energy replenished to each node in each time slot. Specifically, at the beginning of time slot t, the MC solves a modified LP formulation (i.e., replacing Ec with δt Λc to limit the maximal charging energy in a slot, and replacing Es with wi,t to use the latest energy information), and charges the nodes according to the solution to this modified LP formulation. The AEA heuristic aims to maximize the network lifetime through dynamically updating the charging decisions with the assumption that future routes would change accordingly. Essentially, if the actual network flows yielded by the routing algorithm are similar to the flow solution to the LP formulation, charging according to the energy solution to the LP formulation would approximate the upper bound of the maximum network lifetime well. However, if the actual flows differ much from the flow solution to the LP formulation, the performance of AEA would degrade inevitably.

4 Performance Evaluation 4.1 Simulation Settings In the simulation, the node battery capacity Es is set to 15000J, the MC battery Ec is set to 2000000J, the charging efficiency η is 0.01, Λc is 3W , and all nodes are randomly placed in a 400m∗400m area. We study the performance of different charging schemes as the workload, the time slot length and the density of source nodes vary. The performance metric is the network lifetime. The evaluated charging schemes include LRE, LEL-0 (i.e., LEL with α = 0), LEL-0.8 (i.e., LEL with α = 0.8) and AEA. The performance of these schemes is also compared with that of NoC (i.e., no charging), and the lifetime upper bound computed with the LP formulation presented in Section 2.3. 4.2 Simulation Results Lifetime with Varying Workload. As the workload changes, the network-wide energy distribution may be different, which may impact the performance of the charging heuristics. To evaluate the impact, we fix the number of nodes in the network (i.e., 100) but vary the number of source nodes from 25 to 75. Results are plotted in Figure 2. As shown in the figure, all the proposed charging heuristics yield longer network lifetime than NoC. In addition, though the lifetime achieved with the charging heuristics declines as the workload increases, the ratio of the achieved lifetime to the theoretic upper bound keeps almost the same. For example, LRE, LEL-0 and LEL-0.8 achieve around 80% and AEA achieves 90% of the upper bound, when u = 100. Moreover, as the routing parameter u increases from 1 to 100, the achieved lifetime also increases because, as discussed in Section 3.1, the route selection tends to balance energy consumption among all nodes as u increases. Figure 2(a) shows the results when u = 1. It can be seen that the LEL schemes (i.e., LEL-0 and LEL-0.8) outperform other charging heuristics, while the AEA scheme yields the worst performance among all heuristics. This is due to the following reasons.

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When u = 1, the routing algorithm is reduced to that with the minimum energy metric. In this case, the routes are stable and not adjusted frequently to changes in the energy distribution among nodes. As this situation resembles the assumption of the LEL schemes well but is much different from that of the AEA scheme, the LEL schemes outperform the AEA scheme. On the other hand, due to the stability in route selection, factoring the historical workload more (as in LEL-0.8) or less (as in LEL-0) into estimating the future workload of sensor nodes do not make significant difference, which contributes to the performance similarity between LEL-0 and LEL-0.8. 1800

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Figure 2(b) shows the results when u = 100. In this case, AEA yields the best performance among all, followed by LEL-0.8, LEL-0 and LRE in order. In contrary to the case of u = 1, with u = 100, the routing algorithm tends to actively adjust routes to balance the energy consumption among all nodes. This situation resembles the assumption of the AEA scheme well but is much different from that of the LEL schemes; therefore, the AEA scheme outperforms the LEL schemes. For the LEL schemes, LEL-0.8 uses more historical information in estimating future workload, which results in more accurate estimation and hence better performance than LEL-0. Regardless of the parameter u, the LEL schemes always outperform the LRE scheme because route selection is considered in LEL but not in LRE. Lifetime with Varying Time Slot Length. In the proposed heuristics, updating routes and charging planning happen only at the beginning of a time slot. Therefore, the frequency of route updating and charging planning decreases as the slot length increases. To investigate how this frequency impacts the network lifetime, we have conducted simulations with the slot length varying from 1h to 120h. Figure 3(a) shows the results when u = 1. We can see that all charging heuristics are insensitive to changes in slot length. This is because, in this case, minimum energy routes are always selected and remain unchanged regardless of the slot length. In comparison, when u = 100, as shown in Figure 3(b), the performance of all heuristics drop as the slot length increases, since the impact of lifetime extension resulted from balanced route selection has been reduced. Among all the heuristics, the performance of

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AEA drops the most rapidly. This is because AEA assumes frequent route updates in the future. However, as the slot length increases, such update happens less frequently. As a result, the performance of AEA degrades significantly. On the contrary, other heuristics either assume fixed routes or do not consider route selection, hence are less sensitive to changes in the slot length. Figure 3(b) also reveals that, when the slot length is in the range of [1, 24], the performance of all charging heuristics remains at a high level. Therefore, it is not necessary to choose too short a time slot, which may incur more computational and movement overhead for the MC. Lifetime with Varying Network Density. Given fixed number of source nodes, increasing the network density (i.e., increasing the number of nodes in the network) provides more options for selecting routes and distributing workload. It is interesting to evaluate whether the proposed heuristics may exploit the more options presented in a larger scale network and how well they may do so. For this purpose, we have conducted simulations with fixed number of source nodes but varying total number of nodes. As shown in Figure 4, as the total number of nodes increases, the network lifetime increases with all heuristics and the performance gain of heuristics over NoC becomes more significant. This demonstrates that the heuristics have the capability of exploiting the more options brought in by the higher network density. Furthermore, comparing Figures 4(a) and 4(b), we can see that, when the parameter u gets larger, the lifetime achieved by the heuristics increases with the network density more rapidly. This is because a larger u implies that the balance of residual energy among all nodes is favored more in path selection. Hence, workload is distributed more proactively among nodes; this way, the advantages brought in by the increased network density can be better exploited. Among all the heuristics, AEA performs the best, followed by LEL-0.8, LEL-0 and LRE when u = 100. The difference is due mainly to the following reasons. As the network density increases, routes are updated more frequently as there are more available routes with dynamically changing costs. This situation resembles the assumption of AEA better than that of the LEL schemes; hence, AEA outperforms the LEL. LEL-0.8 estimates the future workload more accurately than LEL-0 when the routes are updated more frequently; hence, LEL-0.8 outperforms LEL-0. LRE does not consider the effect

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of path selection and therefore performs worse than other heuristics. Figure 4 also shows that, the lifetime upper bound ascends faster than all heuristics as the network density increases. This is because the routing and charging strategies are scheduled jointly in the LP formulation, while the routing in all heuristics is planned without considering the charging activity of the MC. Summary: We obtain the following insights and preliminary conclusions from the simulation results: (1) With the proposed charging heuristics, the achieved network lifetime can reach a decent fraction of the upper bound, which indicates that the application of wireless charging may prolong the network lifetime even with simple charging strategies. Nevertheless, the performance gap between the proposed charging heuristics and the upper bound is still noticeable in some scenarios, which motivates us to design more delicate joint routing and wireless charging algorithms to further approach the upper bound. This is part of our future work. (2) As the routing algorithm favors more balanced energy distribution among nodes, the AEA heuristic is the preferred choice. On the other hand, the LEL heuristic is more desired when the route selection is mainly affected by the link cost rather than the balance among nodal residual energy. (3) Simulation results also indicate that effective joint routing and charging schemes should be adaptive to the changes in network density.

5 Conclusions In the paper, we study a new type of sensor networks which consists of sensor nodes with replenishable energy supplies and a mobile charger that is able to charge the batteries of sensor nodes wirelessly. We formulate the problem of maximizing the sensor network lifetime via joint routing and charging (ML-JRC), prove its NP-hardness nature, and derive the upper bound of the maximum network lifetime that is achievable with ML-JRC. We also present a set of heuristics to determine the energy charging strategies for the mobile charger under different routing schemes. Simulation results demonstrate the effectiveness of applying the wireless charging technology to prolong the sensor network lifetime, even with simple charging heuristics.

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References 1. Chang, J.H., Tassiulas, L.: Energy conserving routing in wireless ad-hoc networks. In: INFOCOM 2000 (2000) 2. Fan, K.W., Zheng, Z., Sinha, P.: Steady and fair rate allocation for rechargeable sensors in perpetual sensor networks. In: SenSys 2008 (2008) 3. Karalis, A., Joannopoulos, J.D., Soljacic, M.: Efficient wireless non-radiative mid-range energy transfer. Annals of Physics 323, 34 (2006) 4. Li, Q., Aslam, J., Rus, D.: Online power-aware routing in wireless ad-hoc networks. In: MobiCom 2001 (2001) 5. Lin, L., Shroff, N.B., Srikant, R.: Asymptotically optimal energy-aware routing for multihop wireless networks with renewable energy sources. IEEE/ACM Transactions on Networking 15(5), 1021–1034 (2007) 6. Mainwaring, A., Culler, D., Polastre, J., Szewczyk, R., Anderson, J.: Wireless sensor networks for habitat monitoring. In: WSNA 2002 (2002) 7. Park, C., Chou, P.: Ambimax: Autonomous energy harvesting platform for multi-supply wireless sensor nodes. In: SECON 2006 (September 2006) 8. Park, J., Sahni, S.: An online heuristic for maximum lifetime routing in wireless sensor networks. IEEE Transactions on Computers 55(8), 1048–1056 (2006) 9. Powercast: Online link, http://www.powercastco.com 10. Tong, B., Li, Z., Wang, G., Zhang, W.: On-demand node reclamation and replacement for guaranteed area coverage in long-lived sensor networks. In: QShine 2009 (November 2009) 11. Tong, B., Wang, G., Zhang, W., Wang, C.: Node reclamation and replacement for long-lived sensor networks. In: SECON 2009 (June 2009) 12. Xu, N., Rangwala, S., Chintalapudi, K.K., Ganesan, D., Broad, A., Govindan, R., Estrin, D.: A wireless sensor network for structural monitoring. In: SenSys 2004 (2004)

Page Size Optimization for Code Dissemination in Wireless Sensor Networks Wei Dong1 , Xi-bin Zhao2, and Min Xi3 1

2

Zhejiang University & HKUST Key Laboratory for Information System Security of Ministry of Education, School of Software, Tsinghua University 3 Department of Computer Science, Xi’an Jiaotong University [email protected]

Abstract. Wireless sensor networks (WSNs) have recently gained a great deal of research attention, with a wide range of applications being explored. In most applications, WSNs are deployed in inaccessible areas for a long lifetime. Software maintenance and update in WSNs are challenging. Network reprogramming is an important way to address this challenge. Code dissemination is a critical service to enable network reprogramming. Most code dissemination protocols employ segmentation and pipelining to improve the reprogramming efficiency. As we show in this paper, the choice of the page size in these segmented and pipelined dissemination protocols is of vital importance to the overall dissemination time. Hence, we explore the tradeoff in determining the optimal page size in terms of the overall dissemination time. We investigate the impact of page size for two typical code dissemination protocols in WSNs. Results show that the optimal page size decreases when the maximum hop count from the source node increases; and the optimal page size increases when the program size increases. The absolute value of the optimal page size is determined by the network scale, program image size, and protocol details.

1 Introduction Wireless sensor networks (WSNs) have recently gained a great deal of research attention [1,2,3,4,5,6], with a wide range of applications being explored, such as military surveillance, habitat monitoring, and infrastructure protection, etc. WSN applications need to be updated after deployment for a variety of reasons, such as reconfiguring parameters, modifying tasks of individual nodes, and patching security holes. Many sensor networks, however, are deployed in environments where physically collecting previously deployed nodes is either very difficult or infeasible. Network reprogramming is an important way to address such challenges [7]. The critical service required to enable network reprogramming is a code dissemination protocol. Example protocols include Deluge [8] and MNP [9], which distribute new program binaries into a network, enabling complete system reprogramming. Segmentation is a common technique that is employed in most existing code dissemination protocols [7]. There are two major benefits of segmentation. First, without segmentation, a large program breaks into thousands of packets. As a consequence, G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 136–145, 2010. c Springer-Verlag Berlin Heidelberg 2010

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each node needs a large number of states to record the packet information (received or not) [7]. Second, taking advantage of spatial multiplexing, segmentation and pipelining could increase the overall throughput significantly for a large program [7]. With segmentation and pipelining, a program is divided into several pages, each of which contains a fixed number of packets. Instead of receiving a complete program image before forwarding it, a node becomes a source node after it receives one complete page and uses per page negotiation for serving other nodes. Previous works on dissemination protocol design [8,9,10] use fixed page sizes. For example, the default page size in Deluge is fixed as 48 packets (48 pkts×23 bytes/pkt = 1104 bytes). In this work, we explore the tradeoff in determining the optimal page size in terms of the overall dissemination time (and also the dissemination throughput). A large page size (which translates to small number of pages for a given program image) will limit pipelining for spatial multiplexing while a small page size (which translates to large number of pages for a given program image) adds per page negotiation overheads. To investigate the impact of the page size on the overall dissemination time, we use the analytical model presented in [8]. This model fits well with TOSSIM [11] simulation results. Results show that the optimal page size decreases when the hop count from the source node increases; and the optimal page size increases when the program size increases. The results also indicate that, for disseminating a large program image in large-scale sensor networks, page size optimization is of vital importance because the choice of page size has a great impact on the overall dissemination time. The rest of this paper is structured as follows. Section 2 gives the background and describes the related work. Section 3 presents the solution for page size optimization for two typical dissemination protocols—Deluge [8] and MNP [9]. Section 4 shows the optimization results. Finally, we conclude this paper in Section 5.

2 Backgrounds and Related Work This section gives a brief overview of code dissemination in WSNs. We divide prior work into two categories—dissemination protocol design and dissemination performance analysis, which will be described in the following two subsections, respectively. 2.1 Dissemination Protocol Design Deluge [8] is perhaps the most popular code dissemination protocol used for reliable code updates in WSNs. It uses per page negotiation (i.e., ADV-REQ-DATA three way handshake) for reliability, and employs segmentation (into pages) and pipelining for spatial multiplexing. It is highly optimized and can achieve one ninth the maximum transmission rate of the radio supported under TinyOS [12]. MNP [9] shares many common points with Deluge, e.g., segmentation/pipelining, and per page negotiation. Besides, in order to avoid message collisions and to reduce energy consumption, it provides a detailed sender selection algorithm to choose a local source of the code which can satisfy the maximum number of nodes. There are many other code dissemination protocols proposed recently, e.g., Stream [13], Rateless Deluge [10]. Segmentation/pipelining and per page negotiation are two

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common techniques employed in these code dissemination protocols for improved efficiency and reliability. – Segmentation/pipelining. Segmentation and pipelining allow parallel transfers of data in networks, thus improving the dissemination efficiency. Consider disseminating n pages in a network of d hops. Without pipelining, the total time for transmission is O(d · n). With pipelining, the total time for transmission is reduced to O(d + 3(n − 1)) (note that the parallel transfers should take place with at least three-hop spacing as illustrated in Figure 1). – Per page negotiation. Per page negotiation uses message exchanges in the controlplane for data reliability. It is also used to avoid the broadcast storm problem [7]. A simple negotiation-based protocol contains three types of messages [7]: (1) ADV: the source node uses the ADV message to advertise its received object; (2) REQ: the receiver node sends back REQ messages after receiving the ADV message, notifying the source node about which objects are needed; (3) DATA: the source node then sends out the DATA messages of the requested objects. An important problem in these dissemination protocols is the determinization of the page size. When considering pipelining alone, small page size is favored. On the other hand, small page size adds per page negotiation overhead which decreases the dissemination efficiency. Clearly, there is a tradeoff in determining the optimal page size in terms of the overall dissemination time (also the dissemination throughput). 2.2 Dissemination Performance Analysis Experimental analysis helps protocol design and evaluation with high fidelity. Various tools have been proposed in the literature to aid system design and evaluations [14,15,11]. However, testbed experiments require at least a prototype system to be deployed. High-fidelity simulators (e.g., TOSSIM [11]) provides a fairly good resemblance of the real environment. However, simulation with a high fidelity can be really time-consuming. Mathematical analysis, on the hand, is inexpensive in terms of time and resource consumption. Hence, it can investigate various parameters in a larger space. In [8], the authors provide an analytical model for analyzing the performance of Deluge by considering protocol details. The authors demonstrate that the analytical model fits well with TOSSIM simulation results. There are some other analytical models, e.g., [13,16]. In [13], the authors do not consider protocol details. In [16], the authors do not consider concurrent transmission of pipelined pages. Compared to the analysis in [8], they are less accurate.

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More recently, we have proposed an accurate analytical model [17] which considers topological information, impact of contention, and impact of pipelining. For simplicity of analysis, the analytical model used in this paper is based on [8]. However, it should be noted that our contribution is not to produce a more accurate analysis. Rather, we aim to illustrate the impact of page size on the overall dissemination time, which, to the best of our knowledge, has not been performed before.

3 Page Size Optimization In this section, we present our optimization method based on the metric of dissemination time for two typical code dissemination protocols—Deluge [8] and MNP [9]. The optimization results will be shown in the next section. 3.1 Optimization Metric We present our optimization method based on the metric of dissemination time. The optimal page size should minimize the total dissemination time. This also translates to maximize the dissemination throughput. It should also be noted that, for the Deluge protocol [8], minimizing the dissemination time also means minimizing the dissemination energy consumption because all nodes remain in the radio receiving mode during dissemination, which consumes approximately the same energy. For example, for the Telos motes [18], the current draw in the transmitting mode is approximately 19.5mA while in the receiving mode is approximately 21.8mA [18]. 3.2 Optimization Problem For a given program image of S bytes, the number of pages (for Deluge [8] and MNP [9]) is, S n= (1) N ·P where N is the page size in terms of packets number, and P is packet payload size. According to [8], the expected time to transmit an object of n pages across a d hop network is (2) E[Tob j ] = min(d · n, d + 3(n − 1)) · (E[TtPage] + DnewReq) where DnewReq is the expected delay between completing a page and requesting a new page, and E[TtPage ] is the expected time required to transmit a page across a single hop. The page size optimization problem is thus formulated as follows, Given: S, P, . . . Find: N Minimize:

(3)

E[Tob j ]

The program image size, S, varies with different program images to be disseminated. The packet payload size, P, equals to 23 bytes in the current implementation of Deluge [8].

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We can see from Eq. (1) and Eq. (2) that, – with a large page size (i.e., small n), the pipelining factor (i.e., min(d · n, d + 3(n − 1))) will be small, and it is lower bounded by the hop count. On the other hand, with a large page size, the expected time for transmitting a page will be large and cannot be upper bounded because the current implementation of Deluge simply uses padding to transmit a complete page [8], which can be avoided with a better protocol design [19]. – with a small page size (i.e., large n), the pipelining factor will be large. On the other hand, with a small page size, the expected time for transmitting a page will be small, and it is lower bounded by the page negotiation overhead. The optimal page size depends on the image size (i.e., S), the network hop count (i.e., d), and the single-hop transmission time of a page (i.e., E[TtPage ], which again depends on the page size). The single-hop transmission time of a page (i.e., E[TtPage ]) varies for different protocols. In the following two subsections, we will describe the analytical methods to capture this factor for Deluge [8] and MNP [9], respectively. 3.3 Optimization for Deluge The single-hop transmission time of a page, i.e., E[TtPage ], for Deluge is computed as follows [8], (4) E[TtPage ] = E[TrAdv ] + E[Treq] + E[TtGiveup ] + E[Ttx ] where E[TrAdv ] is the expected time for receiving advertisements, which is given by E[TrAdv ] = E[NtPkt ] ·

τl · (1 + E[Nsupp]) 2

(5)

1 where E[NtPkt ] = PRR is the expected number of transmissions for a given packet. As reported in [8], τl = 2 and E[Nsupp ] = 1. After receiving the advertisements, a node makes request to the source node for the given page. The time for request, i.e., E[Treq ] is given by

E[Treq ] = E[NtPkt ] · E[Nreqs ] · E[τr ]

(6)

where E[Nreqs ] = 5.4 and E[τr ] = τ2r = 0.5 2 = 0.25 as reported in [8]. Additionally, when a node exceeds its limit of λ = 2 requests, it must transit to the MAINTAIN state and wait for another advertisement before making additional requests. The expected time spent waiting for additional advertisements, i.e., E[TtGiveup ], is given by E[Nreqs ] E[TtGiveup ] = E[NtPkt ] · − 1 · E[TrAdv ] (7) λ Finally, the expected time required to transmit just the data packets, i.e., E[Ttx ], is given by E[Ttx ] = E[NtPkt ] · TtPkt · N (8) where TtPkt = 0.032 is the time to transmit a single packet [8] (for Mica2 motes), and N is the page size (in terms of number of packets).

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3.4 Optimization for MNP The single-hop transmission time of a page, i.e., E[TtPage ], for MNP is computed as follows, E[TtPage ] = E[TrAdv ] + E[Ttx ] (9) The time for requests is not included in. The reason is that in MNP the advertising node sends a maximum number of κ advertisement messages to allow sender selection, thus by the end of the waiting interval E[TrAdv ], the advertising node has received at least one request message and is ready to serve a receiving node. The expected time for κ advertisement messages is given by, E[TrAdv ] = E[NtPkt ] · κ · Δ τ

(10)

1 is the expected number of transmissions for a given packet, κ = 4 where E[NtPkt ] = PRR is the advertisement threshold, and Δ τ = 2 is the average duration between two advertisements. After E[TrAdv ], the source node sends the data in a page. It sends Start Download and End Download messages signifying the start and the end of each page. Thus, the expected time for data transmission of a page is given by,

E[Ttx ] = E[NtPkt ] · (N + 2) · E[TtPkt ]

(11)

where TtPkt is the time to transmit a single packet, and N is the page size (in terms of number of packets).

4 Results In this section, we show the optimization results for Deluge [8] and MNP [9]. Our results reveal the following findings, – For both Deluge and MNP, the results show that the optimal page size decreases when the maximum hop count from the source node increases. – For both Deluge and MNP, the results show that the optimal page size increases when the program size increases. – The dissemination time of MNP is larger than that of Deluge because MNP requires a more complicated negotiation mechanism to allow sender selection. However, the optimal page sizes for both protocols are very similar. – The results also indicate that, for disseminating a large program image in largescale sensor networks, page size optimization is of vital importance because the choice of page size has a great impact on the overall dissemination time. 4.1 Parameter Settings We show the optimization results with the parameter settings as summerized in Table 1. Note that in order to investigate the impact of the hop count and the program image size, we vary the corresponding parameters, i.e., d, S. The hop count varies from 12

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Table 1. Parameter settings Parameter Meanings P packet payload size DnewReq expected delay between completing a page and requesting a new page PRR expected packet reception rate NtPkt expected number of transmissions for a given packet τl lower bound of the advertisement interval in Deluge E[Nsupp ] expected number of advertisement suppressions E[Nreq ] expected number of request messages for a page E[τr ] expected backoff delay for transmitting requests TtPkt expected time for transmitting a packet λ advertisement threshold in Deluge κ advertisement threshold in MNP d hop count from the source node S program image size

Value 23 bytes 1 second 0.9 ≈ 1.11 2 seconds 1 5.4 0.25 second 0.032 second 2 4 12, 24, 48, 64 2KB, 6KB, 12KB, 24KB 1 PRR

hops to 64 hops. Note that it is practical for long-distance and large-scale sensor networks [3,20]. The image size varies from 2KB to 24KB. Program of 2KB represents simple applications or configurable parameters while program of 24KB represents more complex applications, e.g., the implementation of lwIP (i.e., lightweight TCP/IP) on the AVR platform is approximately 22KB [21]. 4.2 Impact of Hop Count In order to investigate the impact of the maximum hop count on the optimal page size, we vary the hop count from 12 to 64 hops. Figure 2(a–d) show the relationship of the dissemination time and the page size with 12, 24, 48, and 64 hops respectively. We fix the program image size as 2KB. We can see from Figure 2 that the optimal page size decreases when the hop count increases. For example, Nopt ≈ 90 pkts when d = 12, Nopt ≈ 30 pkts when d = 64. We also observe that the page size has a greater impact when the maximum hop count increases because the propagation delay accumulates over multiple hops. This indicates that the selection of page sizes is very important for large-scale sensor networks. In addition, the dissemination time of MNP is larger than Deluge because MNP requires a more complicated negotiation mechanism to allow sender selection. However, the optimal page sizes for both protocols are very similar. 4.3 Impact of Program Image Size In order to investigate the impact of program image size on the optimal page size, we vary the program image size from 2KB to 24KB. Figure 3(a–d) show the relationship of the dissemination time and the page size with program image sizes 2KB, 6KB, 12KB, and 24KB respectively. We fix the network hop count from the source node as 24. We can see from Figure 3 that the optimal page size increases when the program image size

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increases. For example, Nopt ≈ 48 pkts when S = 2KB; Nopt ≈ 90 pkts when S = 6KB; Nopt ≈ 135 pkts when S = 12KB; Nopt ≈ 220 pkts when S = 24KB. This is because a large page size is more efficient for a large program as it decreases per page negotiation overhead. We also observe that the page size has a greater impact when the program image size increases because there will be more negotiation overheads when the number of pages increases. As described above, the dissemination time of MNP is larger than Deluge, and the optimal page sizes for both protocols are similar.

5 Conclusions In this work, we explore the tradeoff in determining the optimal page size in terms of the overall dissemination time. We investigate the impact of the page size for two typical code dissemination protocols in WSNs—Deluge and MNP. Results show that the optimal page size decreases when the hop count from the source node increases; and the optimal page size increases when the program size increases. The absolute value of the optimal page size is determined by the network scale, program image size, and protocol details. As a future work, we will implement page size optimization in typical dissemination protocols to improve the dissemination efficiency.

Acknowledgement This work is supported by the National Basic Research Program of China (973 Program) under Grant No. 2006CB303000, HKUST Fok Ying Tung Graduate School, China NSFC Grants 60933011, and Technology Major Project of China under Grant No. 2009ZX03006-001.

References 1. Akyildiz, I.F., Su, W., Sankarasubramaniam, Y., Cayirci, E.: Wireless Sensor Networks: A Survey. Computer Networks 38, 393–422 (2002) 2. Li, M., Liu, Y.: Underground Coal Mine Monitoring with Wireless Sensor Networks. ACM Transactions on Sensor Networks (TOSN) 5(2) (2009) 3. Mo, L., He, Y., Liu, Y., Zhao, J., Tang, S., Li, X., Dai, G.: Canopy Closure Estimates with GreenOrbs: Sustainable Sensing in the Forest. In: Proceedings of ACM SenSys (2009) 4. Liu, K., Li, M., Liu, Y., Li, M., Guo, Z., Hong, F.: Passive Diagnosis for Wireless Sensor Networks. In: Proceedings of ACM SenSys (2008) 5. Liu, Y., Li, M.: Iso-Map: Energy-Efficient Contour Mapping in Wireless Sensor Networks. In: Proceedings of IEEE ICDCS (2007) 6. Yang, Z., Liu, Y.: Quality of Trilateration: Confidence based Iterative Localization. IEEE Transactions on Parallel and Distributed Systems (TPDS) 21(5) (2010) 7. Wang, Q., Zhu, Y., Cheng, L.: Reprogramming wireless sensor networks: Challenges and approaches. IEEE Network Magazine 20(3), 48–55 (2006) 8. Hui, J.W., Culler, D.: The dynamic behavior of a data dissemination protocol for network programming at scale. In: Proceedings of ACM SenSys (2004) 9. Kulkarni, S.S., Wang, L.: MNP: Multihop Network Reprogramming Service for Sensor Networks. In: Proceedings of IEEE ICDCS (2005)

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10. Hagedorn, A., Starobinski, D., Trachtenberg, A.: Rateless Deluge: Over-the-Air Programming of Wireless Sensor Networks using Random Linear Codes. In: Proceedings of ACM/IEEE IPSN (2008) 11. Levis, P., Lee, N., Welsh, M., Culler, D.: TOSSIM: Accurate and Scalable Simulation of Entire TinyOS Applications. In: Proceedings of ACM SenSys (2003) 12. TinyOS: http://www.tinyos.net 13. Panta, R.K., Khalil, I., Bagchi, S.: Stream: Low Overhead Wireless Reprogramming for Sensor Networks. In: Proceedings of IEEE INFOCOM (2007) 14. Werner-Allen, G., Swieskowski, P., Welsh, M.: MoteLab: A Wireless Sensor Network Testbed. In: Proceedings of ACM/IEEE IPSN/SPOTS (2005) 15. Titzer, B.L., Lee, D.K., Palsberg, J.: Avrora: Scalable Sensor Network Simulation with Precise Timing. In: Proceedings of ACM/IEEE IPSN (2005) 16. De, P., Liu, Y., Das, S.K.: An Epidemic Theoretic Framework for Evaluating Broadcast Protocols in Wireless Sensor Networks. In: Proceedings of IEEE MASS (2007) 17. Dong, W., Chen, C., Liu, X., Bu, J., Liu, Y.: Performance of Bulk Data Dissemination in Wireless Sensor Networks. In: Proceedings of IEEE/ACM DCOSS (2009) 18. Polastre, J., Szewczyk, R., Culler, D.: Telos: Enabling Ultra-Low Power Wireless Research. In: Proceedings of ACM/IEEE IPSN/SPOTS (2005) 19. Panta, R.K., Bagchi, S., Midkiff, S.P.: Zephyr: Efficient Incremental Reprogramming of Sensor Nodes using Function Call Indirections and Difference Computation. In: Proceedings of USENIX Annual Technical Conference (2009) 20. Kim, S., Fonseca, R., Dutta, P., Tavakoli, A., Culler, D., Levis, P., Shenker, S., Stoica, I.: Flush: A Reliable Bulk Transport Protocol for Multihop Wireless Networks. In: Proceedings of ACM SenSys (2007) 21. Dunkels, A.: Full TCP/IP for 8-Bit Architectures. In: Proceedings of ACM MobiSys (2003)

Dynamic Routing Algorithm for Priority Guarantee in Low Duty-Cycled Wireless Sensor Networks Guodong Sun and Bin Xu Department of computer science, Tsinghua University, Beijing, China {sgdcs,xubin}@tsinghua.edu.cn

Abstract. It is a new challenge to provide priority-based delivery in low duty-cycled sensor networks where there are not always-awake communication paths and the wireless links are very time-varying and unreliable. In this paper, we propose a Dynamic Routing Algorithm for priority Guarantee(called DRAG) in low duty-cycled sensor networks. Both schemes of dynamic forwarding decision making and priority-based schedule are used in DRAG to achieve priority guarantee in low dutycycled sensor networks. We evaluate DRAG via extensive simulations and the results show that DRAG can achieve good performance in delivery ratio and network delay.

1

Introduction

The nodes of wireless sensor networks [5,19] are battery driven and the replacement or recharge of battery is not an easy task for the sensor networks with thousands of nodes embedded physically in a large sensing area. The most important objective of wireless sensor networks is to extend the system lifetime as long as possible. To bridge the gap between the system lifetime requirement of applications and the constrained energy of nodes, one popular method of increasing longevity of wireless sensor networks is to duty cycle the nodes and let them sleep most of their operation time[17]. The duty cycle of nodes are often set to be 5%, 1%, or less. Even though low-duty-cycling the nodes can improve the system lifetime, the increased longevity comes at the cost of decreased application ﬁdelity: smaller delivery ratio and longer network delay that severely damage the performance of priority-critical applications such as movement tracking and alarm sensor networks. Two features make priority-based routing in low duty-cycled sensor networks more challenging. First, to transmit a packet, the node has to wait a long time until some forwarder wakes up. Second, the wireless link varies with time and is unreliable, leading to frequent retransmissions and impacting the system performance.

This work is supported in part by the NSF China grant 60803124 and the National Basic Research Program of China(973 Program) grant 2006CB303000.

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In this paper, we propose a Dynamic Routing Algorithm for priority Guarantee (called DRAG) in low duty-cycled sensor networks. DRAG aims at more quickly delivering as many high-priority packets as possible. Totally, the main contributions of this study are as follows. First, to the best of our knowledge, this is the ﬁrst work on the priority guarantee issue in low duty-cycled sensor networks with varying and unreliable links. Second, the proposed algorithm DRAG dynamically selects forwarders in a forwarding candidate set and maintains the link qualities at a low cost, based on which an optimal delay can be achieved. Finally, DRAG improves priority-based delivery in both levels of packet schedule and MAC back-oﬀ. Simulation results show that DRAG achieves good prioritybased delivery with high delivery ratio and small network delay. The rest of the paper is organized as follows. Section 2 describes the study background and motivation. Section 3 presents the system model and some related denotations and assumptions. The detailed design is described in section 4 followed by simulations in section 5. Section 6 brieﬂy summarizes some significant work related to this study. We conclude this work in section 7.

2

Background and Motivation

In wireless sensor networks, the nodes are of energy- and bandwidth-constrained. Thus a large amount of work have been proposed to optimize the energy usage in communication, but most of them assume always-available paths, i.e., there are always nodes ready to receive whenever there is data to be transmitted. But to maintain always-awake paths needs frequent idle listening, which consumes signiﬁcantly the energy of nodes. Raghunathan et al [14] found that for most nodes in WINS project, the energy consumption in idle listening mode is almost equal to that in the receiving mode. Additionally, the CC2420 RF chip requires a 19.7mA energy level to perform idle listening, but a 17.4mA energy level to transmit data [2]. Therefore, in order to eﬀectively reduce the energy consumed by idle listening, it is necessary for nodes to turn oﬀ their radios most of their time, i.e., operate under a low duty cycle. However, it is is more challenging to achieve good priority guarantee in low duty-cycled sensor networks. In many priority-critical sensor network scenarios, diﬀerent data sources may present information with diﬀerent importance. For example, in a movement-tracking application, the data generated nearby a moving object is more valuable in target localization; in a coal-mine monitoring application, the data describing collapse events obviously ought to be delivered in priori [3]. Even for a general sensor network application, the data incurred by some system functions such as network diagnosis [4] need possibly to be delivered in priority. In those applications, whether the data can be delivered timely and whether the high-priority data can take preference in delivery are very critical to the application ﬁdelity. We use simulation to investigate the channel contention and priority-based delivery in the low duty-cycled sensor network under two routing algorithms DSF [7] and DSPR(a dynamic shortest-path based routing algorithm which always select the earliest-awake forwarder without considering link qualities). In

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simulations, 200 nodes are uniform-randomly deployed within a 150×150 square with a single sink centered. The duty cycles of all nodes are 5% and the MAC layer works under a CSMA/CA-like scheme. The simulation results in Fig.1 show that channel contention can not be neglected even though only 6.6% nodes generate packets in a sensor network of 5% duty cycle because sensor networks are essentially of many-to-one communication. Note that the channel contention under DSPR is more frequent than that under DSF, because DSPR does not consider the link qualities of next-hop, resulting in more retransmissions and then more contention. Fig.2 plots three delivery ratios with high-, middle- and low-priority traﬃc, respectively, under DSPR(the simulation under DSF obtains similar results). For each duty cycle, those three delivery ratios seem like without obvious diﬀerentiation in terms of priorities. Analyzing the simulation results, we ﬁnd that (1) channel contention is easy to happen to sensor networks even with a very low duty cycle; (2) the varying, unreliable links make it hard to optimize the delivery ratio; and (3) a high-priority packet without a proper control may wait a long time until some forwarder wakes up, impacting the performance of the priority-critical applications. These observations motivate us to design a new routing algorithm for improving the priority guarantee in low duty-cycled sensor networks.

3

System Models

In this study, the duty cycles of nodes are decided directly by the upper layer such as power management or event coverage control [1], rather than by the MAC layer. The priority can be measured in terms of diﬀerent requirements, such like sensing data type, node location, remaining energy level, and so on. In this study, we generalize the priorities into positive integers. We consider a sensor network consisting of N nodes that are randomly deployed within a square in a uniform distribution. All duty-cycled nodes can be only in one of two states at a give time: awake or sleeping. In awake states, the node can sense, or receive, or send, or process data. A duty-cycled node can wake up for transmission when it has data ready to be

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sent, but it can sense or receive data only in awake states. Specially we assume the sink node is always awake. In this paper, the operation time of nodes are divided into equal-size slots each with a time span of τ (a time unit), and the awake states of each node are periodic due to the periodicity of sensing task requirement. So the node’s time can be divided into continuous working schedules. If the length of a single working schedule of a node is 100 slots and contains 10 awake slots, the node’s duty cycle is 10%. We assume that for each node, at most one data unit can be transmitted over one hop in a single slot. Fig.3 illustrates a 4-node network where every node is of duty cycle 20% with the working schedule length of 10 slots; the whole awake slots of A is {A1 , A8 , A11 , A18 , . . .} where Ai means A wakes up at slot i. Due to the periodicity, we abbreviate the awake slots of A as sA = {A1 , A8 }∗ . We can see that at time 0, node C generates or receives a packet which can be transmitted until time 2 when node D wakes up. The onehop delay from C to D equals to 2-0=2τ . Similarly, the two-hop delay from A to D is 8-1=7τ . In order to know neighbors’ working schedules exactly, local time synchronization is assumed. In this study, we use a CSMA/CA [6] like scheme to control the wireless medium access.

4

Design of DRAG

The basic idea of DRAG is to quickly deliver as many packets as possible meanwhile taking preference to higher-priority packets. DRAG consists of two components: (1) the dynamic selection of forwarders that makes the network deliver packets as soon as possible; and (2) the priority-based packet schedule in both levels of queue management and channel access that oﬀers more chances for higher-priority packets to be transmitted. In DRAG, each node dynamically maintains a set of forwarding candidates, from which the node selects a forwarder in terms of a delay-aware metric, and then schedules a packet with the highest priority. When there are multiple nodes contending a common channel, a priority-based back-oﬀ scheme in MAC layer is used. 4.1

Delay-Aware Routing Policy

Determination of Optimal Forwarding candidates. We ﬁrstly deﬁne by construction the concept of forwarding candidate set of a node and then describe the distributed method of calculating forwarding candidate sets of the node. Deﬁnition 1. Forwarding Candidate Set. First, calculate a union set v sv where v is any neighbor of u; and then sort ascendantly this union set by the wake-up time of neighbors, and then remove some elements from this union set such that all the remaining wake-up time is not earlier than t. Now the remaining set is called u’s forwarding candidate set at time t, denoted as Fu (t). For instance, at time t0 node X has three potential forwarders A, B and C that can relay packets for X; all their duty cycles are 2% where sA = {A2 , A40 }∗ , sB = {B1 , B60 }∗ and sC = {C50 , C80 }∗ . After ascendantly sorting sA ∪ sB ∪ sC

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by the wake-up time, we have FX (t1 ) = {B1 , A2 , A40 , C50 , B60 , C80 }∗ . Diﬀerent from the forwarding sequence deﬁned in [7], our forwarding candidate set allows the repeated selection of a forwarder, e.g., node A is selected two times in FX (t0 ). Suppose node X wants to transmit a packet to B at time t1 . If this packet is successfully received by B, X completes one transmission; otherwise, X can retry a transmission to some node waking up later than t1 , e.g., to A at time t2 or C at time t50 . Note that a maximum allowed transmission χ is often set by applications in order to save node’s energy and bandwidth. Thus we can reduce the inﬁnite set Fu (t) in Deﬁnition 1 into a ﬁnite set Fu (t) = {x1 , x2 , . . . , xχ }. Denote the expected delivery delay of node u as Eud . If no considering the unreliable links, for optimizing delay, u selects just a single forwarder v in Fu (t) which has the minimum (dv + Evd ) among all potential forwarders(dv is the time for u to wait v’s wake-up). But the earliest-awake link is not necessarily the 100% reliable one; the delivery ratios of forwarders impact Eud . We use a dynamic programming based method(introduced in [7]) to optimize the expected sourceto-sink delay of u; the following equations show the expression of Eud Eud = Eut =

m psucc (vi ) (dvi + Evdi ) t E u i=1 m

psucc (vi )Evt i

(1) (2)

i=1

where psucc (vi ) is the probability that u (re)transmits to vi successfully after i − 1 times failures and Eut is the delivery ratio of u. Since the forwarder with a less delay is not necessarily having high link quality, DRAG uses a dynamic programming based method to select an optimal subset Fuopt from Fu such that a minimum Eud can be achieved. In order to optimize Eud , we have to try every potential forwarder in Fu as the last node in the optimal subset, i.e., we have to calculate the expected delay for u with respect to any subset {x1 , x2 , . . . , xk }(1 ≤ k ≤ χ) and then ﬁnd the minimum expected delay value as Eud . For a given subset {x1 , x2 , . . . , xk }, we ﬁrst initialize a set Fu (k) with a empty set; we secondly insert xk into Fu (k); and next we examine every forwarder from xk−1 to x1 . If xi (1 ≤ i ≤ k−1) decreases Eud , then we insert xi into the front of Fu (k); otherwise we examine xi−1 . Finally, we select the Fu (k) resulting in the minimum Eud as the optimal forwarding candidate set Fuopt . We will not give the correctness proof due to the space constraint. Note that the expected delay per hop is timedependent. We calculate a Eud for each awake state of a node, because a node can receive data only when it wakes up. Therefore, once a node receives or generates a packet upon an awake slot t, it can calculate Fuopt according to the E d values of its forwarders that will wake up later than t. Since the packet needs not to be relayed again when they arrive at the sink, d t Esink is obviously equal to 0, and Esink = 1, both of which are the initial conditions of optimizing all nodes’ expected delay; and then in DRAG implementation, we use a ﬂooding algorithm based on Bellman-Ford scheme to

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distributively calculate Eut and then Eud for all awake slots of node u. The related implementation details are omitted due to the limited space. Link-quality Update. Note that the delay optimization in section 4.1 is based on such a fact that every node knows the link qualities of all its forwarding candidates. Consequently we need some link quality updating method to keep the optimal solution. However, updating link quality information needs extra communication overhead. In this section we present a piggyback-based link quality updating method. Technically, when a node receives a packet, it can extract the link quality information in terms of RSSI or LQI or PRR; and then the node can piggyback the link quality information, say LQI in message ACK. In this way, almost no extra costs of computation and communication are in need. The link quality update model for node u is shown in Eq.(3) lu,v (ti+1 ) = (1 − α)lu,v (ti ) + α lu,v (tack i )

(3)

where lu,v (ti ) represents the link quality between u and v maintained by u, lu,v (tack i ) the new link quality piggybacked in the ACK replied by v, lu,v (ti+1 ) the link quality to be used when a next packet will be sent, and α is used to describe the dynamic characteristic of radio links. Parameter α should be function of time. As indicated in [12], the wireless link quality changes continuously over time; larger α means quicker variation of links. 4.2

Priority-Based Schedule

The priority-based schedule which works on both levels of queue management and MAC layer is the other critical technique in DRAG to provide priority guarantee. Priority-based Queue Management. When a packet is received or generated by a node, it enters the node’s buﬀer queue to wait the chance to be sent. In a FIFO queue, a new-arriving packet is inserted at the end of queue, while the queue’s front packet is scheduled to be sent once the channel is seized. The FIFO queue management does not consider and diﬀerentiate the priorities of buﬀered packets. In DRAG, we use a simple, priority-based queue management policy to improve the priority guarantee. When a packet arrives at a node’s queue, the node checks the packet’s priority to determine an appropriate place relative to the previously sorted packets, i.e., the node always maintains a queue sorted in terms of packets’ priorities. Furthermore, a node always selects the highestpriority packet to send as long as the node obtains the channel. If a new-arriving packet makes a node overﬂow, the node will drop the lowest-priority packet in its queue. Priority-based Back-oﬀ. Besides using the priority-based queue schedule, another way to improve the priority guarantee is adopted in DRAG: the prioritybased back-oﬀ scheme. Diﬀerently than the above queue management, the

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priority-based back-oﬀ scheme employs the back-oﬀ operations in MAC layer to achieve the diﬀerentiated transmissions in terms of data priority. Before starting a transmission, a DRAG node needs ﬁrstly to check whether the channel is clear or not. If the channel is busy, i.e., there is data in propagation through this channel, the node can switch immediately to sleep without waiting until the current slot ends, because we have assumed that at most one transmission can be carried out in a slot. If a node, say u determines the channel is clear, it will back oﬀ a duration Tbk (u) calculated by Tbk (u) =

max Tbk + trand max{pu (i)}

1 ≤ i ≤ lQ (u)

(4)

where lQ (u) is the length of u’s queue, pu (i) is the priority value of the i-th packet in u’s queue, trand is a short duration randomly chosen within [0, Tr ], and max Tbk the maximum back-oﬀ time set by applications. By Eq.(4), the node with the highest-priority packet will seize the channel. The parameter trand is used to avoid multiple simultaneous transmissions of packets with the same priority. Note that the back-oﬀ scheme in DRAG delivers high-priority packets in priori but can not avoid entirely the dead lock of low-priority packets in any scenario. One potential solution is to dynamically update the packet priority in delivery according to the travel time or waiting time of packets. But the priority setting upon generation and updating in delivery will be our future work.

5

Simulation

In this section we simulate to evaluate the performance of our work. In our simulation, we focus mainly on two performance metrics as follows: (1) Network Delay. It is measured as the average per-hop delay of a packet traveling for its source to the sink. (2) Priority-diﬀerentiated Delivery Ratio. It is measured as the delivery ratio of packets with diﬀerent priorities. We compare the performance of DRAG with DSF under diﬀerent duty cycles and packet generation rates. 5.1

Simulation Setup

In simulation, 200 nodes are randomly deployed with a uniform fashion in a 150 × 150m2 square, and a single sink centers the square. The slot length τ , max Tbk and Tr are set to be 20ms, 5ms and 5ms, respectively. and each working schedule is of length 100τ . Each simulation runs 105 τ ; and the maximum allowed retransmissions is set to be 4. To simulate faithfully dynamic radio links, we use the CC1000-based radio model described in [20]. It is shown in each simulation that the average number of neighbors per node is about 12 and the minimum-hop count from any node to the sink is 3.5 in average. The simulation under each conﬁguration repeats 20 times with diﬀerent random topologies and the data is reported with the 95% conﬁdence.

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5.2

Performance Comparison

For DRAG and DSF, we evaluate the priority-diﬀerentiated delivery ratio, perhop delay and retransmissions under diﬀerent duty cycles and packet generation rates. The energy eﬃciency of DRAG will be examined in the future work. Diﬀerent Duty Cycles. The packet generation rates are all 0.01 in the following simulations. And each source node assigns any packet with one of three priorities(high, middle and low) with the probability of 0.5, 0.3 and 0.2, respectively. Fig.4 plots the average number of retransmissions of two algorithms under diﬀerent duty cycles. With the duty cycle increasing, the traﬃc load increase and then brings more contention. The numbers of retransmissions of duty cycles 4 and 5 under DRAG are both less than those under DSF, because DSF does not immediately update link qualities to maintain a optimal solution. Fig.5 shows that the per-hop delay under each algorithm decreases as the duty cycle of nodes increases from 1% to 5%. And it is seen that DRAG achieves the per-hop delay of 4 time unit smaller than DSF when the duty cycle is 5%. Considering the average delivery hop of 3.5, the network delay under DRAG is 13 time units smaller than that under DSF. These results illustrate the DRAG’s advantage of delay over DSF because of the dynamic forwarding candidate selection and link update scheme in use. The simulation in Fig.6 evaluates the performance of priority guarantee under two algorithms. DSF, not using any priority guarantee policy delivers all packets with equal likely, while the stacked histograms in Fig.6(b) show that almost all the high-priority packets generated with probability of 0.5 under DRAG can be sent to the sink. As the duty cycle of nodes increases, the delivery ratio under each algorithm decreases a little. Due to the usage of a low traﬃc load(0.01 packet generation rate), the total delivery ratio under both algorithms seems similar. Diﬀerent Packet Generation Rates. In the simulations with diﬀerent packet generation rates, let the duty cycles of all nodes be 5%. In each simulation, 20 nodes are randomly chosen as the sources, and each source generates a packet with a probability when it wakes up. The priority assignment is identical to that in section 5.2.

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Fig.7 shows that the average number of retransmissions under DRAG is slightly less than that under DSF. Fig.8 shows that the per-hop delay under DSF and DRAG both increase as the packet generation rate increases. And when the packet generation rate is beyond 0.6, the per-hop delay under DRAG is clearly smaller than DSF. At the packet generation rate of 0.7, the per-hop delay of DRAG is 32 time units smaller than DSF. Fig.9 compares the prioritybased delivery as well as the total delivery ratio. Similar to Fig.6, DRAG also achieves a good priority guarantee in packet delivery. Specially, it can be seen from Fig.9 that DSF drops more packets than DRAG. It is further to be illustrated that the schemes of link update and priority-based packet schedule used in DRAG contribute to the higher total and the priority-based delivery ratio.

6

Related Work

As so far, there are two major kinds of designs for low duty-cycled sensor networks: MAC and routing algorithms. SCP[18] is a MAC scheme designed for ultra-low duty cycle(0.1%) sensor networks. SCP employs LPL(low-power listening) technique and extends the system lifetime of a sensor network by a factor of 3 to 6 times. Kim et al [11] investigated the priority-based service diﬀerentiation scheme for 802.15.4 sensor networks by adjusting the contention window size and the back-oﬀ exponent according to diﬀerent priority requirement. We will next brieﬂy summarize some major work related to our algorithm. SPEED [10] includes three types of services: realtime unicast, realtime areamulticast and realtime area-anycast; and it uses end-to-end feedback control and non-deterministic QoS-aware geographic forwarding to achieve realtime communication. Lu et al [13] design to reduce the delivery delay by using energy-eﬃcient periodic sleep schedule for sensors. Seada et al [15] investigate the geographic routing in sensor networks and show that the product of PRR and the hop distance can be used to select energy eﬃcient paths. Three performance metrics has been proposed in DSF[7]: EDR(expected delivery ratio), EED(expected end-to-end delay) and EEC(expected energy consumption). DSF uses dynamic programming based methods to respectively optimize the above three metrics. We get the heuristic from DSF in performance optimization. But we extend

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DSF by employing repeated nodes in forwarding candidate set and introducing a link update scheme, both of which further reduce the per-hop delay. Moreover, DSF examines the cases only with a few sources; the consequent results can not reveal some common communication problems such as channel contention or queue build-up, each of which aﬀects the performances of system and application. Guo et al [9] propose an opportunistic ﬂooding scheme for low dutycycled sensor networks which reduces the transmission redundancy and achieves fast dissemination. Similarly, ADB, another ﬂooding algorithm for asynchronous duty-cycled sensor networks was designed in [16].

7

Conclusions

In this study, we propose a dynamic routing algorithm, called DRAG for improving the priority-guarantee in low duty-cycled sensor networks. This work considers both the low duty cycle of nodes and the varying, unreliable wireless links. In DRAG design, each node forwards packets based on a dynamic forwarding candidate selection and two priority-based schedule schemes. Extensive simulations with a CC1000-based radio model are carried out to evaluate our algorithm. The simulation results show that DRAG achieves good performance over DSF in priority-based delivery ratio and per-hop delay.

References 1. Cardei, M., Thai, M.T., Li, Y., Wu, W.: Energy-eﬃcient target coverage in wireless sensor networks. In: INFOCOM, Miami, USA (2005) 2. CC2420: http://www.ti.com 3. Li, M., Liu, Y.: Underground coal mine monitoring with wireless sensor networks. ACM Trans. on Sensor Networks 5, 2 (2009) 4. Liu, K., Li, M., Liu, Y., et al.: Passive diagnosis for wireless sensor networks. In: SenSys, Raleigh, USA, pp. 113–126 (2008) 5. Culler, D., Estrin, D., Srivastava, M.: Overivew of sensor networks. IEEE Computer Magazine (2004) 6. Gast, M.S.: 802.11 wireless networks: the deﬁnitive guide, 2nd edn. OReilly, Southeast University Press, Nanjing (2006) 7. Gu, Y., He, T.: Data forwarding in extremely low-duty-cycle sensor networks with unreliable communication links. In: SenSys, Sydney, Australia, pp. 321–334 (2007) 8. Gu, Y., He, T., Lin, M., Xu, J.: Spatiotemporal delay control for low-duty-cycle sensor networks. In: RTSS (2009) 9. Guo, S., Gu, Y., Jiang, B., He, T.: Opportunistic ﬂooding in low-duty-cycle wireless sensor networks with unreliable links. In: SenSys, pp. 133–144 (2009) 10. He, T., Stankovic, J., Lu, C., Abdelzaher, T.: Speed: a real-time routing protocol for sensor networks. In: ICDCS, USA, pp. 46–55 (2003) 11. Kim, E.J., Kim, M., Youm, S.K., Choi, S., Kang, C.H.: Priority-based service diﬀerentiation scheme for ieee 802.15.4 sensor networks. International Journal of Electronics and Communications 61, 69–81 (2007) 12. Lin, S., Zhang, J., Zhou, G., Gu, L., He, T., Stankovic, J.A.: Atpc: adaptive transmission power control for wireless sensor networks. In: SenSys, pp. 223–235 (2006)

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13. Lu, G., Sadagopan, N., Krishnamachari, B., Goel, A.: Delay eﬃcient sleep scheduling in wireless sensor networks. In: INFOCOM, Miami, USA (2005) 14. Raghunathan, V., Schurgers, C., Park, S., Srivastava, M.: Energy-aware wireless microsensor networks. IEEE Signal Processing Magazine 19, 40–51 (2002) 15. Seada, K., Zuniga, M., Helmy, A., Krishnamachari, B.: Energy-eﬃcient forwarding strategies for geographic routing in lossy wireless sensor networks. In: SenSys, Baltimore, USA, pp. 108–119 (2004) 16. Sun, Y., Gurewitz, O., Du, S., Tang, L., Johnson, D.B.: Adb: an eﬃcient multihop broadcast protocol based on asynchronous duty-cycling in wireless sensor networks. In: SenSys, pp. 43–56 (2009) 17. Ye, W., Heidemann, J., Estrin, D.: Medium access control with coordination adaptive sleeping for wireless sensor networks. IEEE/ACM Trans. on Networking 12, 493–506 (2004) 18. Ye, W., Silva, F., Heidemann, J.: Ultra-low dudty cycle mac with scheduled channel polling. In: SenSys (2006) 19. Yick, J., Mukherjee, B., Ghosal, D.: Wireless sensor network survey. Computer Networks 52, 2292–2330 (2008) 20. Zuniga, M., Krishnamachari, B.: Analyzing the transmissional region in low power wireless links. In: IEEE SECON, pp. 517–526 (2004)

Heterogeneity of Device Contact Process in Pocket Switched Networks Ye Tian1,2 and Jiang Li3 1

Anhui Province Key Laboratory on High Performance Computing, 2 School of Computer Science and Technology, University of Science and Technology of China, Hefei, Anhui 230026, China [email protected] 3 Department of Systems and Computer Science, Howard University, Washington DC 20059, USA [email protected]

Abstract. Understanding device pair’s contacts is essential in pocket switched networks (PSN). However, most of the studies on this issue are focused on the empirical distribution aggregating inter-contact times from all the device pairs, and seeking to ﬁnd common characteristics of their contact processes. In this paper, we present an insightful analysis on both the aggregated and the pair-wise inter-contact times obtained from three real-world datasets. We ﬁnd that device pairs are heterogeneous in many aspects, including not only their contact frequencies, but also their contact patterns. More surprisingly, even for those frequently contacting pairs, their behaviors are diverse, and could not be described with a universal model. Finally, implication of the observed heterogeneity on PSN’s message forwarding algorithm is discussed, and we show that with the awareness of the device pair’s heterogeneous contact pattern, the network’s message relaying service could be improved considerably. Keywords: Delay tolerant networks, pocket switched networks, intercontact times, message forwarding algorithm.

1

Introduction

With the advance of wireless technologies and prevalent use of portable wireless devices, in recent years, the idea of pocket switched network (PSN), which is a special case of the delay tolerant network (DTN), has been proposed (e.g. the Haggle project [1]). In PSN, portable wireless devices such as cell phones and PDAs carried by human beings form an ad-hoc network. In such a network, contacting among devices is the only opportunity for communication, therefore, it is highly important for people to understand the device pair’s contact pattern, especially the intervals between their consecutive contacts (referred to as

This work is funded by the Specialized Research Fund for the Doctoral Program of Higher Education of China 20093402120020, and was funded in part by US NSF grant CNS-0832000 and the Mordecai Wyatt Johnson Program at Howard University.

G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 157–166, 2010. c Springer-Verlag Berlin Heidelberg 2010

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inter-contact times). Previous studies [2][3] on this topic are mainly based on the empirical distribution aggregating inter-contact times of all the device pairs, with an assumption that the aggregated distribution could represent individual pair’s contact process. However, [4] suggests that device pairs are in fact heterogeneous regarding their contacting behavior. In this work, we study both the aggregated inter-contact time distribution in percentiles and the individual pair’s distribution.Three real-world datasets with mobility and contact information of large numbers of portable wireless devices are exploited to identify and understand the heterogeneity of device pairs’ contact processes. Finally, we discuss the implication of the observed heterogeneity on the design of the PSN’s message forwarding algorithms. The remainder part of this paper is organized as follows: in Section 2, related works are surveyed and their relationships with our work are discussed; in Section 3, real-world datasets are analyzed, in particular, the aggregated distributions of the device pairs’ inter-contact times are studied in percentiles; we investigate and classify frequently contacting pairs, and give some interpretations for the heterogeneity observed in Section 4; in Section 5, the implication of our observation is discussed; ﬁnally in Section 6, we conclude this paper and discuss the future work.

2

Related Work

Our work focuses on studying the contact pattern of device pairs in PSN networks, and we are especially interested in pair’s inter-contact time between consecutive contacts. Previous studies on this topic are mostly based on the empirical distribution aggregating the inter-contact times from all the device pairs. In [2], the authors study aggregated distributions from a number of real-world datasets, and ﬁnd that the inter-contact time follows power law and could be modeled with a truncated Pareto distribution. This ﬁnding contradicts the assumption in many works (e.g. [5]) that a pair’s contact process is Poisson with exponential inter-contact times. In [3], it is reported that a dichotomy exists in the aggregated inter-contact time distribution: the distribution is power law in certain range, but it has an exponential tail. A random walk model with inﬁnite sites is used to explain the observed power-law inter-contact time. While in [6], a random walk model in an unbound domain is applied for the same purpose. This work diﬀers with these works in that we only study the inter-contact time distribution of an individual pair or the aggregated distribution of pairs in a small group, and we interpret our ﬁndings with well-founded theories and widely recognized observations. Only a few works address pair-wise inter-contact times. In [7], the authors use the Dartmouth dataset for analyzing device pair’s contact process, and draw a conclusion that for majority of the device pairs, their inter-contact times are exponentially distributed. In [4], it is shown that the mean inter-contact time is heterogeneous, and the inter-contact times for most of the pairs could be described with a lognormal distribution. Our work diﬀers from these previous

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works in that we classify the pairs based on their contact patterns and only consider the distributions backed up with theoretical or experimental supports for describing their inter-contact times, and we also discuss the implication of our observations on the design of PSN’s message forwarding algorithm.

3

A First Look at the Contact Process

For our study, we select three datasets containing long-time mobility and contact information of a large number of wireless devices, which are the dataset from the MIT Reality Mining project [8], the Dartmouth dataset [9], and the dataset from UCSD Wireless Topology Discover project [10]. For the remainder part of this paper, we simply refer to them as Reality, Dartmouth, and UCSD respectively. Among the three datasets, devices contacts were recorded in Reality, while for Dartmouth and UCSD, we considered a contact between two devices happened if they were associated with a same AP simultaneously, as assumed in the previous works[2][3]. Table 1. Pairs and contacts in diﬀerent percentiles Percentile Reality Dartmouth UCSD

0-10 26.77% 0.79% 3.83%

10-20 23.88% 15.30% 23.90%

20-30 15.98% 32.86% 22.40%

30-40 11.01% 21.70% 15.25%

40-50 7.36% 12.78% 9.31%

50-60 5.18% 6.71% 7.64%

60-70 3.57% 4.72% 4.55%

70-80 2.93% 2.67% 4.62%

80-90 2.06% 1.61% 4.87%

90-100 1.26% 0.84% 3.63%

By investigating the three datasets, we are trying to answer the question that whether or not the contact processes for all the device pairs could be viewed homogenous, and if not, in what aspects the heterogeneity exists. For this purpose we study the aggregated distribution of the pairs’ inter-contact times. However, unlike previous works (e.g., [2] and [3]), we do not take all the device pairs into consideration, but group them according to their contact frequencies and study inter-contact times within each group. Concretely, we sort all the pairs in an ascending order regarding their mean inter-contact times, and group the pairs in each ten percentiles. For example, by denoting the percentile group of “10-20”, we mean the pairs with their mean inter-contact times between the ﬁrst 10 and the ﬁrst 20 percents among all the pairs in this order. We list in Table 1 the percentage of the contacts made by each group for the three datasets. One can see that in these datasets, some percentile groups make much more contacts than other groups. Moreover, it is observed that the contact number is decreasing very sharply in groups with lower percentiles for each dataset, but the decreasing becomes smooth in higher percentiles. This observation indicates that we could roughly categorize these device pairs into frequently contacting pairs and infrequently contacting ones, and majority of the contacts are made by the former. For example, pairs in the three most contacting percentile groups contribute 66.63%, 69.86%, and 61.55% of the total contacts in Reality, Dartmouth, and UCSD respectively.

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In Figure 1, we plot the aggregated inter-contact time distribution in complementary cumulative distribution function (CCDF) for pairs in each percentile group of the three datasets. Please note that for percentiles between 0-50, we use the log-log scale, as in Figure 1(a-c), while for percentiles between 50-100, we use the linear-log scale, as in Figure 1(d-f). We also plot the aggregated distribution of all the pairs for comparison. From these ﬁgures, we ﬁnd that for pairs in diﬀerent percentiles, they diﬀer not only in their contact frequencies, but also in the shape of the empirical distributions of their inter-contact times: From Figure 1(d-f) one could see that the distribution curves are approximated straight lines in the linear-log scale, suggesting that the inter-contact times are exponential. On the other hand, Figure 1(a-c) show that the aggregated distributions of the inter-contact times have near straight-line curves within the delays shorter than one day under the log-log scale, indicating that the distributions are more power-law like. The observation of the distribution shapes suggests that pairs in diﬀerent percentile groups may have heterogeneous contact patterns in addition to their contact frequencies. Summarizing our ﬁndings from Table 1 and Figure 1, we can conclude that the device pairs in all the three datasets are highly heterogeneous in at least two aspects: ﬁrst, we ﬁnd that contact frequencies of the device pairs are heterogeneous, for example, Table 1 shows that more than 60% of the contacts are made by only 30% of the most contacting device pairs; more importantly, we ﬁnd that pairs have diﬀerent contact patterns, where the inter-contact times for pairs in lower percentiles are more power-law like, while the inter-contact times for pairs in higher percentiles are more exponential. The observed heterogeneity suggests that it may be inappropriate to use a universal contact process model based on the aggregated inter-contact time distribution for interpreting all the pairs’

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behaviors. On the other hand, as we have seen that a small part of frequently contacting pairs contribute majority of the contacts, it is highly important for people to understand these pairs. With focus narrowed on these frequently contacting pairs, a natural question arises as whether or not these pairs’ behaviors are same. We will answer this question in the next section.

4 4.1

Contact Process of Frequently Contacting Pair Statistical Analysis

In this section, we focus on the frequently contacting device pairs and try to understand their behaviors. For the ﬁrst step of our study, we ﬁlter out those infrequently contacting pairs with a threshold. That is, we only consider the pairs with their contacts more than the threshold. We then use the CramerSmirnov-Von-Mises test[11] to study the frequently contacting pairs ﬁltered out. The Cramer-Smirnov-Von-Mises test is a statistical method to testify whether or not the sampled data is compatible with a given distribution function. In our test, the sampled data is the inter-contact times of the pair under study, and we consider the following candidate distribution functions. – Exponential distribution: For this candidate, we use a CDF function as F (x) = 1 − e−λij x , x ≥ 0 where λij is a constant contact rate between the two devices of the pair. – Pareto distribution: For this candidate, we use a CDF function as F (x) = 1 −

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where βij is the minimum inter-contact time observed for this pair and αij is the slope rate of the empirical CDF curve in the log-log scale. For obtaining the parameter λij of the empirical exponential distribution, we simply let λij be the inversion of the mean inter-contact time. While for the parameters of the empirical Pareto distribution, we use the maximum likelihood estimator of αij [12] as n α ˆ = n xi i=1 ln β where n is the number of the samples and xi is the ith data sample. The reason that we choose the exponential and Pareto distributions is that they are either supported by well-known theory or widely observed in similar activities. For the exponential distribution, it is well known that if a node’s location process is independent, stationary and ergodic, the long-term contact process between any two nodes is Poisson[5]. Moreover, it is proved that the exponential inter-contact time could be generated with popular mobility models such as the

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Random Way Point model and the Random Direction model [13]. On the other hand, recent studies show that for many human activities, the inter-event time is heavy-tailed and follows a Pareto distribution. These activities include composing emails[14], visiting websites[15], responding surface mails[16], and performing ﬁnancial transactions[17]. Although human contacting is diﬀerent from these activities, however, it is highly possible that there are some similarities, therefore we also consider the Pareto distribution as our candidate distribution function. The Cramer-Smirnov-Von-Mises test is a statistical method which compares the sampled data with the hypothetic candidate distribution, and based on a rejection level α, it returns a result of positive or negative on whether or not the sampled data is compatible with the candidate distribution. Here the rejection level α is the probability of the test to make false positive errors. As we have two candidate distributions, there are two tests for each pair, which are referred to as the exponential test and the Pareto test respectively, and for each test we will have two results. We categorize the pairs into four types based on their testing results: Type 1 pairs are whose pairs that have passed only the Pareto test; 2) Type 2 pairs are those passing only the exponential test; Type 3 pairs which have passed both tests; and 4) Type 4 pairs that have passed neither of the tests. As we are trying to categorize pairs’ behavior patterns, the rejection level α must be selected carefully: if α is set too large, the tests are very loose, and many pairs will pass both tests, suggesting that we are actually failed to diﬀerentiate them; if α is too small, the tests are very selective, and we may ﬁnd many pairs failing to pass any test just because of the raw nature of the data samples. In our study we choose α = 0.01 for Reality and Dartmouth, and α = 0.05 for UCSD. We also change the threshold of the contact number for each dataset for ﬁltering out infrequently contacting pairs, and show the test results in percentages of the pairs falling in each type for the three datasets in Figure 2. From the ﬁgure, one can see that for each dataset, when the threshold is small, there are considerable numbers of pairs falling in all the four types; however, when the threshold gets increased, Type 3 pairs are ﬁltered out rapidly. Recall that Type 3 pairs are the pairs passing both tests, it is reasonable to consider these pairs as infrequently contacting pairs, as they pass both tests simply because there is no suﬃcient sampled data for the tests.

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We then focus on the other three types, i.e., Type 1, 2, and 4. From the ﬁgure one can see that each type has a persistent portion under test, and the pairs in the three types are not easily ﬁltered out by increasing the threshold. Based on this observation, we could consider the Type 1, 2, and 4 pairs as representative frequently contacting pairs. In other words, we could categorize the frequently contacting device pairs into the three types, i.e. pairs with exponential intercontact time, pairs with Pareto inter-contact time, and pairs not belonging to the above two. Reality, Threshold = 30

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To have a further understanding of the Type 1, 2, and 4 pairs, we also estimate a pair’s contact rate by averaging its inter-contact times and taking the inversion. The distribution of the contact rates for each type of all the three datasets are plotted in Figure 3. From the ﬁgure one can see that although pairs of all the three types are considered as frequently contacting, there are still some diﬀerences: Type 1 and Type 4 pairs contact obviously more frequently than Type 2 pairs. We also study the empirical Pareto distribution’s slope rate for Type 1 pairs in the three datasets, and ﬁnd that all the slope rates are smaller than 1 and are concentrating around 0.3 ∼ 0.5. 4.2

Discussion and Interpretation

From the above statistic test results, it is implied that there are at least two contact patterns for the frequently contacting pairs, i.e. the contact pattern producing Pareto inter-contact time and the one producing exponential time. As shown in [13], exponential inter-contact time could be caused by device’s independent, stationary, and ergodic location process, therefore we mark these contacts as “unintended”, as this kind of contact is a byproduct of a node’s independent visiting to some locations. For example, neighbors at home or working place may often make “unintended” contacts, but people do not make these contacts on purpose. For this reason, we refer to the relationship between the two parties of a pair producing exponential inter-contact time as “familiar strangers”. While for the Pareto inter-contact time, we believe they are caused by similar reasons as the Pareto inter-event time observed in many human activities, such as the task priority[14] and the human interest[18]. In other words, we believe that

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the Pareto inter-contact times could also be explained with objective reasons of human beings, as human has the interest or urgency to perform the contact on purpose. Therefore we could mark the contacts with Pareto inter-contact time as “intended” contacts and could refer to the relationship between the two parties of such pair as “friends”. Finally, for Type 4 pairs, we explain that these pairs are both “familiar strangers” and “friends”, and their contacts are a combination of the two types of the contacts, but none of them dominates. Finally, we observe that Type 1 pairs contact more frequently than Type 2 pairs, which conforms to the intuition that close friends meet more often than familiar strangers.

5

Implication on PSN Message Forwarding

From the statistical studies in the above section, we ﬁnd that frequently contacting device pairs may have their inter-contact times following exponential or Pareto distribution. It is well known that for exponential inter-contact times, a pair’s contacting frequency indicates its expected next contacting time, due to the memoryless property of the exponential distribution. However, for the Pareto inter-contact times, as the distribution has memory, a device pair that has just contacted is likely to make a contact in near future, while a pair that contacted long ago is not likely to make a contact very soon. Therefore, to predict how soon a pair of devices may contact, we should consider the last recent contact age for the Pareto inter-contacting device pairs as well as the contact frequency for the exponential inter-contacting one. To testify our point, we simulate a PSN network using the UCSD dataset, and examine the success ratios and delivery delays of the network’s message delivery jobs when nodes apply diﬀerent forwarding strategies. Figure 4 shows that when both the node pair’s contact frequency and the last contact age are considered (corresponding to 0 < ρ < 0 in the ﬁgure), performance of PSN’s message delivery job is better than the cases when only the contact frequency (corresponding to ρ = 0 in the ﬁgure) or the last contact age (corresponding to ρ = 1 in the ﬁgure) is concerned. Please note that when ρ = 0, it is exactly

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the “FRESH” algorithm proposed in [19], and for ρ = 0, it is the “Greedy” algorithm studied in [20]. Moreover, as we have categorize the node pairs into “familiar strangers” and “friends”, and classify the contact processes into “intended” and “unintended”, if the relationship of the node pair, or even more, the type of the contact process is available, PSN’s message forwarding algorithms should make much more accurate predictions in selecting the next hop for message forwarding. Unfortunately, although we have statistically identiﬁed the types for some node pairs and contact processes of the datasets studied in this work, precisely knowing the exact type of each node pair and contact process is still infeasible using any existing datasets, therefore we are unable to examine the exact beneﬁts brought by the awareness of the node pair relationship and contact process type experimentally.

6

Conclusion and Future Work

In this paper, we study three datasets containing contact and mobility information of wireless devices, for understanding the device contact process under the context of PSN networks. We ﬁrst group pairs into diﬀerent percentiles based on their contact frequencies and study their aggregated inter-contact time distributions. We ﬁnd that pairs are heterogeneous in many aspects, including their contact frequencies as well as their contact patterns. We then study the pair-wise inter-contact times, and ﬁnd that even for the frequently contacting pairs, they are behaving diversely. We categorize the frequently contacting pairs into three types by using the Cramer-Smirnov-Von-Mises test, and apply diﬀerent theories for interpreting their diﬀerent contact patterns. Furthermore, we discuss the implication of our observation on the contact processes for the PSN message forwarding algorithm, and show that with the awareness of the heterogeneity, better performance on PSN’s message delivery should be expected. Our future work is in two directions. First, more insightful studies are required to meet the gaps between the empirical observation and the theoretical model, such as explaining the Type 4 pairs. More importantly, we need to work on further exploiting the inferred contact pattern for solving critical problems, such as routing, security, resource management, and quality of service, on PSN networks. For example, “friends” type node pairs should be considered more reliable in some mission critical applications.

References 1. Haggle Project, http://www.haggleproject.org 2. Chaintreau, A., Hui, P., Diot, C., Gass, R., Scott, J.: Impact of human mobility on the design of opportunistic forwarding algorithms. In: Proc. of IEEE INFOCOM 2006, Barcelona, Spain (April 2006) 3. Karagiannis, T., Boudec, J.Y.L., Vojnovic, M.: Power law and exponential decay of inter contact times between mobile devices. In: Proc. of ACM MobiCom 2007, Montreal, Canada (September 2007)

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4. Conan, V., Leguay, J., Friedman, T.: Characterizing pairwise inter-contact patterns in delay tolerant networks. In: Proc. of ACM Conference on Autonomic Computing and Communication Systems, Rome, Italy (October 2007) 5. Grossglauser, M., Tse, D.N.C.: Mobility increases the capacity of ad hoc wireless networks. IEEE/ACM Trans. Networking 10(4), 477–486 (2002) 6. Cai, H., Eun, D.Y.: Crossing over the bounded domain: from exponential to powerlaw inter-meeting time in manet. In: Proc. of ACM MobiCom 2007, Montreal, Canada (September 2007) 7. Conan, V., Leguay, J., Friedman, T.: The heterogeneity of inter-contact time distributions: its importance for routing in delay tolerant networks (arXiv:cs/0609068v2 [cs.NI]) (January2007) 8. MIT Reality Mining Project, http://reality.media.mit.edu 9. CRAWDAD, http://crawdad.cs.dartmouth.edu 10. UCSD WTD Project, http://sysnet.ucsd.edu/wtd 11. Eadie, W.T.: Statistical Methods in Experimental Physics. Elsevier Science, Amsterdam (1983) 12. Asrabadi, B.R.: Estimation in the pareto distribution. Physica 37(1), 199–205 (1990) 13. Groenevelt, R., Nain, P., Koole, G.: The message delay in mobile ad hoc networks. Perform. Eval. 62(1-4), 210–228 (2005) 14. Barabasi, A.L.: The origin of bursts and heavy tails in human dynamics. Nature 435, 207–211 (2005) 15. Dezso, Z., Almaas, E., Lukacs, A., Racz, B., Szakadat, I., Barabasi, A.L.: Dynamics of information access on the web. Phys. Rev. E 73, 066132 (2006) 16. Oliveira, J.G., Barabasi, A.L.: Human dynamics: the correspondence patterns of Darwin and Einstein. Nature 437, 1251 (2005) 17. Plerou, V., Gopikrishnan, P., Amaral, L.A.N., Gabaix, X., Stanley, H.E.: Economic ﬂuctuations and anomalous diﬀusion. Phys. Rev. E 62, 3023–3026 (2000) 18. Han, X.P., Zhou, T., Wang, B.H.: Modeling human dynamics with adaptive interest. New J. Phys. 10, 073010 (2008) 19. Dubois-Ferriere, H., Grossglauser, M., Vetterli, M.: Age matters: eﬃcient route discovery in mobile ad hoc networks using encounter ages. In: Proc. of ACM MobiHoc 2003, Annapolis, MD, USA (June 2003) 20. Erramilli, V., Chaintreau, A., Crovella, M., Diot, C.: Diversity of forwarding paths in pocket switched networks. In: Proc. of ACM IMC 2007, San Diego, CA, USA (October 2007)

Delay Minimization of Tree-Based Neighbor Discovery in Mobile Robot Networks Heejun Roh, Kyunghwi Kim, and Wonjun Lee Dept. of Computer Science and Engineering, Korea University, Seoul, Korea [email protected]

Abstract. In this paper, delay minimization schemes for tree-based neighbor discovery in mobile robot networks are proposed and analyzed. Depending on the tree construction scheme, the expected value of neighbor discovery delay is changed. In our study, we focus on M -ary and M -Binary tree-based neighbor discovery. Regarding the number of neighboring robots, M -ary tree-based neighbor discovery has low but steady performance whilst M -Binary tree-based neighbor discovery shows better performance for optimal M . The simulation results provide performance comparisons of these schemes.

1

Introduction

In a mobile robot network, as a promising application of mobile ad hoc network, the reason that each mobile robot needs the information of neighboring robots is twofold. First, for the robots to perform complex and various tasks cooperatively and eﬃciently, information of one-hop neighbor robots is required. Second, because generally the network has no infrastructure, it is essential to collect the information for conﬁguring the network environment. However, explicit neighbor discovery may not be performed in typical ad hoc networks currently [1]. That is, it is assumed that all transmissions are listened to by neighboring robots in the link layer level. This assumption is not suitable for practical network implementation, because it is energy-ineﬃcient and robots in the network may be assumed to operate with batteries. To reduce energy consumption, energy-eﬃcient explicit neighbor discovery should be considered. Moreover, because the number of follower robots may be changed frequently, an eﬃcient neighbor discovery scheme is required, which implies that average delay minimization of neighbor discovery schemes in mobile robot networks is an important problem. Recently, numerous studies on neighbor discovery for mobile robot networks have been proposed in the literature [2], [3], [4]. In F. Santos’ work [2], an adaptive TDMA protocol for mobile autonomous robots is proposed. However, it does not support a dynamic number of robots. The improved protocol proposed in [3] has dynamic reconﬁguration of the TDMA round, which supports changes of

The corresponding author.

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the number of robots. But it is not suitable to our leader-follower model assumption, because each robot in a group maintains a state machine and the whole member information of the group. J. Arai [4] proposed an Adaptive ReservationTDMA (AR-TRMA) MAC protocol for real-time robot network which supports dynamic time slot allocation schemes during robot intercommunication and the packet collision avoidance method in the joining procedure. However, when a packet collision occurs, the protocol uses ﬁxed-size mini-slots to resolve the collision. This scheme suﬀers from lack of scalability, because it may not support dynamic situations of networked mobile robot systems, which include a situation that a large number of robots and groups move around frequently. With this motivation, we have studied tree-based neighbor discovery schemes in mobile robot networks, which guarantee the discovery of whole neighbors in each leader robot. In this study, our objective is to reduce the average delay of tree-based neighbor discovery. According to our study, the delay depends on the tree construction scheme. We focus on M -ary and M -Binary tree-based neighbor discovery, because these algorithms have simplicity and optimality, respectively. In summary, our contributions are two-fold. First, we apply tree algorithm variations to neighbor discovery. To the best of our knowledge, our study is the ﬁrst work applying tree algorithms to neighbor discovery in mobile robot networks. Second, we show that the M -Binary tree-based scheme with optimal M is suitable for the neighbor discovery process.

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We assume that mobile robots are uniformly distributed in a ﬁeld and the carrier sensing range is twice the transmission range. This model consists of the leader robot (LB) which manages and controls a group of robots, follower robots (FBs) which are in the group of robots, neighboring robots (NBs) which is not a FB and can communicate with a LB in 1-hop, and non-neighboring robots. Note that a LB has the number of required robots for performing a cooperative task. A group following a LB uses TDMA with a channel distinct from the default channel. Because the frame structure of TDMA guarantees a delay bound and transmission opportunities, each robot of the group has a chance to transmit its packet fairly. Fig. 1 shows an example of a TDMA frame. In this example, LB broadcasts a beacon message in a beacon period. In a joining period, LB requests its NBs to join the group under the number of required robots via the proposed schemes. LB broadcasts a control message containing the FBs’ data period schedules during a control period, and each of the follower robots can send its own data according to the schedule. Neighbor discovery is performed in the joining period. Depending on the neighbor discovery scheme, the time interval of the joining period may be variable, which causes a signiﬁcant delay. Therefore, a joining scheme to minimize the total expected delay in the period is required. In this paper, it is called the Joining Delay Minimization Scheme (JDMS), and in the next section, two tree-based JDMSs are discussed.

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Fig. 1. An example of TDMA frame structure

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Tree-Based JDMSs

In this section, M -ary tree-based JDMS (M T-JDMS) and M -Binary tree-based JDMS (M -BT-JDMS) are proposed, where M is the maximum number of children nodes (branches) in a tree diagram. Before the speciﬁc description of each scheme, we explain the common description. First, assume all neighbor nodes have anonymous ID (i.e., empty string) at the beginning of the joining period and the joining period consists of one or more request-response slots. Speciﬁcally, in each request-response slot, a LB requests that NBs with the same ID that LB wants respond ﬁrst. Then, there are three cases (states) for the response: idle (no response), success (1 response), and collision (2 or more responses). In the success slot, LB can discover one NB and include it into the group which LB leads. Note that it is impossible for LB to decode two or more response at a slot due to the characteristic of wireless medium. Therefore, to avoid the collision, each responded NB concatenates its ID and an integer which is randomly chosen in the interval R. If LB discovers the required number of NBs for performing a cooperative task, the joining period ends. For convenience, the concatenation of A and B is denoted by A ⊕ B and the current wanted ID is denoted by CID. In M T-JDMS, the decision algorithm of the wanted ID of LB in each slot is as follows. At the beginning, the wanted ID of LB is anonymous ID. In each

Fig. 2. (a) The M -ary tree-based JDMS (when M = 3); (b) The M -Binary tree-based JDMS (when M = 5); C=collision slot, I=idle slot, S=success slot

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collision, LB schedules to search CID ⊕ 0, CID ⊕ 1, ..., CID ⊕ M − 1. The searching order is the depth-ﬁrst-search and increasing order, where the depth equals the length of the wanted ID. Finally, let R be the interval [0, M − 1]. The above description can be visualized by the M -ary tree. A visualized example for M T-JDMS is shown in Fig. 2 (a). In M -BT-JDMS, the decision algorithm of the wanted ID of LB in each slot is as follows. At the beginning, the wanted ID of LB is anonymous ID. When CID is anonymous ID and collision is occurred, LB schedules to search CID⊕0, CID⊕1, ..., and CID ⊕ M − 1. However, if CID is not anonymous ID and collision is occurred, LB schedules to search CID ⊕ 0 and CID ⊕ 1. The searching order is the depth-ﬁrst-search and increasing order, where the depth equals the length of the wanted ID. In addition, each NB has diﬀerent R, depending on its ID. When its ID is anonymous ID, R is the interval [0, M − 1]. Otherwise, R is the interval [0, 1]. The above description can be visualized by the M -Binary tree, which is a tree that the root node has at most M children but other nodes have at most 2 children. A visualized example for M -BT-JDMS is shown in Fig. 2 (b).

4

Simulation Results

Some simulation experiments were performed to compare the two JDMSs. To validate each simulation result, we used the values of the 802.11 system parameters listed in [6]. In addition, since a success/collision slot needs one request (with 2-bit payloads for 3 slot types), 2 SIFSs and 1 response (with 32-bit ID ﬁeld), it takes about 543 μs while the idle slot that includes one request and one aSlotT ime takes 388 s. Therefore, the ratio of the collision slot time to idle slot time α is set to 1.371. The number of required NBs for a task is denoted by k. First, Fig. 3 (a) shows (N is ﬁxed to 60) the average total delay for k, where M = 4 and M = 8. In this ﬁgure, when M is 8, M -BT-JDMS has a smaller delay then M T-JDMS, but when M is 4, M T-JDMS has a smaller delay then M BT-JDMS. This shows that choosing a suitable value of M has a considerable eﬀect on the performance of each scheme. Fig. 3 (b) shows the average total

Fig. 3. (a) Performance Comparison of M T-JDMS and M -BT-JDMS with M = 4, 8; (b) Performance Comparison of JDMSs with k = 25

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delay for N with ﬁxed k (k = 25) when M T-JDMS, M -BT-JDMS, and M ∗ -BTJDMS is performed with the optimal value of M , respectively, where M ∗ -BTJDMS is an adaptive scheme that the optimal value of M is selected depending on N . As shown in the ﬁgure, the delay of 3T-JDMS is generally longer than that of M -BT-JDMS, which implies the performance of M -BT-JDMS is better. However, depending on the N , the average total delay is dynamic in M -BTJDMS. Therefore, in the context of stability, we can conclude that M T-JDMS may have better performance.

5

Conclusion

We have discussed the joining scheme of the TDMA MAC protocol for robot cooperation. This paper proposed two joining schemes for minimizing the total delay of the joining period: M T-JDMS and M -BT-JDMS. Based on the simulation result, we found that when it is hard to estimate the number of neighbors and stability is important, M T-JDMS is suitable, while for a small estimation error when quick joining is required, M -BT-JDMS with dynamic M calculation is suitable. As a future work, we will study and analyze a scheme to solve the problem that arises from the estimation error.

Acknowledgment This research was jointly supported by MEST, Korea, under WCU (R33-2008000-10044-0), a KOSEF Grant funded by the Korean Government (MEST) (No. R01-2007-000-11203-0), a KRF Grant (KRF-2008-314-D00354), and MKE, Korea under ITRC NIPA-2010-(C1090-1021-0008).

References 1. McGlynn, M.J., Borbash, S.A.: Birthday Protocols for Low Energy Deployment and Flexible Neighbor Discovery in Ad Hoc Wireless Networks. In: Proc. of ACM MOBIHOC (2001) 2. Santos, F., Almeida, L., Pedreiras, P., Facchinetti, T.: An adaptive TDMA Protocol for Soft Real-time Wireless Communication among Mobile Autonomous Agents. In: Proc. of WACERTS 2004 in Conjunction with RTSS (2004) 3. Santos, F., Almeida, L., Lopes, L.S.: Self-configuration of an Adaptive TDMA wireless communication protocol for teams of mobile robots. In: Proc. of IEEE ETFA (2008) 4. Arai, J., Koyama, A., Barolli, L.: Performance Analysis of an Adaptive Medium Access Control Protocol for Robot Communication. In: Proc. of ICPADS (2005) 5. Bianchi, G., Tinnirello, I.: Kalman Filter Estimation of the Number of Competing Terminals in an IEEE 802.11 Network. In: Proc. of INFOCOM (2003) 6. Bianchi, G.: Performance Analysis of the IEEE 802.11 Distributed Coordination Function. IEEE JSAC 18(3), 535–547 (2000)

Two-Stage Target Locating Algorithm in Three Dimensional WSNs under Typical Deployment Schemes Lei Mao1, Junzhao Du1,2, Hui Liu1, Deke Guo3, Xing Tang1, and Ning Wei1 1

Software Engineering Institute and 2 State Key Lab of ISN, Xidian University, Xi’an 710071, Shannxi, P.R. China 3 Key lab of C4ISR Technology, College of Information Systems and Management, National University of Defense Technology, Changsha 410073, Hunan, P.R. China

Abstract. Target locating is a kernel and challenging function of many applications based on wireless sensor networks. Many existing approaches of target locating employ the binary detection model. In this paper, we first propose a probabilistic-based detection model for wireless sensor networks in a threedimensional region, which is more realistic in reality. We further design a probability-based target locating algorithm and evaluate this algorithm under four typical deployment schemes of wireless sensor networks. Simulation results show that our simple and effective algorithm can locate the target and save much energy during the tracking process of a moving target. Keywords: WSNs, target localization, probability-based target locating.

1 Introduction Target localization has emerged as a fundamental problem representing a very important class of sensor network applications. For example, in the emergency rescue and disaster relief scenarios, a sensor network deployed in disaster areas can detect the locations of medical staffs. In military conflicts, a sensor network can monitor and track one even multiple targets of the enemy. Current sensor nodes still suffer limited processing capability and low power. Thus, it is very essential to study novel algorithms, which can rapidly locate a target in an energy efficient way once the target enters the coverage area of a sensor network. In this paper, we propose a probability-based target locating algorithm to accurately locate a target in a cluster-based sensor network. In a cluster, each cluster sensor is responsible to detect emerged targets and report the detecting results to the cluster head. It will consume much energy if each cluster sensor transmits detailed locations of targets to the cluster head. In addition, the transmission from those cluster sensors to the cluster head might be delayed if large numbers of cluster sensors compete for limited wireless bandwidth concurrently. To address this issue, we propose a two-step transmission scheme between the cluster head and cluster nodes. First, each cluster sensor only reports the appearance of a target to the cluster head. Once the cluster head receives a report from at least one sensor, it will instruct a group of sensors to report the detailed locating information according to some specific rules. Second, all selected sensors transmit detailed target location information to the cluster head, and hence the cluster head can further locate the target. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 172–181, 2010. © Springer-Verlag Berlin Heidelberg 2010

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In this paper, we assume all sensors are deployed in a three dimensional area, which is characterized as a three-dimensional grid. The granularity of the grid can be adjusted to make the better trade-off between the computation complexity and the effectiveness of the coverage measure. The target detecting model for each sensor node is modeled as a sphere. The centre and radius of the sphere denote the location of a sensor node and its detection range, respectively. We study two different target detecting models. Firstly, we consider a binary detection model. A target is detected (or not) with certainty by a sensor node if the target is inside (or outside) the sphere of the sensor node. Secondly, we investigate a realistic probabilistic model in which the probability that the sensor detects a target depends on the relative position of the target within the sphere. We then propose the probability-based target locating algorithm and evaluate its performance. The remainder of the paper is organized as follows. In section 1, we introduce the background and problem this paper addresses. In section 2, we summarize the related work. We propose the probability-based target locating algorithm in section 3. Section 4 evaluates the algorithm through stimulations. Section 5 concludes this paper and discusses our future work.

2 Related Work Recently, many efforts have been done to cover, locate and track targets using wireless sensor networks. Wan et. al. address the k-covers problem in the deployment field [1]. They assume the sensors are deployed as ether a Poisson point process or a uniform point process in a square or disk region. They study how the probability of k-coverage changes with respect to the sensing radius or the number of sensors. Cheng et. al. propose the sweep coverage with mobile sensors problem in wireless sensor networks [2]. Kumar et. al. propose the k-barrier coverage problem of a belt region using wireless sensor networks [3]. They establish the optimal deployment scheme to ensure the k-barrier coverage when deploying sensors deterministically. Liu et. al. research on Passive Diagnosis for Wireless Sensor Networks [4]. Bai et. al. study the problem of constructing full-coverage in three dimensional networks with multiple connectivity [5]. They prove their optimality among regular lattice deployment patterns, irrespective of the ratio of the communication range over the sensing range. Yang et. al. improve the quality of trilateration using confidence based iterative localization [6]. Brooks et. al. design a distributed tracking system based on the extended Kalman filter [7]. Aslam et. al. propose a particle filtering style tracking algorithm using binary sensors which can detect whether a target is approaching or not [8]. Liu et. al. use radio frequency to minimize frequent trajectory patterns for activity monitoring [9]. Zhang et. al. design a tree-based algorithm to facilitate the collaborative tracking of moving targets [10]. The difference of our work from the aforementioned approaches is that we propose an energy efficient distributed algorithm to track a target.

3 Probability-Based Target Locating In this section, we describe the probability-based target locating algorithm in a threedimension wireless sensor network.

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3.1 Assumptions In this paper, we assume that the exact positions of all sensors are known by some self-positioning approaches. All nodes consume the same amount of energy in detecting, sending and receiving signals. And the consumed energy will cover detecting, sending signals and receiving signals, others are omitted. All sensor nodes within the same cluster are able to communicate with the cluster head. Due to the short-distance transmission, the possibility of a transmission error is very low. The detection area of sensor networks locates in a given three-dimensional area. A two-step communication scheme is used between the cluster head and other nodes within a cluster. At the first step, a sensor node detecting a target uses a bit to report whether there is a target. If a target appears, it reports 1. Otherwise, it reports 0. After the cluster head receives the report information, it will pick up a group of sensor nodes to transmit the detailed information of the target according to specific rules. At the second step, the sensor nodes chosen by the cluster head transmit the detailed location information to the cluster head and the cluster head will locate the target. 3.2 The Detection Model This section describes the sensor detection model. We first define the distance between a pair of sensor and target in a three-dimension space. Given a sensor si ( xi , yi , zi ) and a target p( x, y, z ) , we denote the distance between si and p as: d ( si , p ) = ( x − xi ) 2 + ( y − yi ) 2 + ( z − zi ) 2 .

Let r denotes the detection range of any sensor, re denotes the measurement of the uncertainty of r. When the distance between a target and a sensor is less than r − re , the target can be detected with possibility 1. When a target is r + re away from a sensor, it can’t be detected by the sensor. A target will be detected with a certain probability when it is at the distance greater than r − re but less than r + re . We improve the target detecting model in literature [11][12], and redefine the target detecting probability as ⎧ 0 β ⎪ c p ( si ) = ⎨e − λa ⎪ 1 ⎩

d ( si , p) ≥ r + re r − re < d ( si , p ) < r + re d ( si , p ) ≤ r − re

(3-1)

where a = d ( si , p ) − (r − re ) while λ and β are two parameters of a sensor network. 3.3 Target Localization Detection Probability Table. Localization algorithm can be used to determine the location for each sensor node. Each cluster head generates a table of detection probability for every grid point in the three dimensional space. The detection probability table contains entries for all possible detection reports from those sensors that can detect a target at this grid point. We assume there are k sensors, and a grid point p(x, y, z)

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can be detected by a set of kxyz sensors, denoted as Sxyz, i.e., | s xyz |= k xyz , 0≤kxyz≤k. k xyz

There exists 2 cases when the kxyz sensors monitor the area of the same grid point, including the case that none of the sensors detect anything and the case that all sensors detect the same target. Thus the detection probability table for a grid point p(x,y,z) contains 2

k xyz

entries, defined as:

pxyz (i ) = ∏s

j ∈s xyz

pxyz ( s j , i )

(3-2)

For the ith detection case, if a sensor sj detects a target at the grid point p(x,y,z), then pxyz ( s j , i ) = c p ( s j ) . Otherwise, pxyz ( s j , i ) = 1 − c p ( s j ) . Assume a grid point (3, 3, 3), which is detected by sensors s1, s2 and s3 with probabilities 0.78, 0.78, 0.61, respectively. For a binary string 110, the conditional probability is given by p333 ( s1 , 6) p333 ( s2 .6) p333 ( s3 , 6) =0.78*0.78*(1-0.61) = 0.242. Score-based Ranking. After the probability tables have been generated for all grid points, the cluster head can calculate the location of a target using an inference method once the target is detected by one or more sensors. Assume the cluster head receives reports from k(t) sensors at time instant t. Detail information about the target involves large amount of data, which consumes more energy and bandwidth. Therefore, the cluster head cannot afford to query every sensor. In addition, there exists inherent redundancy among the detected information of sensors, so it is unnecessary for the cluster head to query all the sensors. Using the score-based ranking approach, the cluster head checks the detection probability table for a grid point around the target, and pick up the most suitable sensors to query more detailed information about the target. As shown in Fig.1, there are 3 sensors deployed in a 5×5×5 grid, where k=3, r=3, and re=1. The zigzag shape line A→B is a moving path of a target. The target starts from point A at time slot tA and reaches point B at time slot tB. Fig.2 illustrates a score report at the time slot tA when a target appears at point A.

Fig. 1. A moving target in a region covered by a wireless sensor network.

Fig. 2. The scoring result for a target in the sensor field at time slot tA. (s2 reports the target in this case).

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Assume that R(t) is the set of sensors that have reported the detection of the target, S R , xyz (t ) is the set of sensors that can detect the point P(x, y, z) and have also reported the detection of the target. Obviously, S R , xyz (t ) ⊆ R and S R , xyz (t ) ⊆ S xyz (t ) . The weight of the grid point P(x,y,z) at time slot t is defined as: Wxyz (t ) = k R , xyz (t ) / k R (t )

(3-3)

Where k R (t ) = R (t ) and k R , xyz (t ) = S R , xyz (t ) .The score of the grid point P(x,y,z) at time slot t is calculated by:

SCORExyz (t ) = pxyz (i(t )) × wxyz (t ) Where i(t) is the index of the

(3-4)

pxyz (i (t )) at the time slot t. The parameters i(t) can be

calculated from S xyz (t ) and S R , xyz (t ) , while pxyz(i(t)) corresponds to the conditional probability that the cluster head receives a report representing that there was a target at a grid point P(x,y,z). For example, given a grid point (1, 2, 1) at the time tA as shown in Fig 2, S xyz (t ) = S R , xyz (t ) ={s2,s3},we have i (t A ) =11 p121 (i(t A )) =0.7152 and w121 (t A ) =1, then SCORE121 (t A ) = 1×0.7152 = 0.7152. Selection of Sensors to Query. Assume the maximum number of sensor nodes used to report an event is kmax, and the cluster head select a set of sensors to query at time slot t, i.e. Q(t). There is Q(t ) ⊆ R(t ) ⊆ {s1 , s2 ,..., sk } . If kmax ≥ k R (t ) at time slot t, all sensor nodes reporting the event will be queried by the cluster head. Otherwise, we select the sensors based on the score-based ranking approach. The distance between selected sensors and the grid point with the highest score should be shortest. The rule is defined as:

where si

∈R(t) and p

Q(t ) : d (Q (t ), pMS ) = min{d ( si , pMS )}. MS

(3-5)

denotes the grid points with the highest score.

Estimating the Position of the Target. Note that all sensors need to be queried have been determined in the last step. Here, a multilateral algorithm is needed to estimate the location of the target. If only one sensor needed to be queried, the reported location of the queried sensor is used to indicate the location of the current target. If two sensors are queried, the midpoint of line of the reported location of the two sensors indicates the location of the current target. If three or more sensors are involved, we use the multilateral algorithm to estimate the location of the current target. Suppose n senor nodes have reported the target information at time t in the monitoring area, their coordinates are ( x1 , y1 , z1 ), ( x2 , y 2 , z 2 ),..., ( xn , y n , z n ) , and the

distances measured to the moving target are d1 , d 2 ,..., d n , respectively. Here,

d i = d i '+ Δd , i = 1,2,..., n , where d i ' is the actual distance, Δd is a zero-mean Gaussian variable indicating measurement error, Δd ∈ N (0, s 2 ) . The position to be

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determined of the target is ( x, y, z ) .The model of multilateral algorithm is expressed by the following formula: ⎧( x1 − x) 2 + ( y1 − y ) 2 + ( z1 − z ) 2 = d1 2 ⎪ ⎨# ⎪ 2 2 2 2 ⎩( x n − x ) + ( y n − y ) + ( z n − z ) = d n

(3-6)

Using the method that every equation minus the final equation starting from the first equation, we can get a new equation which is expressed by the matrix AX = b , where 2( y1 − yn ) 2( z1 − z n ) ⎤ ⎡ 2( x1 − xn ) ⎡ x⎤ ⎢ ⎥ ⎢ ⎥ A=⎢ # # # ⎥, X = ⎢ y ⎥, ⎢⎣ z ⎦⎥ ⎣⎢2( xn −1 − xn ) 2( yn −1 − y n ) 2( z n −1 − z n )⎦⎥ ⎡ x12 − xn 2 + y12 − yn 2 + z12 − z n 2 + d n 2 − d12 ⎤ ⎢ ⎥ b=⎢ # ⎥ ⎢ xn −12 − xn 2 + yn =12 − yn 2 + z n −12 − z n 2 + d n 2 − d n −12 ⎥ ⎣ ⎦

(3-7)

We use the least square method to obtain X = ( AT A) −1 AT b , hence the position of the target can be estimated. The Pseudocode of the Algorithm Algorithm 1 Generate Probability Table(P(x,y,z),{s1,s2,…,sk}) 1 /*Find Sxyz the set of sensors that can detect the grid point P(x,y,z) */ 2 For si ∈ {s1,s2,…,sk} 3 If d(si,P(x,y,z))≤r+re 4

∪{s };

Sxyz = Sxyz

i

5 End 6 End 7 /* Build up the detection probability table*/ 8 For i,0≤i≤kxyz, kxyz = |Sxyz|; 9 If sj detects P(x,y,z) 10 Set pxyz(sj,i) = cxyz(sj); 11 Else 12 Set pxyz(sj,i) = 1 – cxyz(sj); 13 End 14 Set pxyz(i) = ∏sj ∈ Sxyz pxyz(sj,i); 15 End Fig. 3. The pseudo-code generates the probability table for each grid point

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Algorithm 2 Localization (Grid,{s1,s2,…,sk},TargetTrace) /* kmax is the maximum number of sensors that are allowed for querying; pr is the threshold for each sensor to determine whether report an event to the cluster head. The moving of a target starts from time slot tA and ends at time slot tB*/ 1 Set t = tA; 2 While (t ≤ tB) 3 /* current target location*/ 4 Set Target = TargetTrace(t); 5 /* calculate the score */ 6 Calculate R(t) from {s1,s2,···,sk},Target(t),pR; 7 Set kR(t) = |R(t)|; 8 For P(x,y,z) in Grid; 9 Calculate Rxyz(t) from R(t) and P(x,y,z); 10 Calculate index i(t) of pxyz from R(t) and Rxyz(t); 11 Set kR,xyz(t)=|Rxyz(t)|; 12 Set wxyz(t)=kR,xyz(t)/kxyz(t); 13 Set SCORExyz(t) = pxyz(i(t))×wxyz(t); 14 End 15 /* select sensors for querying */ 16 Calculate Q(t) from SCORExyz(t) and kmax; 17 /*Predict the position of the target ProTarget(t)*/ 18 Set qtlen=|Q(t)| /* how many sensors*/ 19 If qtlen==1 20 21 22 23

∈Q(t);

Sa

Set ProTarget(t)=ReportLocation(Sa); Else if qlen==2 Sa,Sb

∈Q(t)

24 /*Sab is the midpoint of line of ReportLocation of Sa Sb*/ 25 Calculate Sab from ReportLocation(Sa,Sb); 26 Set ProTarget(t)=Sab; 27 Else 28 / *use multilateral algorithm to estimate the position of the target */ 29 Calculate Sc from ReportLocation(Q(t)); 30 Set ProTarget(t)=Sc; 31 End 32 /* next time instant */ 33 Set t = t+1; 34 End Fig. 4. The Pseudo-code to locate a moving target

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Assume a region covered by k sensors is divided as a m × n × l grid. The computa-

tional complexity of generating the detection probability table is O ( mnl2k ) . The

computational complexity of locating algorithm is O(mnlkmax ) where k max ≤ k . Therefore, the complexity of the probability-based locating algorithm k k is max{O ( mnlkmax ), O ( mnl2 )} = O ( mnl2 )} .

(a) Random deployment in 3D region

(b) Random deployment in two vertical walls

(c) Grid deployment in two vertical walls

(d) Grid deployment in two vertical walls with some nodes between walls

Fig. 5. Four typical deployment schemes of sensors

4 Performance Evaluation Results We implement the probability-based target locating algorithm in Matlab and conduct simulations in typical deployment schemes. Assume 100 sensors are deployed in a 10 × 10 × 10 grid field, where r=3, re=1, λ=0.5, and β=0.5. We consider four typical deployment schemes, as shown in fig 5, where a red ball denotes a sensor. In Fig 5(a), all sensors are randomly distributed in the whole space. In Fig 5(b), all sensors are randomly distributed in two vertical walls. In Fig.5(c), all sensors are uniformly distributed on grid points in two vertical walls. In Fig. 5 (d), we add several sensors between the two walls in Fig. 5 (c). Suppose that an object moves only one grid unit in one time slot, the distances measured to the target follow Gaussian distributions. We assume measurement error Δd ∈ N (0,0.5) and then use the probability-based target locating algorithm to estimate

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the position of the moving target. Fig 5 shows the results, black fold line indicates the real moving trace of an object, while the blue line shows a predicated moving trace using the probability-based locating algorithm. For the four deployment schemes, Fig 6 shows the distance between a pair of predicated point and real point. The x coordinate represents the time slots when the object moves, while the y coordinate represents distance.

(a) Error in Fig 5 (a)

(b) Error in Fig 5 (b)

(c) Error in Fig 5 (c)

(d) Error in Fig 5 (d)

Fig. 6. The predication error between a pair of the predicated point and the real point

By analyzing the simulation results, we can know that the algorithm can locate the target with less energy consumption for the four typical deployment schemes. The precision of predicated position of a target is not well in current stage. We will improve our locating algorithm to make it more accuracy in future work.

5 Conclusions In this paper, we focus on the target locating problem in wireless sensor networks deployed in a three dimensional space. Based on the traditional binary detecting model of a moving target, we propose a probability-based detecting model and a related target locating algorithm. Furthermore, we conduct expensive simulations to evaluate our model and algorithm under four typical deployment schemes of wireless sensor networks. The results show that our simple and effective algorithm can save much energy during the tracking process of a moving target. Acknowledgements. The authors are grateful for a variety of valuable comments from the anonymous reviewers. This work is supported in part by the NSFC under

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grants No. 60803152 and No. 60903206, the Key Research Project of Ministry of Education under grant No. 2009-144, the Open Funds of ISN National Lab under grant No. ISN-9-09, the Fundamental Research Fund for the Central Universities, the Pre-research Fund under grant No. 9140A06050610KG0117, and Research on Key Technologies of Electromagnetic Spectrum Monitoring Based on Wireless Sensor Network under grant No. 2010ZX03006-002-04.

References [1] Wan, P., Yi, C.: Coverage by randomly deployed wireless sensor networks. IEEE Transactions on Information Theory (2006) [2] Cheng, W., Li, M., Liu, K., Liu, Y., Li, X.Y., Liao, X.: Sweep Coverage with Mobile Sensors. In: IEEE IPDPS, USA (April 2008) [3] Kumar, S., Lai, T.H., Arona, A.: Barrier Coverage With Wireless Sensors. In: Prof. of Mobicom 2005, Cologne Germany, August 28-September 2 (2005) [4] Liu, K., Li, M., Liu, Y., Li, M., Guo, Z., Hong, F.: Passive Diagnosis for Wireless Sensor Networks. In: ACM SenSys 2008, Raleigh, NC, USA (November 2008) [5] Bai, X., Zhang, C., Xuan, D., Jia, W.: Full-Coverage and k-Connectivity Three Dimensional Networks. In: The Proceedings of IEEE INFOCOM 2009 (2009) [6] Yang, Z., Liu, Y.: Quality of Trilateration: Confidence based Iterative Localization. IEEE Transactions on Parallel and Distributed Systems 21(5), 631–640 (2010) [7] Brooks, R.R., Ramanathan, P., Sayeed, A.: Distributed target tracking and classsification in sensor networks. In: Proceedings of the IEEE (2002) [8] Aslam, J., et al.: Tracking a Moving Object with a Binary Sensor Network. In: First ACM Conference on Embedded Networked Sensor Systems (2003) [9] Liu, Y., Chen, L., Pei, J., Chen, Q., Zhao, Y.: Mining Frequent Trajectory Patterns for Activity Monitoring Using Radio Frequency Tag Arrays. In: IEEE PerCom 2007, White Plains, NY, USA (March 2007) [10] Zhang, W., Cao, G.: Optimizing tree reconfiguration for mobile target tracking in sensor networks. In: IEEE INFOCOM 2004 (2004) [11] Elfes, A.: Occupancy Grids: A Stochastic Spatial Representation for Active Robot Perception. In: Proc. 6th Conference on Uncertainty in Al, pp. 60–10 (July 1990) [12] Zou, Y., Chakrabarty, K.: Sensor deployment and target localization based on virtual forces. In: IEEE INFOCOM, pp. 1293–1303 (2003)

Interference Analysis for FH-Based Multi-radio Wireless Mesh Networks Davis Kirachaiwanich and Qilian Liang Department of Electrical Engineering University of Texas at Arlington Arlington, TX 76019-0016, USA [email protected], [email protected]

Abstract. In this paper, we study the interference analysis in a Nocoherent Frequency-Hopping (NC-FH) MFSK rural infrastructure Wireless Mesh Networks (WMNs) with each router node being equipped with multiple radio interfaces. Our choice of the FH/MFSK modulation technique here is not just to satisfy the security requirement in military communications but also to provide easy implementation for each router nodes; since FH/MFSK modulation technique has been speciﬁed in IEEE 802.11 standard, these router nodes can be implemented also using IEEE 802.11(FH) equipments. The performances of noncoherent slow frequency-hopping system with M -ary frequency-shift-keyed modulation (NC-FH/MFSK) with AWGN channel and Rician fading under independent multitone jamming (independent MTJ) are investigated in this paper. The expressions for calculating the exact BER performances of the system under the eﬀect of the jamming strategies are derived. We apply the analyses to channel assignment (CA) in multiradio rural WMNs. We obtain a new interference model combining interfence tone and partial band noise, which would be incorporated into the CA algorithm to assign the most appropriate channel (or hopping pattern, in our case) to links in the mesh. Because it takes into account both the intranetwork and the coexisting-network interferences, the new interference model thus reﬂects a very realistic interference situation in WMNs.

1

Introduction

Wireless mesh networks (WMNs) consist of mesh routers and mesh clients, where mesh routers have minimal mobility and form the backbone of WMNs. They provide network access for both mesh and conventional clients. The integration of WMNs with other networks such as the Internet, cellular, IEEE 802.11, IEEE 802.15, IEEE 802.16, sensor networks, etc., can be accomplished through the gateway and bridging functions in the mesh routers. Mesh clients can be either stationary or mobile, and can form a client mesh network among themselves and with mesh routers [1]. The IEEE 802.11b/g and IEEE 802.11a standards deﬁne 3 and 12 non-overlapping frequency channels, respectively. Using multiple channels in multi-radio WMNs greatly improves the network throughput [5]. One G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 182–191, 2010. c Springer-Verlag Berlin Heidelberg 2010

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of the most important design questions for a multi-radio WMN is the channel assignment problem, i.e., how to bind each radio interface to a radio channel [2]. In this paper, we will study the channel assignment in an Nocoherent FrequencyHopping (NC-FH) MFSK rural infrastructure WMN with each router node being equipped with multiple radio interfaces. Our choice of the FH modulation technique here is not just to satisfy the security requirement in military communications but also to provide easy implementation for each router nodes; since FH/MFSK modulation technique has been speciﬁed in IEEE 802.11 standard, these router nodes can be implemented also using IEEE 802.11(FH) equipments. Generally, these IEEE standard FH equipments uses 83.5MHz of ISM frequency bandwidth (2.4GHz-2.4835GHz) as the operational bandwidth and divide it into 79 FH bands, each with 1 MHz, to support the FH modulation technique. Subsequently, to enable multiple transmissions, three non-colliding hopping patterns are established, each with 26 FH bands [3]. The rest of this paper is organized as follows. In Section 2, the BER expressions for NC-FH/MFSK system with AWGN channel and Rician fading under independent multi-tone jamming (MTJ) will be derived. Section 3 will be devoted for the study of the channel assignment in rural NC-FH/MFSK WMN. Section 4 will conclude the paper.

2

NC-FH/MFSK under Independent Multitone Jamming

In this paper the FH system is assumed to be slow hopping over N nonoverlapping FH bands, i.e. a hop period is a multiple of symbol period (Th =kTs , where k=1,2,3,..). Each FH band is comprised of M = 2K signal frequencies of the M -ary FSK modulation. Hence, there are N M possible frequency bins for a signal tone to be transmitted. If all FH bands are contiguous, the total communication bandwidth thus equals to BT = N M/Ts . Also, we will assume that the transmission bit rate of the system is Rb = KRs = K/Ts , where Rs = 1/Ts denotes the symbol rate. And, the average received power for every symbol transmitted, regardless of the channels eﬀect, is assumed to be Ps or an average symbol energy of Es = Ps Ts . Therefore, the received bit energy can be calculated by Eb = Es /log2 M . The jamming source of interest is assumed to possess the complete description of the signal transmitted from the NC-FH/MFSK system and able to transmit multiple signal-like interference tones simultaneously within one symbol time, Ts . Also it is assumed that the power source in the jammer is an ideal source and can supply the power constantly at all time. While q, 1 ≤ q ≤ N M , is the number of simultaneously transmitted interference tones, which are distinct and uniformly distributed over the entire bandwidth BT , if we assume that the total interference tone power at the receiver is PjT and that each interference tone equally shares this power, the received power for each interference tone is therefore equal to Pj = PjT /q or the received energy of Ej = Pj Ts = PjT Ts /q. The transmitted signal and interference tones are assumed to undergo an independent fading channel before arriving at the receiver with noncoherent

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detection scheme and all fading channels in this study are modeled as slow fading, frequency non-selective Rician processes, whose PDF are of the form 2 αk xk xk x + α2 fXk (xk ) = 2 exp − k 2 k I0 (1) u(xk ) σk 2σk σk2 where I0 (·) denotes the zeroth order modiﬁed Bessel function and u(·) is the unit step function. α2 and 2σ 2 are the average power of the LOS (Line-Of-Sight) and the scattering rays of the fading channel. We can also deﬁne another parameter to determine the Rician fading channel by using the ratio of α2 and 2σ 2 . This ratio is called the Rician K factor, Kk = α2k /2σk2 where k=1,2 is used for signal tone and interference tone respectively. It is used to determine how severe the multipath eﬀect is. At the receiver, the received signal will be noncoherently detected and it is assumed that each symbol of the M -ary FSK is equally likely. The receive signal can be represented as ri (u, t) = xk 2Pi cos(ωm t + φi ) + n(u, t) (2) where xk is a Rician random variable representing the envelope of the fading channel and its PDF can be represented as (1). Pi is the average received power of the tone if the eﬀect of fading channel is not accounted. The subscript i=s,j denotes signal tone and interference tone respectively. ωm , m=1,2,..,M , is angular frequency for an MFSK symbol and φ is unknown phase. n(u, t) is AWGN, Gaussian-process thermal noise, with total power σn2 = N0 /Ts . Since the interferers are distinct and uniformly distributed, in any hop there can be as few as none and as many as min(q, M ) interference tones. Hence, the probability of symbol error (or symbol error rate, SER) of the system can be calculated by min(q,M−1)

Ps (e) = P0 · Ps0 (e|no int. tone) +

Pn · Psn (e|n int. tones)

n=1

+PM · PsM (e|M int. tones) · u(q − M )

(3)

where Ps (e) is the probability of symbol error. P0 , PM , and Pn are the probablities that a hop is jammed by zero, M , and n interference tones, where 1 ≤ n ≤ min(q, M − 1). Ps0 (e), PsM (e) and Psn (e) are the probablities of symbol error corresponding to the speciﬁed number of interference tones in the hop. It should be noted that u(·) in the last term on RHS is the unit step function included to account for the fact that the probability of having M interference tones in a hop is equal to zero if the total number of interference tones in the entire spectrum bandwidth is less than M , i.e. q < M . Finally, for MFSK system, the probability of bit error can be calculated from the probability of symbol error by Pb (e) =

M/2 Ps (e) M −1

(4)

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For probability of symbol error with no interference tone, readers can refer to [6] [4]. 2.1

Probability of Symbol Error with n Interference Tones Given the Signal Branch Is Jammed

With total q interference tones simultaneously transmitted from the jammer, if a hop is actually jammed, the possible number of interference tones in the hop can range from 1 up to min(q, M ). For an NC-FH/MFSK system with N M possible frequency bins, the probability that a hop will be jammed by n interference tones, where 1 ≤ n ≤ min(q, M − 1) is given as Pn =

n−1

(

k=0

M−1 q−k q−n ) ) (1 − N M − k j=n NM − j

(5)

Given that a hop is interfered by n interference tones, the probability of symbol error can further be divided into two subcases. First is the case when the signal tone is jammed by one of n interference tones in the hop and the probability that one out of n interference tones will jam the signal tone is n/M . Second is when none of n interference tones is located in the same frequency bin as the signal, i.e. the signal tone is not jammed. The probability of the second case is 1 − n/M = (M − n)/M . Hence, the probability of symbol error given a hop is interfered by n interference tones can be expressed as Psn (e|n int. tones) n = ( )Psn (e|signal is jammed) M M −n +( )Psn (e|signal is not jammed) M

(6)

To evaluate the probability of symbol error when the signal branch is jammed, [8] has provided a computational-eﬃcient method based on the use of phasor representations and noncentral chi-squared PDF’s. It can be shown for any two 2 2 > R02 ). Rician random variables, say R01 and R02 , that P (R01 > R02 ) = P (R01 In (3) of [8], the probability is given as

√ 4K01 K02 b ·I0 b+1

2 > R2 ) P (R01 > R02 ) = P (R01 02 b K01 + K02 b 2K01 2K02 b =Q , − exp − b+1 b+1 b+1 b+1

(7)

where Kl = α2l /2σl2 , l = 01, 02, are Rician factor for R01 and R02 respectively, 2 2 b=σ02 /σ01 , and Q(x,y) is the Marcum’s Q function. Without loss of generality we can assume that the signal is present in the ﬁrst output branch of the detector,

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as well as one of the n interference tones that jam the signal tone. And we will assume also that the rest n-1 interference tones are in the next consecutive branches. Therefore, 1st to nth output branches of the envelope detector will have their PDF follow Rician distributions and the output of the rest M − n branches will follow just a Rayleigh distribution. Furthermore, when the signal tone in the ﬁrst output branch is jammed by an interference tone, the averaged LOS power in the output branch can be expressed as α2sj = α2s + α2j + 2αs αj cos ϕ

(8)

ϕ is the random phase diﬀerence between received signal tone and interference tone and assummed to uniformly distributed between 0 to 2π. So, the probability of symbol error given the signal tone is jammed can be calculated by Psn (e|signal is jammed) = 1 − P (R01 > R02 ∩ R01 > R03 ∩ R01 > R04 ...) =1−

n

P (R01 > R0k )

M

P (R01 > R0j )

j=n+1

k=2

= 1 − P (R01 > R02 )n−1 · P (R01 > R0M )M−n

(9)

The second equality in (9) is obtained by using the fact that each output of the detector branches is independent of each other. Now, we will consider seperately the ﬁrst product term on RHS of the third equality in (9). For given ϕ, the conditional probability P (R01 > R02 |ϕ) can be evaluated easily by substituting the following parameters into (7) K01 =

α2sj 2σs2 + 2σj2 + σn2

K02 =

α2j 2σj2 + σn2

b=

2σj2 + σn2 2σs2 + 2σj2 + σn2

(10)

In the same manner, if we treat a Rayleigh random variable only as a special case for Rician random variable, we can also obtain the probability for the second product term with a condition on ϕ as

α2sj σn2 · exp − (11) P (R01 > R0M |ϕ) = 1 − 2(σs2 + σj2 + σn2 ) 2(σs2 + σj2 + σn2 ) Substituting (10) in (7) to obtain the conditional probability of the ﬁrst product term, using (11) for the conditional probability of the second product term, and integrating (9) over ϕ, we then obtain the complete expression for the symbol error probability for the case.

Interference Analysis for FH-Based Multi-radio Wireless Mesh Networks

Psn (e|signal is jammed) 2π α2sj α2j 1 =1− Q , 2π 0 σT2 + σj2 σT2 + σj2

n−1 2σj2 + σn2 α2sj + α2j 2αsj αj − exp − · I0 2(σT2 + σj2 ) 2(σT2 + σj2 ) σT2 + σj2 M−n−1

α2sj σn2 · 1 − 2 exp − 2 dϕ 2σT 2σT

187

(12)

where σT2 = σs2 + σj2 + σn2 2.2

(13)

Probability of Symbol Error with n Interference Tones Given the Signal Branch Is Not Jammed

When all n interference tones in the hop are not in signal branch, n + 1 output branches of the envelope detector will follow Rician distribution and the rest M − n − 1 branches will follow a Rayleigh distribution. Similarly, we will assume that the signal tone is present in the ﬁrst output branch and each of the n interference tones is in the next consecutive branches. In this case the probability of symbol error can be expressed as Psn (e|signal is not jammed) = 1 − P (R01 > R02 ∩ R01 > R03 ∩ R01 > R04 ...) =1−

n+1

P (R01 > R0k )

M

P (R01 > R0j )

j=n+2

k=2

= 1 − P (R01 > R02 )n · P (R01 > R0M )M−n−1

(14)

Once again, the probability P (R01 > R02 ) for the third equality of (14) can be calculated by substituting the following parameters into (7) α2s 2σs2 + σn2 α2j = 2σj2 + σn2

K01 = K02

b=

2σj2 + σn2 2σs2 + σn2

(15)

and, by the same mean, the conditional probability for the second product term is found to be α2s σn2 exp − P (R01 > R0M ) = 1 − (16) 2(σs2 + σn2 ) 2(σs2 + σn2 )

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To obtain the complete expression for the symbol error probability for the case, we then substitute (15) in (7) to obtain the probability of the ﬁrst product term of (14) and use (16) for the probability of the second product term. Finally, the symbol error probability for the case when signal branch is not jammed can be shown as follow, where σT2 is given in (13). ⎧ ⎛ ⎞ ⎨ 2 2 2 α2j α s ⎠ − 2σj + σn Psn (e|signal is not jammed) = 1 − Q ⎝ , 2 2 2 ⎩ σT σT 2σT

α2s + α2j ·exp − 2σT2

2.3

I0

2αsj αj σT2

n

M −n−1 α2s σn2 exp − 1− (17) 2(σs2 + σn2 ) 2(σs2 + σn2 )

Probability of Symbol Error with M Interference Tones

Once q ≥ M , it is possible that we can have all M branches jammed by the interference tones and the probability of having M interference tones in a hop can be calculated easily by PM =

M−1

(

k=0

q−k ) NM − k

(18)

With M interference tones, every output branch of the envelope detector will have their PDF’s now followed only Rician distributions with the averaged LOS power equal to α2j . An exception is made only for the branch where signal tone is also present, in which the averaged LOS power is equal to the summation of α2s and α2j , as shown previously in (8). Without loss of generality, we will also assume in here that the ﬁrst detector output branch is one where the signal tone is present. Thus, the ﬁrst output branch will have both the signal and an interference tone. In this case, the probability of symbol error can be found by PsM (e|M int. tones) = 1 − P (R01 > R02 ∩ R01 > R03 ∩ R01 > R04 ...) =1−

M

P (R01 > R0k )

k=2

= 1 − P (R01 > R0M )M−1

(19)

The conditional probability P (R01 > R0M |ϕ) of the squared term in the equation can be computed by applying same parameters in (10) to (7) and, in the end, we can obtain the probability of symbol error given M interference tones in the hop as 2π α2sj α2j 1 , PsM (e|M int. tones) = 1 − Q 2π 0 σT2 + σj2 σT2 + σj2

M−1 2σj2 + σn2 α2sj + α2j 2αsj αj − exp − ·I0 dϕ (20) 2(σT2 + σj2 ) 2(σT2 + σj2 ) σT2 + σj2

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where σT2 can be calculated from (13). Finally, we can combine together every equation we have derived so far to compute for the total probability of symbol error. The probability of symbol error given n interference tones can be obtained ﬁrst by substituting (12) and (17) into (6). Then, using the substitution result together with (5), (20), and (18) in (3), we will obtain the complete analysis for the probability of symbol error for FH/MFSK system with the independent multitone interference. And, we can easily convert it to the probability of bit error (BER) using (4).

3

Development of a Combined Interference Model for Rural Wireless Mesh Network

In this paper, we take on an example of the channel assignment for an NCFH/MFSK rural infrastructure WMN with each router node being equipped with multiple radio interfaces. In the entire communication bandwidth BT , we assume that there are N FH bands available and, to achieve the greatest beneﬁt of having multiple interfaces on nodes, we will divide the FH bands into b interleaving and non-colliding hopping patterns to enable concurrent links. Each hopping pattern thus contains equal number of Np = N/b FH bands, each with M possible signal frequency bins. So, there will be the total of Np M possible signal frequencies on any link. And, because these patterns are interleaving and non-colliding, a link in one hopping pattern will not interfere with that on the others. Hence, it is possible for the nodes with multiple NC-FH/MFSK interfaces to support concurrent information links without having them interfering with each other, if the links are performed using diﬀerent hopping patterns. As an illustration of how interferences should be taken into account in the algorithm , let us consider an example scenario shown in Fig. 1. In Fig. 1, six router nodes were used to setup a simple mesh network and each node is labeled corresponding to its name and number of its information interfaces, for example A-2 means node A is equipped with two NC-FH/MFSK interfaces. Here, we assumed that the total of N FH bands was divided into two hopping patterns and node A was chosen as the PAC node. After information of nodes had been forwarded, node A thus began to create the map and the MCG of the mesh. At the time being, we assumed that node A has already started running the CA algorithm and some of the links have already been assigned with hopping patterns; A↔B (the link between node A and node B) and C↔E are assigned to pattern 1 while A↔C and B↔D are allocated to pattern 2. The algorithm is now running for C↔F at node C. First, let us consider if C↔F is to be assigned with pattern 1. Because hopping patterns of each link are further shuﬄed to make hopping sequences of each link distinct and independent from each other, therefore, it appears that the interference model for the independent MTJ in Section 2 has once again proven useful. Recalling from Section 2, however, we can see that the interference model was developed, at that time, mainly based on the assumption that all interference tones are received with equal power and no two interference tones exist on the same frequency bin. But, from the situation in Fig. 1, it is obvious that, when

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D-1 2

1 2

C-3

PAC

1

A-2

B-2

F-1

Fig. 1. An Example of Interferences in WMN Table 1. Input values for calculating BER of WMN in Fig. 1 (if C↔F is to be assigned with pattern 2) Variable N Ps Pj q K1 K2

Deﬁnition Value Total FH bands in the pattern Np Received power of signal tone PF C Avg. received power of int. tone 12 (PAC + PBC ) Total int. tones in the pattern q2 = 2 Rician factor for signal fading 100dB Rician factor for int. fading 100dB

signal tone Total interference tones (q) = 3

interference tone

interference noise

f FH band in pattern 1

Bj B

FH band in pattern 2

T

Fig. 2. A frequency-domain illustratation of interference tones (intra-network interferences) and interference noise (external interference) of the mesh network with 2 hopping patterns

C↔F is to assigned with pattern 2, the interference tones from node A and node B will be received at diﬀerent power due to their diﬀerent distances from node C. Besides, the interference tone from node A and node B can also be transmitted on the same frequency bin because their hopping sequences are random and independent of each other. Nevertheless, it should also be noted that, because hopping patterns are established from the interleaving and non-colliding FH bands within the entire bandwidth BT , each hopping pattern then share NJ jammed FH bands equally; therefore, the jamming ratio ρi of hopping pattern i will be the same as the network jamming ratio ρ.

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For ease of representation, let us ﬁrst deﬁne A as the event that the signal hop is jammed by the interference noise and A as the complementary event or the event that the signal hop is not jammed by the interference noise. We can calculate the probability of symbol error by Ps (e) = P (A) · Ps (e|A) + P (A ) · Ps (e|A )

4

(21)

Conclusions

In this paper, we studied the interference analysis in a NC-FH MFSK rural infrastructure WMNs with each router node being equipped with multiple radio interfaces. The performances of noncoherent slow frequency-hopping system with M -ary frequency-shift-keyed modulation (NC-FH/MFSK) with AWGN channel and Rician fading under independent multitone jamming (independent MTJ) were investigated in this paper. The expressions for calculating the exact BER performances of the system under the eﬀect of the jamming strategies are derived. We applied the analyses to channel assignment (CA) in multiradio rural WMNs. We obtained a new interference model combining interfence tone and partial band noise, which would be incorporated into the CA algorithm to assign the most appropriate channel (or hopping pattern, in our case) to links in the mesh. Because it takes into account both the intra-network and the coexistingnetwork interferences, the new interference model thus reﬂects a very realistic interference situation in WMNs.

References 1. Akyildiz, I.F., Wang, X., Wang, W.: Wireless mesh networks: a survey. Computer Networks 47, 445–487 (2005) 2. Avallone, S., Akyildiz, I.F.: A Channel Assignment Algorithm for Multi-Radio Wireless Mesh Networks. Computer Communications 31(7), 1343–1353 (2008) 3. Crow, B.P., Widjaja, I., Kim, J.G., Sakai, P.T.: IEEE 802.11 wireless local area networks. IEEE Comm. Magazine 35(9), 116–126 (1997) 4. Proakis, J.G.: Digital Communications, 2nd edn. McGraw-Hill, New York (1989) 5. Raniwala, A., Chiueh, T.: Architecture and Algorithms for an IEEE 802.11-Based Multi-Channel Wireless Mesh Network. IEEE INFOCOM 3, 2223–2234 (2005) 6. Robertson, R.C., Sheltry, J.F.: Multitone interference of frequency-hopped noncoherent MFSK signals transmitted over Ricean fading channels. IEEE Trans Commun. 44(7), 867–875 (1996) 7. Schwartz, M.: Mobile Wireless Communications. Cambridge University Press, Cambridge (2005) 8. Tjhung, T.T., Chai, C.C.: Multitone jamming of FH/BFSK in Rician channels. IEEE Trans Commun. 47(7), 974–978 (1999)

Interference-Aware Gossiping Scheduling in Uncoordinated Duty-Cycled Multi-hop Wireless Networks Xianlong Jiao1 2 , Wei Lou2 , Xiaodong Wang1 , Junchao Ma2 , Jiannong Cao2 , and Xingming Zhou1 1 School of Computer, National University of Defense and Technology, Changsha, China {xljiao,xdwang,xmzhou}@nudt.edu.cn 2 Department of Computing, The Hong Kong Polytechnic University, Kowloon, Hong Kong {csweilou,csjma,csjcao}@comp.polyu.edu.hk

Abstract. Gossiping is to broadcast the message of every node to all the other nodes in multi-hop wireless networks (MWNs). This operation plays an important role and is widely used in MWNs. Interference-aware gossiping scheduling (IAGS) aims to provide an interference-free scheduling for gossiping with the minimum latency. Previous work on IAGS mostly assumes that nodes are always active, and thus is not suitable for duty-cycled scenarios. In this paper, we investigate the IAGS problem in uncoordinated duty-cycled multi-hop wireless networks (IAGS-UDC problem) under protocol interference model and unbounded-size message model. We prove that the IAGS-UDC problem is NP-hard. We propose a novel approximation algorithm called MILD for this problem. The MILD algorithm achieves an approximation ratio of 3 2 ( 6) T , where is 23 ( 2), denotes the ratio of the interference radius to the transmission radius, denotes the maximum node degree of the network, and T denotes the number of time-slots in a scheduling period. Moreover, the number of transmissions scheduled by the MILD algorithm is at most 3 times as large as the minimum number of transmissions. Keywords: Gossiping scheduling, interference, duty cycle, multi-hop wireless networks.

1 Introduction Uncoordinated duty-cycled multi-hop wireless networks (UDC-MWNs) consist of nodes with limited transmission ranges and different duty cycles. Nodes in UDCMWNs are powered by batteries, so they often switch between the active state and the sleep state to conserve the energy, and this switching operation is uncoordinated. Broadcast is one of the most important communications in UDC-MWNs, and one kind This work is supported in part by Hong Kong GRF grants (PolyU 5236/06E, PolyU 5102/08E), PolyU ICRG grant 1-ZV5N, National Basic Research Program of China grant 2006CB30300. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 192–202, 2010. c Springer-Verlag Berlin Heidelberg 2010

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of the broadcast communications is gossiping, which is to broadcast the message of every node to all the other nodes. Gossiping is widely used in UDC-MWNs for data collection and code update, etc. In many time-critical applications of UDC-MWNs, gossiping must be completed with low latency. There are many interference models in UDC-MWNs, such as graph-based interference model and protocol interference model. Under the graph-based interference model, the interference is treated as the collision, and if two nodes send messages to their common neighboring node concurrently, the common neighboring node will receive neither of the two messages. Under the protocol interference model, if one node lies in the interference range of one transmitter node, it cannot receive the messages from other nodes when this transmitter node is transmitting messages. Two common message models in UDC-MWNs are unit-size message model and unbounded-size message model. Under the unit-size message model, one node cannot combine its received messages as one message. Under the unbounded-size message model, one node can combine its received messages as one message, and can broadcast this message in one time-slot. Interference aware gossiping scheduling (IAGS) aims to provide an interference-free scheduling for gossiping with the minimum latency. The IAGS problem in conventional multi-hop wireless networks (MWNs) is known to be NP-hard no matter whether the networks are modeled as general graphs [1] or unit disk graphs [2]. Many efficient approximation algorithms [2,3,4,5,6,7], which follow the assumption that all nodes always keep active, have been presented for this problem. Unlike in conventional MWNs, one node in UDC-MWNs may require transmitting several times to inform all its neighboring nodes with different active time. Hence, these algorithms are not suitable for the IAGS problem in UDC-MWNs. In this paper, we investigate the IAGS problem in UDC-MWNs (IAGS-UDC problem) under both protocol interference model and unbounded-size message model. To the best of our knowledge, this is the first work to study this problem under these two models. Our main contributions include: 1) We prove that the IAGS-UDC problem is NP-hard; 2) We propose a novel approximation algorithm called MILD for this problem; 3) We show the correctness of the MILD algorithm, and prove that the approximation ratio of this algorithm is at most 3 2 ( 6) T , where is 23 ( 2), denotes the ratio of the interference radius to the transmission radius, denotes the maximum node degree of the network, and T denotes the number of time-slots in a scheduling period; 4) We prove that the number of transmissions scheduled by the MILD algorithm is at most 3 times as large as the minimal number of transmissions.

2 Related Work Since gossiping plays a very important role in MWNs, a lot of studies have been done on this problem [1,2,8,3,4,5,6,7,9]. Gossiping is known as the all-to-all broadcast, and the simplest implementation of broadcast is flooding, which may cause a large amount of contention and collision [8]. The multi-hop wireless network is often modeled as a unit disk graph (UDG) when the nodes have the same transmission radius. The IAGS problem, which aims to provide an interference-free gossiping scheduling with the

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minimum latency, is known to be NP-hard in both the general graphs [1] and the unit disk graphs [2]. Much work [3,4,5] has focused on the gossiping problem under graph-based interference model and unbounded-size message model. Chrobak et al. [3] propose a gossiping algorithm for radio networks with unknown topology, and show that their algorithm can finish in O(n3 2 log2 n) time. They further present a gossiping algorithm in [4], which is a randomized algorithm and can finish in O(n log4 n) time. For networks with diam eter Æ O(n ) ( 1), an efficient gossiping algorithm running in O( Æn log2 n) is proposed in [5]. Many algorithms [2, 6, 7, 9] have been presented for the gossiping problem under unit-size message model. Recently, Gandhi et al. [2] investigate the IAGS problem under graph-based interference model and unit-size message model, and propose an approximation algorithm of a constant ratio. In [6], Huang et al. show that this ratio is more than 1000, and give a 27-approximation algorithm. This ratio is further improved to 20 by Gandhi et al. in [7]. Wan et al. [9] propose a constant approximation algorithm to tackle the IAGS problem in multi-channel MWNs under protocol interference model and unit-size message model. None of the work mentioned above, however, has taken the active/sleep cycles into consideration. The broadcast problems in duty-cycled scenarios have been extensively studied in [10, 11, 12, 13]. The only work to study the IAGS-UDC problem is [13], which investigates this problem under the graph-based interference model and both two message models, and presents two approximation algorithms with a ratio of 17 T 20 and ( 22) T respectively. To the best of our knowledge, none of previous work has focused on the IAGS-UDC problem under both protocol interference model and unbounded-size message model. In this paper, we will investigate this problem under these two models and give an efficient approximation algorithm for this problem.

3 Preliminaries 3.1 Network Model We model the uncoordinated duty-cycled multi-hop wireless network as a UDG G (V E), where V contains all the nodes in the network, and E is the set of edges, which exist between any two nodes u and v if their Euclidean distance d(u v) is no larger than the transmission radius r. We consider protocol interference model as the interference model, and regard unbounded-size message model as the message model. We denote by r f the interference radius, and by the ratio of r f to r. A node cannot send and receive the message at the same time. We denote by n the number of nodes in the network and by NG (u) the set of neighboring nodes of node u. We assume that nodes independently determine the active/sleep time in advance. The duty cycle is defined as the ratio of the active time to the whole scheduling time. The whole scheduling time is divided into multiple scheduling periods of the same length. One scheduling period T is further divided into fixed T unit time-slots 0 1 T 1. Every node v independently chooses one time-slot in T as its active time-slot A(v).

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A node can transmit the message at any time-slot, but is only allowed to receive the message at its active time-slot. 3.2 Problem Formulation This paper studies the gossiping problem in UDC-MWNs. In this IAGS-UDC problem, every node has a message to send to all the other nodes. The gossiping task completes when every node receives the messages from all the other nodes. We model the gossiping scheduling as assigning the transmitting time-slots for every node, i.e., assigning a function T T S : V 2 . The objective of gossiping scheduling is to minimize the largest transmitting time-slot. If we set T 0 and 1, then the IAGS-UDC problem reduces to the conventional IAGS problem which has been proved to be NP-hard in [2], so the IAGS-UDC problem is also NP-hard. To schedule the transmissions efficiently, we construct a shortest path tree as follows. If we choose one node w as the source node and this node starts broadcasting its message at time-slot 0, the latency Lat(u v) of every edge (u v) E is:

A(v) 1 if u w; Lat(u v) A(u) if u w and A(v) A(u) 0; A(v) A(v) A(u) T else

(1)

The shortest path tree rooted at node w can be achieved by applying Dijkstra’s algorithm with this latency. The broadcast tree is constructed based on the shortest path tree, and the gossiping is scheduled according to the broadcast tree. To distinguish the parent nodes of node v in the shortest path tree and in the broadcast tree, we call the parent node of node v in the shortest path tree as the father node of node v, and denote it by F(v); we denote by P(v) the parent node of node v in the broadcast tree. 3.3 Graph-Theoretic Definitions We denote by G[U] the subgraph of G induced by a subset U of V. If there is no edge between any two nodes in G[U], we call the subset U an Independent Set (IS) of G. A Maximal Independent Set (MIS) of G is not a subset of any other IS of G. It is known that a node can be adjacent to at most 5 nodes in an IS. A proper tessellation of hexagons in the whole plane is to partition the plane into hexagons of the same size. Coloring of these hexagons is to assign every hexagon one natural number representing the color of this hexagon. According to [14], for any integer 1, a proper 3 2 coloring of half-open half-closed hexagons can make sure that the distance between two hexagons of the same color is larger than 3 2 radii of the hexagon. If we set as 32 ( 2) and set the radius of the hexagon as r2, the distance between two hexagons of the same color will be larger than ( 1)r r f r.

4 Approximation Algorithm Since the IAGS-UDC problem is NP-hard, we propose and detail the MILD algorithm in this section. Recall that one node can combine its received messages as one message

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and send the combined message in one time-slot. Therefore, the MILD algorithm contains two processes. In the first process, the messages of all the nodes are gathered to a special node, which is called as a data aggregation process. In the second process, the special node combines all the messages as one message, and broadcasts this message to all the other nodes. The pseudocode of the MILD algorithm is shown in Algorithm 1. The MILD algorithm starts with coloring all the nodes. We use a proper tessellation and 3 2 -coloring of hexagons with a radius of r2 in the whole area to color these nodes, where 32 ( 2). We use f : V 1 2 3 2 to denote this coloring method. After coloring all the nodes, we find a special node s. The maximum latency D of the shortest path tree T S PT rooted at this node is the minimum. We can build the shortest path trees rooted at all the nodes based on the latency defined in Eq. 1, and find this special node. The tie can be broken randomly. Then all the nodes are divided into different layers L0 , L1 , ..., LD according to the latency of the shortest paths from node s to these nodes in T S PT .

Algorithm 1. MILD algorithm Input: G (V E), s, A, , r, T . Output: Gossiping Scheduling T T S : V 2 . 1: Apply a proper tessellation and 3 2 -coloring of hexagons with a radius of r2 in the whole area to color all the nodes. Use f : V 1 2 3 2 (0 j T 1) to denote this coloring method, where 32 ( 2). 2: Find a special node s such that the maximum latency of the shortest path tree T S PT rooted at this node is the minimum. 3: Set D as MaxLatency(T S PT ), and divide V into L0 , L1 , ..., LD . 4: Apply Algorithm 2 to construct the MIS’es Q0 , Q1 , ..., QT 1 with different active time-slots, and to construct the IS’es M1 , M2 , ..., MD layer by layer. 5: Apply Algorithm 3 to construct the broadcast tree T B rooted at node s and to get the array P to maintain every node’s parent node. 6: Apply Algorithm 4 to achieve the scheduling of aggregating the messages to node s and broadcasting the combined message from node s to all the other nodes.

Next we construct the MIS’es layer by layer. The nodes in V s are partitioned into different subsets U0 U1 U2 UT 1 according to their active time-slots. Recall that every node v in V s can only receive the message at its active time-slot. The latency of the shortest path from node s to this node should be in the form of k T A(v) 1, where k 0 1 2 . So we can find that each subset U j consists of nodes at several layers in the T S PT , i.e., U j iI Li , where I i (i 1) j mod T 1 i D. At each layer Li , we find the independent set Mi of G[Li ] such that Q j Mi is an MIS of G[ i I Li ], where I i (i 1) j mod T 1 i i, and j (i 1) mod T . Finally, we can find the MIS Q j of G[U j ]. The pseudocode of this process is shown in Algorithm 2. ¼

¼

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Algorithm 2. Construct the MIS’es 1: 2: 3: 4: 5: 6: 7: 8:

Divide V s into subsets U0 , U1 , ..., UT 1 . for j 0 to T 1 do Qj for i 1 to D do j (i 1) mod T , I i (i 1) j mod T 1 i i Construct an IS Mi of G[Li ] such that Q j Mi is an MIS of G[ Q j Q j Mi return Q0 , Q1 , ..., QT 1 and M1 , M2 , ..., MD

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Once the MIS’es have been found, we start constructing the broadcast tree T B rooted at node s as shown in Algorithm 3. For nodes in Mi , we choose some of their father nodes in T S PT as their parent nodes in the broadcast tree. The choosing process also proceeds layer by layer. In each layer Li , we pick one of the father nodes of those nodes in Mi as the parent node if this father node v covers the most unassigned nodes in Q j , where j (i 1) mod T . These unassigned nodes are set as its children nodes and collected in C1 j (v). This process continues until all the nodes in Mi have been assigned parent nodes. Since nodes in Li Mi must be adjacent to some nodes in i I Mi , where I i (i 1) j mod T 1 i i, we pick some nodes in this set as their parent nodes. The choosing process is similar to the previous one. Note that the node v which covers the most unassigned nodes in U j Q j will be first chosen. These unassigned nodes are set as the children nodes of node v and collected in C2 j (v). ¼

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Algorithm 3. Construct the broadcast tree 1: T B (V B E B ), V B V, E B 2: for i 1 to D do 3: j (i 1) mod T 4: X1 u P(u) NIL u Mi , Y1 F(u) P(u) NIL u Mi , Z1 5: I i (i 1) j mod T 1 i i 6: X2 u P(u) NIL u Li Mi , Y2 i I Mi , Z2 U j Q j 7: for k 1 to 2 do 8: while Xk do 9: for each node v Yk do 10: Ck j (v) u P(u) NIL u Zk NG (v) 11: Find a node v with the maximum Ck j (v ) . 12: for each node u Ck j (v ) do 13: P(u) v , E B E B (v u), Xk Xk u 14: return T B and P

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Next, we aggregate the messages of all the nodes to the special node s. This process works from the bottom layer to the top layer. In each layer Li , the scheduling contains two phases: the nodes in Li Mi first send their messages to their parent nodes, and then do the nodes in Mi . We collect the children nodes into the set X, and collect the parent nodes into the set Y. The transmissions are scheduled iteratively. In each iteration,

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Algorithm 4. Data aggregation and broadcast the combined message Step I: Data aggregation 1: t 0 2: for i D down to 1 do 3: if Li then 4: j (i 1) mod T , X1 Li Mi , X2 Mi 5: for k 1 to 2 do 6: X Xk 7: while X do 8: Y P(x) x X , t t 9: for each node y Y do 10: Find one of its children nodes x in X. 11: z (k 1)x (2 k)y 12: T T S (x) T T S (x) t ( f (z) 1) T A(y) 13: if t t ( f (z) 1) T A(y) 1 then 14: t t ( f (z) 1) T A(y) 1 15: X X x 16: t t T T Step II: Broadcast the combined message 1: for i 1 to D do 2: if Li then 3: j (i 1) mod T 4: m1i max f (u) u Mi , m2i max f (v) v Mi and C2 j (v) 5: for each node u Mi do j 6: T T S (P(u)) T T S (P(u)) t ( f (u) 1) T 7: t t m1i T 8: for each node v Mi and C2 j (v) do j 9: T T S (v) T T S (v) t ( f (v) 1) T 10: t t m2i T 11: return T T S

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every parent node y in Y chooses one of its children nodes x in X, and receives the message from this child node. Note that, to avoid the interference of the transmissions, we schedule these transmissions based on the chromatic numbers of nodes in Q j , i.e., f (u) of node u Q j , where j (i 1) mod T . If the children nodes are in Li Mi , the scheduling is based on the chromatic numbers of their parent nodes, which belong to Q j according to Algorithm 3. If the children nodes are in Mi , the scheduling is based on the chromatic numbers of these children nodes, which are also in Q j . After each iteration, the current time-slot t advances to multiple times of T when all the transmissions in this iteration can finish. The pseudocode of this process is shown in Algorithm 4 Step I. Finally, after the messages are aggregated to node s, node s combines all these messages as one message, and broadcast this combined message to all the other nodes as shown in Algorithm 4 Step II. The scheduling works from the top layer to the bottom layer. In each layer Li , the message is first delivered to nodes in Mi , and then nodes in Mi broadcast the message to its children nodes. Like the scheduling in the data aggregation process, to avoid the interference, we schedule the transmissions from parent nodes to

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their children nodes based on the chromatic numbers of nodes in Q j , where j (i 1) mod T . We denote by m1i the maximum chromatic number of nodes in Mi , and by m2i the maximum chromatic number of nodes in Mi with children nodes in U j Q j . Recall that the children nodes in U j Q j of node v are collected in C2 j (v). The message is first delivered to each node u in Mi at t ( f (u) 1) T j. The current time-slot t increases by m1i T such that all these transmissions can finish. Each node v in Mi then broadcasts the message to its children nodes in C2 j (v) at t ( f (v) 1) T j. Similarly, the current time-slot t increases by m2i T such that all these transmissions can finish. Example 1. We take an example to illustrate the MILD algorithm. The network consists of ten nodes. The network topology of G is shown in Fig. 1(a). The scheduling period T contains ten time-slots from 0 to 9. The active time-slots of ten nodes are {7, 6, 7, 4, 6, 7, 9, 5, 8, 2}. According to Algorithm 1, we first color all the nodes by a proper tessellation and 27-coloring ( 2) of hexagons as shown in Fig. 1(a), e.g., the color of node 9 is 26. We then find that node 4 is the special node and construct the shortest path tree rooted at node 4 as shown in Fig. 1(b). All the nodes from node 0 to node 9 are divided into several layers L8 L7 L8 L15 L0 L13 according to the latency of the shortest path from node 4 to every node. Afterward, we construct the MIS’es layer by layer, and construct the broadcast tree T B as shown in Fig. 1(c). Next, we aggregate the messages to node 4 from the bottom layer to the top layer according to Algorithm 4 Step I. In this example, the bottom layer L15 only contains node 3, and this node will transmit its message to its parent node 2 in T B at time-slot t ( f (3) 1) T A(2) 0 (21 1) 10 7 207. Then the current time-slot advances to time-slot (207 1)10 10 210, and node 9 in the upper layer L13 is scheduled to transmit its message to its parent node 2 in T B at time-slot t ( f (9) 1) T A(2) 210 (26 1) 10 7 467. Node 4 will ultimately receive all the messages at time-slot 1260. Next, node 4 broadcasts the combined message from the top layer to the bottom layer. According to Algorithm 4 Step II, node 4 first sends the message to node 7 in L6 at time-slot t ( f (7) 1) T (6 1) mod T 1260 (21 1) 10 5 1465, and then the current time-slot advances to time-slot 1260 21 10 1470. The schedule then proceeds to the next layer containing nodes, and so on. Node 3, the unique node in the bottom layer L15 , will receive the message from its parent node 2 at time-slot

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( f (3) 1) T (15 1) mod T 21 10 2510 time-slots.

2504 . Finally, the total broadcast latency is

5 Performance Analysis In this section, we show the correctness of our MILD algorithm, and then give the approximation ratio of our algorithm. We also prove that the number of transmissions of our algorithm is within a constant factor of the minimum number of transmissions. Finally, we give the time complexity of our algorithm. We omit the proof of some theorems due to the space limit. Theorem 1. The MILD algorithm provides a correct and interference-free gossiping scheduling. Proof (sketch). During the data aggregation process, at each layer Li , nodes in Li Mi first aggregate their message to their parent nodes in upper layers or in Mi . Then nodes in Mi aggregate their message to their parent nodes in upper layers. So node s will ultimately receive all the message. During the broadcast process, at each layer Li , the combined message will be first delivered to nodes in Mi , which then deliver this message to their children nodes. Nodes in Li Mi will be informed by their parent nodes in upper layers or in Mi . So all the nodes will receive the combined message. According to the tessellation and coloring method discussed in Section 3.3, the distance between two nodes in an IS with the same chromatic number should be larger than r f r. It is easy to prove that the transmissions to these two nodes or from these two nodes are interference-free. Since all the transmissions are scheduled based on the chromatic numbers of nodes in an IS, these transmissions are interference-free. 2 Lemma 1. The latency of the data aggregation process is at most 3 2 (

4) T D.

Proof. During the data aggregation process, the messages are aggregated to node s layer by layer. At each layer, nodes in Li Mi first aggregate their messages to their parent nodes iteratively. Since each parent node can receive the message of only one of its children nodes during one iteration, the parent node with the most children nodes will always exist in the set Y during all the iterations, and the number of its children nodes in the set X will decrease by one after each iteration. Moreover, this parent node belongs to an IS, which does not include node s, and therefore it should have one parent node in the broadcast tree. So this parent node has at most 1 children nodes, where is the maximum node degree of the network. The total number of iterations is bounded by 1. f (z) is no larger than 3 2 , and A(y) is at most T 1. The latency of transmissions in one iteration is at most (3 2 1) T ( T 1) 1 3 2 T . So the latency of data aggregation from nodes in Li Mi to their parent nodes is at most 3 2 ( 1) T . Nodes in Mi then aggregate their messages to their parent nodes. Since one parent node has at most 5 children nodes in Mi , the total number of iterations is bounded by 5. The latency of data aggregation from nodes in Mi to their parent nodes is at most 5 3 2 T . Combine two kinds of latency as 3 2 ( 1) T 5 3 2 T 3 2 ( 4) T , which is the latency of data aggregation at each layer. Since there are at most D 1 layers, the latency of the entire data aggregation process is at most 3 2 ( 4) T D. 2

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Theorem 2. The approximation ratio of the MILD algorithm is at most 3 2 (

6) T .

Proof. We first claim that D is a trivial lower bound for the IAGS-UDC problem. The MILD algorithm contains two processes. The first process is the data aggregation process, and we have proved that the latency of this process is at most 3 2 ( 4) T D in Lemma 1. The second process is the broadcast process, which works layer by layer. At each layer, the schedule contains two phases. In the first phase, the message is broadcasted to nodes in Mi . Since the maximum chromatic number m1i is at most 3 2 , the latency of this phase is at most 3 2 T . In the second phase, the message is broadcasted from nodes in Mi to their children nodes. Similar to previous phase, the latency of this phase is at most 3 2 T because m2i is bounded by 3 2 . We combine these two kinds 3 2 T 6 2 T . In the worst case, there are D 1 layers, so of latency as 3 2 T the latency of the broadcast process is at most 6 2 T D. Combine the latency of two processes as 3 2 ( 4) T D 6 2 T D 3 2 ( 6) T D. 2 Theorem 3. The total number of transmissions scheduled by the MILD algorithm is at most 3 times as large as the minimum total number of transmissions. Proof. Since each node needs to transmit at least once to broadcast its message to others, the minimum total number of transmissions is at least n. During the data aggregation process, every node combines its received messages as one message, and is only scheduled once to transmit the combined message to its parent node. During this process, only node s does not transmit, so the total number of transmissions is n 1. During the broadcast process, at each layer Li , the message is first broadcasted to each node u in Mi . Only one transmission is required to cover node u, so the number of transmissions during this phase is bounded by Mi . The message is then broadcasted from nodes in Mi to their children nodes with the same active time-slot. Note that, each node in Mi transmits only once to inform its such children nodes, and therefore the number of transmissions during this phase is bounded by Mi . We combine these two numbers of transmissions as Mi Mi 2 Mi . The number of transmissions during the broadcast process is bounded by 1iD 2 Mi , which is at most 2(n 1). So the total number of transmissions scheduled by the MILD algorithm is at most n 1 2(n 1) 3(n 1), which is at most 3 times as large as the minimum total number of transmissions. 2 Theorem 4. The time complexity of the MILD algorithm is O(n2 T

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6 Conclusion In this paper, we investigate the IAGS-UDC problem. We prove this problem is NPhard, and propose an approximation algorithm MILD. This algorithm provides a correct and interference-free gossiping scheduling, and achieves a ratio of 3 2 ( 6) T . The number of transmissions scheduled by this algorithm is at most 3 times as large as the minimum number of transmissions. We also show that this algorithm is a polynomialtime algorithm.

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References 1. Chlamtac, I., Kutten, S.: On broadcasting in radio networks - problem analysis and protocol design. IEEE Transactions on Communications 33, 1240–1246 (1985) 2. Gandhi, R., Parthasarathy, S., Mishra, A.: Minimizing broadcast latency and redundancy in ad hoc networks. In: Proc. of ACM MobiHoc (2003) 3. Chrobak, M., Gasiniec, L., Rytter, W.: Fast broadcasting and gossiping in radio networks. Journal of Algorithms 43, 177–189 (2002) 4. Chrobak, M., Gasiniec, L., Rytter, W.: A randomized algorithm for gossiping in radio networks. In: Wang, J. (ed.) COCOON 2001. LNCS, vol. 2108, pp. 483–492. Springer, Heidelberg (2001) 5. Gasieniec, L., Lingas, A.: On adaptive deterministic gossiping in ad hoc radio networks. Information Processing Letters 83, 89–93 (2002) 6. Huang, S.C.H., Du, H., Park, E.K.: Minimum-latency gossiping in multi-hop wireless networks. In: Proc. of ACM MobiHoc (2008) 7. Gandhi, R., Kim, Y.-A., Lee, S., Ryu, J., Wan, P.J.: Approximation algorithms for data broadcast in wireless networks. In: Proc. of IEEE INFOCOM (2009) 8. Ni, S.Y., Tseng, Y.C., Chen, Y.S., Sheu, J.P.: The broadcast storm problem in a mobile ad hoc network. In: Proc. of ACM MobiCom (1999) 9. Wan, P.J., Wang, Z., Wan, Z., Huang, S.C.H., Liu, H.: Minimum-latency schedulings for group communications in multi-channel multihop wireless networks. In: Liu, B., Bestavros, A., Du, D.-Z., Wang, J. (eds.) WASA 2009. LNCS, vol. 5682, pp. 469–478. Springer, Heidelberg (2009) 10. Wang, F., Liu, J.: Duty-cycle-aware broadcast in wireless sensor networks. In: Proc. of IEEE INFOCOM (2009) 11. Guo, S., Gu, Y., Jiang, B., He, T.: Opportunistic flooding in low-duty-cycle wireless sensor networks with unreliable links. In: Proc. of ACM MobiCom (2009) 12. Hong, J., Cao, J., Li, W., Lu, S., Chen, D.: Sleeping schedule-aware minimum latency broadcast in wireless ad hoc networks. In: Proc. of IEEE ICC (2009) 13. Jiao, X., Lou, W., Ma, J., Cao, J., Wang, X., Zhou, X.: Duty-cycle-aware minimum latency broadcast scheduling in multi-hop wireless networks. In: Proc. of IEEE ICDCS (2010) 14. Huang, S.C.H., Wan, P.J., Deng, J., Han, Y.S.: Broadcast scheduling in interference environment. IEEE Transactions on Mobile Computing 7, 1338–1348 (2008)

A Game Theoretic Approach to Multi-radio Multi-channel Assignment in Wireless Networks Devu Manikantan Shila, Yu Cheng, and Tricha Anjali Illinois Institute of Technology {dmanikan,cheng,anjali}@iit.edu

Abstract. It has been long recognized that the interference among concurrent wireless transmissions plays a crucial role in limiting the performance of wireless networks. Recently, studies indicate that equipping nodes with multiple radios and operating these radios on multiple frequency channels can greatly enhance the capacity of wireless networks. On the other hand, to fully realize the beneﬁts of multi-radio multichannel communication, one may need to design an eﬃcient channel assignment algorithm. In this paper, we study the channel assignment problem by proposing an algorithm that achieves load balancing Nash Equilibrium solution even in a selﬁsh as well as topology-blind environment. Our simulation results also depict the eﬀectiveness of the proposed channel assignment solution.

1

Introduction

One of crucial challenges that aﬀect the performance of wireless networks is the presence of interference among multiple concurrent transmissions. With the motivation of enhancing the performance of wireless networks, recently great attention has been devoted to networks where each node is equipped with multiple radios and can operate on multiple channels [1],[9],[12]. This new degree of freedom has been shown to potentially allow for increased capacity with respect to single-channel single radio networks. On the other hand, to fully realize the beneﬁts of multi-radio multi-channel network, one may need to design eﬃcient channel assignment algorithms. In this paper, we study the so-called channel assignment problem from the perspective of game theory [2],[3],[4],[5] in a competitive wireless network. Though there exists a large body of literatures ([7],[9],[12] and the referenced therein) focusing on the problem of channel assignment in wireless networks, most of these works assume that nodes are collaborative and hence can achieve a high system performance. Nevertheless, this assumption is not true as the users of these nodes are usually selﬁsh and need to maximize their own utilities/performance without necessarily respecting the system goals. Therefore, we focus our attention on a non-cooperative or selﬁsh channel assignment game (game is referred to as selﬁsh since each node is required to agree on sharing a common wireless medium in a distributed manner) and analyze the scenario of a single collision clique where each node’s transmission can interfere G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 203–208, 2010. c Springer-Verlag Berlin Heidelberg 2010

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with the transmission of every other node. In [6], authors study a similar channel assignment game and prove the existence of load balancing Nash Equilibrium of this game. They also present a distributed algorithm based on perfect as well as imperfect information to achieve this load-balancing Nash Equilibrium. Nonetheless the algorithm using imperfect information can sometimes lead to an unbalanced channel allocation state i.e., state where some channels are completely under-utilized, which in turn reduces the throughput that each node can sustain. In this paper, we consider this issue of unbalanced channel allocation and design a novel multi-radio multi-channel allocation algorithm based on imperfect information for single collision clique wireless networks. Our proposed solution operates in three stages, each stage focusing on improving the total achievable data rate of each node. The rest of the paper is organized as follows: In section 2, we discuss our system model along with a game-theoretic description of non-cooperative channel assignment. Section 3 presents the proposed distributed algorithm and some simulation results to highlight the performance of the proposed solution. Finally, we summarize our work in section 4.

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We assume that the available frequency band is split into orthogonal channels of the same bandwidth using the Frequency Division Multiple Access (FDMA) method [1],[12] and channels obtained in this manner are the set of available denoted by C = c1 , c2 , · · · , c|C| , where |.| represents the number of elements in the corresponding set. In this paper, as in [6], we also assume that each node participates in only one communication scenario and communicates with each other over a single hop termed as single collision clique. Further, each node is equipped with k ≤ |C| radio transmitters, all having the same data rate. We consider a ﬁnite set of players 1 denoted by P = p1 , p2 , · · · , p|P | . The key objective of each player is to maximize his total throughput or channel utilization and such players are referred to as selfish players. This work also assumes that there is a mechanism that enables each player to communicate simultaneously by using multiple orthogonal frequency channels [8], [12]. We denote the set of radios of player pi using channel c by kpi ,c for every c ∈ C. For the purpose of suppressing the co-channel interference in a node, we also assume that diﬀerent radios of the same player cannot employ the same channel i.e., kpi ,c ≤ 1, where pi ∈ P and c ∈ C. Next, we formulate the channel assignment problem as a non-cooperative game, which corresponds to a ﬁxed channel allocation among the players. Each player’s strategy consists of deﬁning the number of radios on each of the channels. Hence, we deﬁne the strategy of player pi as its channel assignment vector: spi = (kpi ,1 , kpi ,2 , · · · kpi ,|C| ) The strategy vectors of all players deﬁne the strategy matrix S, where the row i of the matrix represents the strategy vector of player pi . 1

In this paper, we use the terms nodes, users, devices and players interchangeably.

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⎛

⎞ sp1 S = ⎝ ··· ⎠ sp|P | Furthermore, we denote the strategy matrix except for the strategy of player i by S−i (see [3],[5] for more detailson game theory). Let the total number of radios used by player pi be kpi = c k pi ,c . Likewise, kc deﬁnes the number of radios using a particular channel kc = pi kpi ,c . Since the players are assumed to be rational, their main objective is to maximize the payoﬀ i.e., achievable total rate, in the network. We denote the payoﬀ obtained by each player pi as part of the channel allocation process as Upi , i = 1 to |P |. Similar to [6], we also assume that the total rate achievable on channel c, denoted as R(kc ), is a decreasing function of the number of radios kc deployed on this channel. When player pi assigns its radio to channel c, its achievable rate on this channel can be written as Rpi ,c = kpi ,c · R(kc ). Since the total rate on a given channel is equally shared by all the radios assigned to it, we can say that the higher the number of radios in a given channel, the lower the rate per radio. We can obtain the brief, we can rate Rpi for each player pi by Rpi = c Rpi ,c . In write the payoﬀ function of each player pi as follows:Upi = Rpi = c Rpi ,c = c kpi ,c · R(kc ). We next summarize two important theorems from [2],[6] regarding a Nash Equilibrium [3],[4],[5] channel allocation for wireless networks which guide our development of the channel allocation algorithm in Section 3. Theorem 1. S ∗ is a NE iﬀ the following two conditions hold: (1) ηx,y ≤ 1 for any x, y ∈ C and (2) kpi ,c ≤ 1 for any c ∈ C, where ηx,y = |kx − ky |. Theorem 2. A NE channel allocation S ∗ is max-min fair iﬀ c∈Cmin kpi ,c = c∈Cmin kpj ,c for all pi , pj ∈ P , where Cmin is the minimum number of allocated channels. This implies that Upi = Upj , ∀ pi , pj ∈ P .

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Based on the theorems given in Section 2, we next present a distributed channel assignment algorithm (Algorithm 1) that enables the selﬁsh players to converge to the NE from an arbitrary initial conﬁguration. The proposed distributed solution operates in three stages. In the ﬁrst stage, each player, in a distributed manner, allocate its k radios to channels in a sequential manner. The implication is that the ﬁrst k channels i.e., ci ∈ |C| , i = 1 to k, will be occupied by all the k radios of each player. In [6], authors actually consider a random channel assignment, however we note that such random assignments can sometime lead to increased convergence time. After the sequential channel assignment process, in the second stage as well as in third stage, each player improves its total rate by reallocating those radios to the remaining channels. Similar to [6], to avoid that all players change its radios simultaneously, we employ the technique of backoﬀ mechanism seen in the IEEE 802.11 medium access technology [1],[10],[11]. When the backoﬀ counter reaches zero, bw = 0, players compare the number of radios

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Algorithm 1. NE Channel Assignment Algorithm with Local Information Stage 1: Each node, in a distributed manner, assign its radios to channels sequentially. for a given player pi ∈ |P | do for j = 1 to k radios do move each radio j to cj . end for end for Stage 2: After initial channel assignment, each player improves the total rate by switching its radios to |C| \ d = c1 , · · · , ck channels. Each player has knowledge of |·k only the number of radios on its own channel. Let kavg = |P|C| for a given player pi ∈ |P | do if (backoﬀ counter==0) then for j = 1 to k radios do radio j uses channel ∈ d = c1 , · · · , ck if kd − kavg ≥ 1 and kpi ,|C|\d = 0 then move the radio j from channel d to |C| \ d = c1 , · · · , ck with uniform 1 probability |C|\d . end if end for end if end for Stage 3: Convergence to NE. Let the average rate be Ravg = k ·R(kavg ), where R(kavg ) denotes the rate of channel with kavg radios. Let Cpi ∈ C be the channels used by player pi . while (true) do for a given player pi ∈ |P | do if (backoﬀ counter==0) then if R pi < Ravg then Cpi for j = 1 to k radios do if kCpi − kavg ≥ 1 then 1 move radio j to channel |C| \ Cpi with uniform probability |C|\C . pi end if end for end if end if end for end while

on its channel d, where d = c1 , · · · , ck , with the average number of radios, kavg = |P |·k . If the number of radios exceeds kavg , players reallocate those radios to |C| unused channels i.e., to |C| \ d = c1 , · · · , ck . Finally each player, after the second channel allocation stage, compares its observed total rate Rpi with the average estimated total rate. If the observed rate falls below the estimated rate, we need the players to reorganize its radios till an improved rate (i.e., greater than average estimated rate) is achieved. Note that, the second as well as third stage design

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focuses on the theorems 1 and 2 to ensure NE as well as fairness of the solution. Though our algorithm ensures NE channel allocation, sometimes the process is not fair. This is because some players will always get an advantage of channel assignment over others and correspondingly, higher payoﬀ. In Figure 1, we also present the performance of our algorithm for two diﬀerent scenarios (|P | = 10, |C| = 8, k = 3 and |P | = 10, |C| = 8, k = 4). The maximum number of rounds for the algorithm termination is set as 15.

Fig. 1. Payoﬀ of players, |P |, for k = 3 and |C| = 8 and |P |, for k = 4 and |C| = 8, = 10−4 for the proposed solution as well as the existing solution in [6] respectively

4

Conclusion

In this paper, we have devoted our attention on the problem of non-cooperative channel allocation among devices that use multiple radios. We developed a game theoretical solution with imperfect information that balance and maximize all of the players total payoﬀ under the constraints of the number of channels and radios in a selﬁsh as well as a topology-blind environment.

References 1. Kumar, A., Manjunath, D., Kuri, J.: Wireless Networking, pp. 1–448. Morgan Kaufmann Press, San Francisco (2008) 2. Felegyhazi, M., Hubaux, J.P.: Game theory in wireles networks: A tutorial. Technical Report LCA-REPORT-2006-002, EPFL (2006) 3. Fudenberg, D., Tirole, J.: Game Theory. MIT Press, MA (1991) 4. Gibbons, R.: A primer in game theory. Prentice Hall, New York (1992) 5. Osborne, M.J., Rubinstein, A.: A course in game theory. MIT Press, MA (1994) 6. Felegyhazi, M., Cagalj., M., Bidokhti, S.S., Hubaux, J.P.: Non-cooperative Multiradio Channel Allocation in Wireless Networks. In: Proceedings of IEEE INFOCOM, Alaska, USA (2007) 7. Adya, A., Bahl, P., Padhye, J., Wolman, A., Zhou, L.: A multi-radio uniﬁcation protocol for IEEE 802.11 wireless networks. In: Proceedings of Broadnets, Lausanne, Switzerland, pp. 344–354 (2004)

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8. Akyilidiz, I.F., Wang, X., Wang, W.: Wireless Mesh Networks: A Survey. In: Computer Networks, March 2005, vol. 47, pp. 445–487. Elsevier North-Holland, Inc., Amsterdam (2005) 9. Alicherry, M., Bhatia, R., Li, L.: Joint channel assignment and routing for throughput optimization in multi-radio wireless mesh networks. In: Proceedings of ACM/IEEE MOBICOM, Cologne, Germany, pp. 58–72 (2005) 10. Bianchi, G.: Performance analysis of the IEEE 802.11 distributed coordination function. IEEE Journal on Selected Areas in Communication 18(3) (March 2000) 11. Cagalj, M., Ganeriwal, S., Aad, I., Hubaux, J.P.: On selﬁsh behavior in CSMA/CA networks. In: Proceedings of IEEE INFOCOM, Miami, USA (2005) 12. Raniwala, A., Chiueh, T.C.: Architecture and Algorithms for an IEEE 802.11 based multi-channel wireless mesh network. In: Proceedings of IEEE INFOCOM, Miami (2005)

PAPR Analysis for SOFDM and NC-SOFDM Systems in Cognitive Radio Xue Li1 , Chi Zhou2 , Xiangqian Zhou1 , Zhiqiang Wu1 , and Bing Xie3 1

2

Wright State University Illinois Institute of Technology 3 Peking University

Abstract. Noncontiguous OFDM (NC-OFDM) and Noncontiguous Spread OFDM (NC-SOFDM) have been proposed as two candidates for cognitive radio and dynamic spectrum access network. Since the available spectrum in cognitive radio system usually spreads through multiple non-contiguous spectrum holes, multi-carrier based transceivers need to deactivate some of their subcarriers to avoid interference to primary users. However, like OFDM and SOFDM, NC-OFDM and NC-SOFDM also suﬀer from large peak to average power ratio (PAPR). Hence, it is desired to analyze the PAPR performance of NC-OFDM and NCSOFDM schemes with diﬀerent spreading code sets. In this paper, we analyze the PAPR performance of NC-OFDM, NCSOFDM with binary Hadamard-Walsh spreading codes, and NCSOFDM with polyphase Carrier Interferometry (CI) spreading codes. Both analysis and simulation results show that the NCSOFDM with polyphase CI code set provides the lowest PAPR, while the NC-OFDM provides the highest PAPR and NC-SOFDM with Hadamard-Walsh code set in the middle.

1

Introduction

Radio spectrum is a precious resource, and it has long been the dream of wireless communication engineers to maximize the utilization of the radio spectrum. Since the 1920’s, the Federal Communications Commission (FCC) has controlled the radio frequency spectrum. There are just a few narrow unlicensed bands left for anyone to use as long as they follow certain rules. Nevertheless, the unlicensed bands are always crowded. The traditional wireless communication systems employ wasteful static spectrum allocation which signiﬁcantly decreases the eﬃcient usage of spectrum. In November 2002, the FCC released a report generated by the Spectrum Policy Task Force [1] that reshaped the traditional models of spectral allocation and control. Studies in recent years have shown that a very large portion of the radio spectrum is unused or under-used for long periods of time [2]. New FCC policies signiﬁed the beginning of a new spectrum allocation paradigm called dynamic spectrum access (DSA). In this new paradigm, a cognitive radio transceiver can detect and harvest unused frequency bands to transmit its information while avoiding interference to incumbent primary users. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 209–219, 2010. c Springer-Verlag Berlin Heidelberg 2010

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To make better use of spectrum, cognitive radio transceivers need to transmit over multiple non-contiguous frequency “holes”. It is therefore natural to choose multi-carrier based technologies for the physical layer. For example, orthogonal frequency division multiplexing (OFDM) and spread OFDM (SOFDM) have been considered two strong candidates for cognitive radio and dynamic spectrum access. In both OFDM and SOFDM systems in a DSA network, the subcarriers located in the vicinity of an incumbent user need to be deactivated to avoid interference to the incumbent users. In the literature, these schemes are referred to as non-contiguous OFDM (NC-OFDM) or non-contiguous SOFDM (NC-SOFDM) [3][4]. It is a well known problem that all the multicarrier transmission technologies (e.g. MC-CDMA, OFDM) suﬀer from a large peak to average power ratio (PAPR) [5]. A lot of researcher proposed many methods to reduce PAPR for OFDM systems [6][7][8]. For SOFDM system, the PAPR varies for diﬀerent spreading code sets. The most common spreading code set is Hadamard-Walsh code set and the adaptive version for NC-SOFDM system. Besides, our previous work on carrier interferometry OFDM (CI/OFDM) [9] and CI/MC-CDMA [10][11][12][13][14], and other similar technologies such as single carrier OFDM (SC-OFDM) [15] and single carrier transmission with frequency domain equalization (SCFDE) [16] are essentially the same technology that combines the beneﬁt of multi-carrier transmission and single carrier transmission, by employing cyclic preﬁx to single carrier transmission and frequency domain processing. These technologies have received a lot of attention as the alternative transmission technique to the conventional OFDM because of their better performance and lower PAPR property compared with conventional OFDM. However for the NC-SOFDM, due to the non-contiguous subcarriers, the PAPR of the non-contiguous system diﬀers from the PAPR of the contiguous system. Hence, it is highly desired to analyze the PAPR performance of non-contiguous multi-carrier transmission schemes such as NC-OFDM and NCSOFDM. In this paper, we investigate the PAPR performance of SOFDM system and NC-SOFDM with diﬀerent spreading code sets for cognitive radio in a DSA network. With the analysis and simulation results among diﬀerent spreading code sets, we draw the conclusion that the CI code set still provides the lowest PAPR, outperorming both the NC-OFDM and NC-SOFDM with HadamardWalsh codes. The rest of the paper is organized as follows. In Section II, the system models for SOFDM and NC-SOFDM are presented. A brief introduction to the adaptive NC-SOFDM with Hadamard-Walsh spreading code is covered in Section III. The Carrier Interferometry (CI) code set is described in Section IV. Section V presents the analysis of PAPR performance of NC-OFDM and NC-SOFDM systems with non-spreading, Hadamard-Walsh code and CI code. Simulation results are reported in Section VI, followed by the conclusion.

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211

System Model

In a DSA network, portions of the spectrum are occupied by primary users. Multicarrier based cognitive radio transceivers can deactivate subcarriers overlapping with primary users’ transmission in frequency domain to avoid interference to incumbent users. Such a case is shown in Fig. 1. In Fig. 1, there are N subcarriers and one primary user is currently operating on M subcarriers. Deactivated subcarriers are shown in red line to avoid interference to the primary user. fN

f1

f M subcarriers deactivated

N-M non -contiguous active subcarriers

Fig. 1. Dynamic Spectrum Access and Deactivated Subcarriers

2.1

SOFDM

In a traditional SOFDM, assuming N total subcarriers, the transmission of the k th symbol corresponds to (using complex baseband notation) N −1 Eb (k) j2πi·Δf t (k) (k) s (t) = b β e g(t), (1) N i=0 i where b(k) is the k th information symbol (assumed +1 or −1 for ease of presen(k) tation) and βi is the ith value in symbol k’s spreading sequence (e.g. +1 or −1 in accordance with known spreading sequences such as Hadamard-Walsh sequences); i · Δf is the frequency position of the ith carrier component; and g(t) is a rectangular waveform of unity height which time-limits the code to one symbol duration Ts . It is crucial to note that selection of Δf = 1/Ts ensures subcarrier orthogonality over the time-limited duration of interest, i.e., over symbol duration Ts . It is also important to note that, to maintain the orthogonality among all the spreaded information symbols, the spreading code for each information symbol should be orthogonal. The transmitted signal at one SOFDM symbol at carrier frequency fc is N −1 N −1 Eb (k) (k) j2π(fc +i·Δf )t SSOF DM(t) = b βi e g(t). (2) N i=0 k=0

2.2

NC-SOFDM

Now, for an NC-SOFDM system operating in a DSA network with M subcarriers deactivated, only N − M subcarriers are still active. Let us deﬁne a new variable

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to represent if one subcarrier is deactivated or not: Ai = 1 if the ith carrier is active and Ai = 0 if the ith carrier is inactive. The transmitted signal of one NC-OFDM symbol corresponds to SN C−SOF DM (t) =

N −M−1 N −1 Eb (k) b(k) Ai · βi ej2π(fc +iΔf )t g(t). N −M i=0

(3)

k=0

Note that the NC-SOFDM only needs to spread its information over N − M subcarriers. Therefore, the transmission power on each subcarrier is increased accordingly. It is obvious that deactivated subcarriers result in a loss of orthogonality among the spreading codes and raise the irreducible error ﬂoor in NC-SOFDM [?].

3

NC-SOFDM Based on Hadamard-Walsh Code Set

For NC-SOFDM based on Hadamard-Walsh(HW) code set, the spread code is Hadamard-Walsh sequence that contains −1 and +1. As we mentioned earlier, deactivated subcarriers result in a loss of orthogonality among the spreading codes. To minimize or eliminate this loss of orthogonality, the spreading codes of an NC-SOFDM system can be adaptively selected. Let us denote the active subcarriers as f0 , f1 , . . . , fN −M−1 . Theoretically, such a N = N − M subcarrier SOFDM system should be able to support N orthogonal spreading codes, hence N orthogonal information symbols. However, Hadamard-Walsh spreading codes do not exist for every code length. Speciﬁcally, Hadamard-Walsh codes only exist when the code length is 1, 2 or a multiple of four. In the rest of the paper, we concentrate on scenarios when N ≥ 4. In other words, if N = 4m where m is an integer, we can ﬁnd a group of N orthogonal spreading codes and eliminate the loss of orthogonality. However, when N = 4m, such a group of orthogonal Hadamard-Walsh codes does not exist. 3.1

General Hadamard-Walsh Code Set for Maintaining Orthogonality

One easy approach to eliminating the loss of orthogonality is to deactivate more subcarriers. We can determine a group of Hadamard-Walsh codes with code length N4 where N4 is the largest multiple of four which is smaller than N , i.e., N4 = 4m, N4 ≤ N . This approach deactivates N − N4 more subcarriers than necessary, and allows the transmission of up to N4 information symbols. 3.2

General Hadamard-Walsh Code Set for Maintaining Data Rate

For NC-SOFDM based on Hadamard-Walsh code set, the best we can do to maintain the data rate is to determine a group of Hadamard-Walsh codes with code length N4 where N4 is the smallest multiple of four which is larger than

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N , i.e., N4 = 4m, N4 ≥ N . While this approach does not eliminate the loss of orthogonality, it does reduce the loss of orthogonality signiﬁcantly compared with that of Hadamard-Walsh codes with code length N . In subsequent sections, we will use NC-SOFDM with Hadamard-Walsh code set that maintains the data rate as a reference for performance comparison.

4

NC-SOFDM Based on CI Code Sets

In an NC-SOFDM system with CI code, a complex orthogonal Carrier Interferometry spreading code set is employed. The k th symbol’s spreading code for the lth subcarrier corresponds to (k)

βl

= exp(−j

2π k · l). N

(4)

It is important to note that CI spreading codes deﬁned in equation (4) are orthogonal spreading codes for any code length N , which can be easily adapted into NC-SOFDM system.

5

PAPR Performance Analysis

In this section, we provide an analytical comparison of the PAPR performance of OFDM, SOFDM and NC-SOFDM systems with diﬀerent spreading code sets. In [5], a discrete PAPR of an OFDM symbol (or NC-SOFDM symbol) is deﬁned as square of the maximum amplitude divided by the mean of power of discrete symbols in time domain. Assuming the symbol vector in time domain to be s = [s0 , s1 , ..., sN −1 ], the maximum amplitude can be denoted as ||s||∞ = max(|s0 |, |s1 |, ..., |sN −1 |), and the mean of power is ||s||22 = (|s0 |2 + |s1 |2 + ... + |s2N −1 |)/N . And then the PAPR is deﬁned as ||s||2∞ . (5) P AP R = ||s||22 5.1

PAPR Analysis for SOFDM System

For SOFDM system, when using matrix representation to express the transmitted symbols in time domain, we get: Eb s= bCFH (6) N where b = [b(0) , b(1) , ..., b(N −1) ] represents the transmitted symbols in frequency domain and s = [s0 , s1 , ..., sN −1 ] is the transmitted samples in time domain. Ma(k) trix C denotes the spreading code matrix (e.g. C(k, l) = βl ). F is the Discrete H Fourier Transform (DFT) matrix and F presents the conjugate transpose of F.

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In an OFDM or SOFDM system, when the signal in time domain converges to one peak, the worst case PAPR is observed. For traditional OFDM system, the spreading code set reduces to an identity matrix, and the worst case is that when bk = (−1)k , the power converges into one sample in time domain Eb s= bIFH = [0, 0, ..., Eb · N, 0, ..., 0] (7) N then the maximum |s|2 is Eb · N and the mean |s|2 is Eb . Hence PAPR= N . It is obvious that the maximal PAPR of the OFDM system depends on the number of subcarriers. Similar to OFDM system, when applying Hadamard-Walsh code set to the SOFDM system, the PAPR also depends on the number of total subcarriers and the input data symbols. However, the CI code provides the system a very nice property of PAPR. For (k) the CI code, it is important to notice that the spreading code βl in Eq. (4) th th is exactly the k column and l row in the DFT matrix. Hence the spreading code matrix will cancel out the eﬀect of the Inverse DFT (IDFT). Eb Eb Eb H H s= bCF = bFF = b·N (8) N N N Hence, no matter what b is, the PAPR maintains a constant as long as (Phase Shift Keying) PSK modulation is used, because: |s|2 = N · Eb |b|2 = N · Eb [1, 1, 1, ..., 1]

(9)

and the maximum |s|2 is N · Eb and the mean |s|2 is N · Eb . Hence PAPR= 1 which is a constant and the lowest PAPR value can be obtained. 5.2

PAPR Performance Analysis for NC-SOFDM System

For NC-SOFDM system, we also use matrix representation to express the transmitted symbols in time domain, s contains N elements: Eb bCFH (10) s= N −M Here, matrix C denotes the N × N spreading code matrix and F is an N -point DFT matrix. C(k, l) presents the spreading code from the k th symbol to the lth subcarrier. Due to the non-contiguous property, the lth column is zero if the lth subcarrier is deactivated, and there are M zero columns in C. Unlike contiguous SOFDM with PSK modulation, the NC-SOFDM system deactivates some subcarriers, and the PAPR performance will change. Accordingly, the CI code set cannot maintain PAPR= 1 any more. For better analysis we can represent the s by: Eb bCp Fp H (11) s= N −M

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where Cp is partial spreading code matrix, which is (N − M ) × (N − M ) square matrix and does not include the deactivated columns. Fp is the partial N -point DFT matrix which is N × (N − M ) matrix and does not include the deactivated columns. In the SOFDM system with CI code, it is obvious that the spreading matrix compensate with the matrix FH by creating an identity matrix, and then the time domain symbol vector s is simply the transmitted symbol vector b multiplying with a constant. However, for the NC-SOFDM system, it is impossible to build an orthonormal matrix C to compensate the Fp H matrix which has higher dimension ((N − M ) < N ), since only (N − M ) rows in matrix C can not be orthogonal to N columns in matrix Fp H .

6

Simulation Results

In this section, we compare the PAPR performance of contiguous and noncontiguous OFDM, SOFDM with Hadamard-Walsh code set that maintains the data rate and SOFDM with Carrier Interferometry code set. All the systems employ N = 128 subcarriers when no subcarrier is deactivated. To compare the PAPR property of diﬀerent systems, we will analyze the cumulative distribution function (CDF) of the PAPR deﬁned in Eq. (12) according to diﬀerent scenarios. P (P AP R ≤ z) = CDF (z) (12) 6.1

SOFDM System

In the SOFDM system, no subcarriers are deactivated. Fig. 2 presents the CDF of PAPR deﬁned in Eq. (5), and the maximum, minimal and average values are shown in table 1. It is evident that the PAPR

PAPR Comparison for BPSK, 128 out of 128 subcarriers are activate 1 0.9 0.8 0.7

CDF

0.6 OFDM SOFDM with HW SOFDM with CI

0.5 0.4 0.3 0.2 0.1 0

0

2

4

6 PAPR(dB)

8

10

12

Fig. 2. The CDF of PAPR for SOFDM with diﬀerent spreading code sets

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Spread Max(dB) Min(dB) AVG(dB) OFDM 12.5527 3.6072 6.7913 HW 9.9699 3.2984 5.6886 CI 1.6376e-014 3.0079e-015 7.7825e-015

of CI code maintains 0dB, while Hadamard-Walsh code set has much higher PAPR, and the OFDM has the worst PAPR since it does not apply any spreading schemes. 6.2

NC-SOFDM System

In the NC-SOFDM system, for ease of presentation, we assume that one primary user occupies one folder of subcarriers in the middle band and the primary user will cause the deactivation of M subcarriers. Fig. 3 shows the CDF of the PAPR for NC-SOFDM system with diﬀerent spreading code sets, with 23 out of 128 subcarriers are active. And the maximum, minimal and average values are shown in table 2. For non-contiguous case, as analyzed in previous section, since there does not exist an (N − M ) basis to be

PAPR Comparison for BPSK, 23 out of 128 subcarriers are activate 1 0.9 0.8 0.7

NC−OFDM NC−SOFDM with HW NC−SOFDM with CI

CDF

0.6 0.5 0.4 0.3 0.2 0.1 0

0

2

4

6 PAPR(dB)

8

10

12

Fig. 3. The CDF of PAPR for NC-SOFDM with diﬀerent spreading code sets Table 2. PAPR analysis for NC-SOFDM system

Spread Max(dB) Min(dB) AVG(dB) OFDM 11.9578 2.5481 5.8972 HW 9.859 2.4157 5.446 CI 6.755 1.5279 4.4359

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orthogonal with N columns in Fp matrix, the CI code cannot maintain PAPR= 0dB any more. However, it is clear that CI code still provides the lowest PAPR because it uses polyphase DFT code for spreading. The Hadamard-Walsh code set also provides higher PAPR compared to CI code set. Meanwhile, NC-OFDM has the largest PAPR due to the non-spreading nature. Fig. 4 and Fig. 5 illustrate the maximum and average PAPR versus the number of active subcarriers to show the worst and average performance, respectively. As we mentioned before, the PAPR is related to the number of active subcarriers. When the number of active subcarriers increases, the PAPR of these three cases all tend to become larger. No matter the worst case or the average PAPR, the CI code maintains the best compared to the other two spreading schemes and the

PAPR Comparison for BPSK with total 128 subcarriers 14 NC−OFDM NC−SOFDM with HW NC−SOFDM with CI

13

Maximum PAPR (dB)

12 11 10 9 8 7 6 5 4

8

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14 16 18 number of active subcarriers

20

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24

Fig. 4. The maximum PAPR v.s. number of active subcarriers for NC-SOFDM with diﬀerent spreading code sets PAPR Comparison for BPSK with total 128 subcarriers 6

5.5

Average PAPR (dB)

5

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NC−OFDM NC−SOFDM with HW NC−SOFDM with CI

3.5

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14 16 18 number of active subcarriers

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Fig. 5. The average PAPR v.s. number of active subcarriers for NC-SOFDM with diﬀerent spreading code sets

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variation of CI code is smaller too. The PAPR of adaptive Hadamard-Walsh code set is very sensitive to the number of active subcarriers, and when N − M = 4m the maximum PAPR reduces obviously since in these cases no adaptation is needed for the Hadamard-Walsh code; it provides higher PAPR compared to CI code set, but it is still better than the NC-OFDM.

7

Conclusions

In this paper, we analyzed the PAPR performance for diﬀerent spread coding schemes in SOFDM and NC-SOFDM systems for cognitive radio in a DSA network. With the matrix expression, the PAPR of all systems are derived, and the dimension of the matrix expression shows the limitation of NC-SOFDM to reduce the PAPR. Both the analysis and the simulation results illustrate that the CI code provides the lowest PAPR for all the cases, while the PAPR of Hadamard-Walsh code set and adaptive Hadamard-Walsh code set is much higher and sensitive to the number of active subcarriers, and the non-spreading scheme produces the worst PAPR.

References 1. FCC - notice of proposed rulemaking and order, facilitating opportunities for ﬂexible, eﬃcient and reliable spectrum use employing cognitive radio technologies. FCC Document ET Docket No. 03-108 (December 2003) 2. Mitola, J.: Cognitive radio: an integrated agent architecture for software deﬁned radio, Ph.D. dissertation, KTH Royal Institute of Technology, Stockholm, Sweden (2000) 3. Poston, J.D., Horne, W.D.: Discontiguous OFDM considerations for dynamic spectrum access in idel TV channels. IEEE DySPAN (2005) 4. Rajbanshi, R., Chen, Q., Wyglinski, A., Minden, G., Evans, J.: Quantitative comparison of agile modulation technique for cognitive radio tranceivers. IEEE CCNC (Jaunary 2007) 5. Goldsmith, A.: Wireless Communications. Cambridge University Press, Cambridge (2005) 6. Hussain, S., Louet, Y.: PAPR reduction of Software Radio signals using PRC method. In: Sarnoﬀ Symposium, SARNOFF 2009, March 30-April 1, pp. 1–6. IEEE, Los Alamitos (2009) 7. Kim, S.-W., Chung, J.-K., Ryu, H.-G.: PAPR Reduction of the OFDM Signal by the SLM-based WHT and DSI Method. In: TENCON 2006, IEEE Region 10 Conference, November 14-17, pp. 1–4 (2006) 8. Guel, D., Palicot, J.: OFDM PAPR Reduction Based on Nonlinear Functions without BER Degradation and Out-of-Band Emission. In: 2009 International Conference on Signal Processing Systems, May 15-17, pp. 167–171 (2009) 9. Li, X., Zhou, R., Chakravarthy, V., Wu, Z.: Intercarrier Interference Immune Single Carrier OFDM via Magnitude Shift Keying Modulation. In: Global Telecommunications Conference, GLOBECOM 2009, December 2009, pp. 1–6. IEEE, Los Alamitos (2009)

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10. Natarajan, B., Nassar, C.R., Shattil, S., Wu, Z.: High-Performance MC-CDMA via Carrier Interferometry Codes. IEEE Transactions on Vehicular Technology 50(6), 1344–1353 (2001) 11. Natarajan, B., Wu, Z., Nassar, C.R.: Large Set of CI Spreading Codes for HighCapacity MC-CDMA. IEEE Transactions on Communications 52(11), 1862–1866 (2004) 12. Hijazi, S., Natarajan, B., Michelini, M., Wu, Z.: Flexible Spectrum Use and Better Coexistence at the Physical Layer of Future Wireless Systems via a Multicarrier Platform. IEEE Wireless Communications 11(2), 64–71 (2004) 13. Wu, Z., Natarajan, B., Nassar, C.: The Road to 4G: Two Paradigm Shifts, One Enabling Technology. In: IEEE DySPAN 2005 (2005) 14. Wu, Z., Ratazzi, P., Chakravarthy, V., Hong, L.: Performance evaluation of adaptive non-contiguous MC-CDMA and non-contiguous CIMC-CDMA for dynamic spectrum access. In: 3rd Int’l. Conf. on Cognitive Radio Oriented Wireless Networks and Communications (2008) 15. Kobayashi, H., Fukuhara, T., Yuan, H., Takeuchi, Y.: Proposal of single carrier OFDM technique with adaptive modulation method. In: IEEE Vehicular Technology Conference, April 2003, vol. 3, pp. 1915–1919 (2003) 16. Czylwik, A.: Comparison between adaptive OFDM and single carrier modulation with frequency domain equalization. In: IEEE Vehicular Technology Conference, May 1997, vol. 2, pp. 865–869 (1997)

Application of Compressed Sensing for Secure Image Coding Gesen Zhang, Shuhong Jiao, and Xiaoli Xu Information and Communication Engineering College, Harbin Engineering University, Nantong Street 145, Harbin, China {zhanggesen,jiaoshuhong,xuxiaoli}@hrbeu.edu.cn

Abstract. The secure image coding scheme using compressed sensing (CS) is proposed and the secrecy of the scheme is explored. We verify that the CS-based coding scheme can provide a guarantee of secrecy by analysis and simulation. In our approach, random matrices are used as keys of decryption. Based on the feasibility of random symmetric signs matrices in compressed sensing, we obtain a theoretical result that the signal compressed sensing using sparse random binary matrices can be exactly recovered with high probability. Numerical results verify the theory and show matrices proposed in this paper perform equally to the prominent Gaussian matrices when measurement rate is higher than an equivalence threshold. Keywords: compressed sensing, secure image coding, projection matrices, key matrices, sparse random binary matrices.

1

Introduction

In real-time signal and image coding, how to minimize the required number of sampling with high speed has been a subject of extensive research. In the past several years, a new theory tackling this issue named compressed sensing, which seeks to reconstruct a compressible signal from a minimal number of linear measurements acquired by a non-adaptive projection matrix, was ﬁrstly introduced in [1] and [2]. Moreover, encryption became an inherent byproduct of CS which is ensured by the use of a random matrix maintaining the measurements meaningless to the undesired observers [3]. The chief goal of an image coding system is to minimize the bandwidth and storage requirements for transmission in channel while maintaining the reconstruction error below certain speciﬁed maximum allowances. Moreover, security of the transmission must be considered when the contents are private [4]. So far, either some custom design joint compression and encryption schemes or traditional cryptographic algorithms have been developed. The recent proposed compressed sensing theory introduces an innovation with the result that encryption is inherently embedded into image coding with high speed due to the using of random projection matrices in projection process. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 220–224, 2010. c Springer-Verlag Berlin Heidelberg 2010

Application of Compressed Sensing for Secure Image Coding

2 2.1

221

Secure Image Coding via Compressed Sensing Secrecy of Compressed Sensing

In terms of the security, an immediate consideration is the brute force attack; that is an eavesdropper may guess the non-adaptive projection matrices with overwhelming computation. However, the measurements are completely worthless to the eavesdropper on account of the noise like image (reconstructed using wrong keys) and size of the random matrix. Moreover, the eavesdropper may have a certain ratio of the knowledge about the key in some circumstance. By the PSNR’s number in Fig. 1(b) with repeating each experiments 100 times (measurement rate is 40% and the basic scheme goes to particulars in section 2.2), we get that the eavesdropper can hardly identify the image even through having the knowledge with high rate as long as not the complete key matrix. As the size of general images, we have that CS via random matrices, including the matrices proposed in section 3, provides a guarantee of secrecy. Note that we treat the Lena image (256×256) as 256 vectors rather than a long vector of size 65536 to save the computational load. 25

PSNR(dB)

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(a) Original Lena image

0

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(b) Rate of the eavesdropper’s knowledge about the key

Fig. 1. (a) Original Lena image (256×256); (b) rate of the eavesdropper’s knowledge about the key against the PSNR in dB (measurement rate is 40%)

2.2

CS-Based Image Coding Scheme

A desirable coding scheme provides a trade-oﬀ in spatial and temporal resolutions. CS theory allows exact recovery of a compressible signal from a dynamic small number of samples which are very feasible in image coding. In CS-based secure image coding scheme, coding process directly condenses the image into a number of linear digital measurements using a random matrix out of a matrices dictionary. The trade-oﬀ in spatial and temporal resolutions is adapted by the adjustable measurement number and matrix sparsity out of the matrices dictionary. The scheme is especially suitable high-speed image acquisition. In decoding process, there have been a lot of outstanding works on this topic

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[5] [6]. The chief diﬀerence between this scheme and the conventional approach are the asymmetric size of projection matrix and the unity of encryption and compression.

3

Sparse Random Binary Key Matrices

In [7], Shannon puts forward the classical criteria for good secrecy system; one of the criteria is that sometimes the key must be memorized and it is therefore desirable to have the key as small as possible. However, the key matrix of CSbased encryption inherently induces the inconvenience of the use of private key encryption. Then, it is obviously signiﬁcant to explore the performances of simple structure key matrices such as sparse random binary matrices. In terms of the generating of the projection matrices, the restricted isometry property (RIP) is frequently employed as a suﬃcient condition on projection [8]. Some work in [9] has shown the symmetric signs distribution matrices or its related sparse distribution (with sparsity ≤ 2/3) satisfy RIP with overwhelming probability. The result of the relationship between random symmetric signs distribution and RIP is recapitulated as follows. Let Φ be a random n×N matrix according to symmetric signs or its related sparse distribution. Given n, N , and any K ≤c0 n/ log(N/K), there exist constants c0 , c1 > 0 depending only on δ prescribed as δ3k in RIP [8] and Φ satisﬁes RIP with probability ≥ 1 − 2 exp(−c1 n), (1) where c1 ≤ (3δ 2 − δ 3 )/48 − c0 [1 + (1 + log(12/δ))/ log(N/K)].

(2)

Zero setting all the negative numerical values in Φ, we get random binary distribution and denote its entries with the form √ +2 n with probability 1/s, φij = (3) 0 with probability 1-1/s. An n×N matrix drawn from the random distribution in (3) of order 2 < s ≤ 6 is deﬁned by sparse random binary matrix. For the purpose of verifying that the signal vector compressed sensing using sparse random binary matrices can be exactly recovered with high probability, we generate a random sparse symmetric signs matrix Φs (with sparsity 2/3). Obviously, Φs can be decomposed into two sparse binary matrices of order s = 6 with the form Φs = Φb − Φb . Φs as well as its two sparse binary counterparts has random entries drawn independently. We then have that for each measurement yi (x, Φs ) = yi (x, Φb ) − yi (x, Φb ), 1 ≤ i ≤ n.

(4)

From the above, we know that random symmetric signs matrices satisﬁes RIP with high probability; that is this sort is “good” CS matrix with high probability. However, RIP is just a suﬃcient condition of signal exact recovery. As

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an alterative, “good” CS matrices do not damage the information in the signal vector [10]. We may denote an information damage factor which get involved in probability computation; that is if an inner product do not damage the information in the image or signal, λ = 1. Clearly, for the process in (4) of combining signal points, we have λ(Φb ) ≥ λ(Φs ). Each measurement acquired by Φs capture partial (average 1/6) information components in the signal vector. We note that it is enough to prove the result in the case that the sum of information components in the signal equal to 1; so we know that the information components in the signal vector tend to 1 when the length is large enough; that is with probability 1 − (5/6)n = 1 − 5n /6n projection matrix Φs capture all the signal vector points in the signal. Then we know that if Φs satisﬁes RIP, the signal observed with its counterpart Φb can be exactly recovered with high probability. Taking the indicated above into account, we can easily derive result as follows. There exist an N ×N random symmetric signs matrix Φs . Given n, N , and any K ≤c0 n/ log(N/K), there exist constants c0 , c1 > 0 depending only on δ prescribed as δ3k in RIP [8] and Φ satisﬁes RIP with probability (1). Then signal compressed sensing using sparse random binary matrix Φb can be exactly recovered with probability ≥ [1 − 2 exp(−c1 n)](1 − 5n /6n ),

(5)

where c1 according to (2). That is the signal projected by sparse random binary matrices can be exactly recovered with high probability when the signal is in general length. 32 30

PSNR(dB)

28 26 24 22 Gaussian CS sparse random binary CS (s=4) very sparse random binary CS (s=8)

20 18 0.35

0.4

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0.5 0.55 Measurement rate

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Fig. 2. PSNR versus measurement rate for Lena image (256×256) compressed sensing using diﬀerent matrices

In Fig. 2, we illustrate the performances of a typical sparse random binary matrix (s = 4) and a typical very sparse random binary matrix (s = 8). Lena image is reconstructed by the uniform l1 problem solver [6] implemented in the wavelet-domain. We treat the images as 256 vectors rather than a vector of the size 65536 × 1 and compare PSNRs against the measurement rate by repeating

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each experiments 100 times. Numerical results show that the matrices proposed and Gaussian matrices perform equally well when the measurement rate is higher than an equivalence threshold (0.5).

4

Conclusions

In this paper, the CS-based secure image coding scheme is proposed. Analysis and simulation show that the image coding scheme provides an assurance of secrecy. We get that the eavesdropper can hardly identify the image even through having the knowledge with high rate as long as not the complete key matrix. We also get a theoretical result that the signal compressed sensing using sparse random binary matrices can be exactly recovered with high probability. The results of PSNR against the diﬀerent matrices verify the theory and show the matrices proposed in this paper perform equally to the Gaussian matrices when the measurement rate is higher than an equivalence threshold.

References 1. Cand`es, E.J., Romberg, J., Tao, T.: Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information. IEEE Trans. Inform. Theory 52, 489–509 (2006) 2. Donoho, D.L.: Compressed Sensing. IEEE Trans. Inform. Theory 52, 1289–1306 (2006) 3. Duarte, M., Wakin, M., Baron, D., Baraniuk, R.: Universal Distributed Sensing via Random Projections. In: Proceedings of the 5th International Conference on Information Processing in Sensor Networks, pp. 177–185. ACM Press, New York (2006) 4. Lian, S., Liu, Z., Ren, Z., Wang, H.: Secure Advanced Video Coding Based on Selective Encryption Algorithms. IEEE Trans. on Consum. Electr. 52, 621–629 (2006) 5. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear Total Variation Based Noise Removal Algorithms. Physical D 60, 259–268 (1992) 6. Tropp, J.A., Gilbert, A.C.: Signal Recovery from Partial Information via Orthogonal Matching Pursuit. IEEE Trans. Inform. Theory 53, 4655–4666 (2007) 7. Shannon, C.E.: Communication Theory of Secrecy Systems. Bell System Technical Journal 28, 654–715 (1949) 8. Cand`es, E.J.: The Restricted Isometry Property and Its Implications for Compressed Sensing. C. R. Acad. Sci. Paris Sr. I, Math. 346, 589–592 (2008) 9. Baraniuk, R.G., Davenport, M., DeVore, R.A., Wakin, M.: A Simple Proof of the Restricted Isometry Property for Random Matrices. Constructive Approximation 28, 253–263 (2008) 10. Cand`es, E.J., Tao, T.: Decoding by Linear Programming. IEEE Trans. Inform. Theory 51, 4203–4215 (2005)

Eﬃcient Wireless Broadcasting Using Onion Decoding Pei Wang, Qunfeng Dong, Mingjun Xiao, and Liusheng Huang Department of Computer Science and Technology University of Science and Technology of China, Hefei, China 230027 [email protected], {qunfeng,xiaomj,lshuang}@ustc.edu.cn

Abstract. The broadcast nature of wireless transmissions is a twoedged sword for wireless broadcasting. On one hand, it makes broadcasting much more eﬃcient; on the other hand, it causes concurrent transmissions much more likely to collide, deteriorating throughput and delay. In this paper, we propose a novel PHY layer technique called onion decoding, which enables correct decoding of two or more colliding transmissions. Thus, more concurrent transmissions can be scheduled, leading to improved throughput and delay. As we shall see in the paper, achieving optimal broadcast throughput and delay, with and without onion decoding, are non-trivial in multi-rate wireless networks. No eﬃcient algorithm is known. We propose a simple heuristic algorithm for wireless broadcasting using onion decoding, and evaluate its performance through simulations. Simulation results conﬁrm that onion decoding can signiﬁcantly improve throughput and delay of wireless broadcasting. Keywords: wireless broadcast, onion decoding, throughput, delay.

1

Introduction

The broadcast nature of wireless medium is a two-edged sword for broadcasting. The fact that a single wireless transmission can reach multiple neighbors simultaneously also means potentially colliding with more other transmissions reaching these nodes, deteriorating overall throughput and delay (as the transmissions will have to proceed sequentially instead of concurrently). Therefore, being able to decode colliding transmissions would greatly improve performance of wireless broadcasting. Because nodes can parallelize their transmissions more aggressively, leading to improved throughput and delay. While some PHY layer techniques have been proposed, none is well suited for wireless broadcasting. For instance, successive interference cancellation [1] requires one or more of the colliding packets are already known, which is often not the case. ZigZag decoding [7] requires the colliding packets must collide together multiple times. This requirement may be appropriate for unicast packets (which are typically retransmitted multiple times), but makes a poor choice for wireless broadcasting, because wireless broadcast packets are typically not acknowledged and not retransmitted, as is the case in IEEE 802.11. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 225–234, 2010. c Springer-Verlag Berlin Heidelberg 2010

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Fig. 1. A multi-rate example

Fig. 2. A collision packet and duplication packets

In this paper, we propose onion decoding, a novel PHY layer technique for decoding colliding transmissions. In contrast with previous proposals, onion decoding well suits wireless broadcasting, as it does not need to know any colliding packet, and is able to decode even if concurrent transmissions of a packet collide once only. To see how onion decoding works, consider the simple example in Figure 1, where node 1 has a packet to broadcast. The number beside each link is the highest allowable transmission rate on that link. Suppose nodes 2 and 3 forward the packet concurrently, resulting in a collision at node 5. For ease of discussion, we refer to a forwarded packet as duplication packet, and refer to the packet resulting from a collision as collision packet. The collision packet and duplication packets are shown in Figure 2. If the two duplications packets, denoted by A and B, are perfectly aligned in time, they merely result in stronger signals and can be correctly decoded with a standard decoder. However, if one duplication packet, say A, is ahead of B by a certain amount of time (denoted by Δ), both transmissions are considered corrupt and lost. Onion decoding, in contrast, is able to iteratively decode the collision packet piece by piece, like slicing an onion. First, the receiving node 5 can decode the collision-free piece (i.e., the Δ part) using a standard decoder. This collision-free piece 1 of the collision packet is chunk 1 of the original packet, and is used to bootstrap the entire onion decoding process. In particular, the receiving node can subtract chunk 1 from piece 2 of the collision packet to obtain chunk 2 of the original packet. This simple decoding process iteratively proceeds until all pieces of the collision packet have been decoded, and the original packet is obtained. Using onion decoding, throughput and delay of wireless broadcasting can be signiﬁcantly improved. For the example in Figure 1, let us say it takes one time slot to transmit a packet at 11Mbps. With a standard decoder, node 2 and node 3 cannot transmit concurrently; the total time needed to broadcast a packet is

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23 time slots. In contrast, with the support from onion decoding, node 2 and 3 can now transmit concurrently. 12 time slots have been enough. In this paper, we also propose a simple heuristic algorithm for eﬃcient wireless broadcasting using onion decoding, and evaluate its performance through MATLAB simulations. Our simulation results conﬁrm that onion decoding can signiﬁcantly eﬀectively improve both throughput and delay.

2

Onion Decoding

Upon detecting a packet, an onion decoder ﬁrst tries to decode it using a standard decoder, assuming no collision. If the decoded packet passes checksum test, reception is successful. No harm is caused by onion decoding. If the decoded packet fails checksum test, the onion decoder makes additional eﬀorts (i.e., onion decoding) in hope of saving the packet. First, the onion decoder checks whether the packet is a collision packet. If it is, the onion decoder tries to decode the packet piece by piece, using onion decoding. If the decoded passes checksum test, the collision packet is saved. If the decoded packet again fails checksum test, the packet is considered undecodable and discarded. There is no additional performance cost due to the failure, since a standard decoder has failed already. 2.1

Detecting and Locating Collisions

As explained at the beginning of this section, we need to determine whether the packet being received is a collision packet, before starting onion decoding. Moreover, we also need to determine the exact position of collision within the collision packet, so that the oﬀset Δ can be determined for bootstrapping the onion decoding process. One way to detect and locate packet collision is to correlate packet preambles. Packets being transmitted over wireless medium typically start with a known preamble. For example, all 802.11 packets start with such a known preamble. The preamble is a sequence of L samples, which possesses a unique correlation property. Speciﬁcally, if a sequence of these L samples are aligned with another sequence of the same L samples, their correlation spikes. In contrast, if a sequence of these L samples are aligned with any other sequence of L samples, which can be part of a data packet or a misaligned version of the preamble, their correlation will be near zero. Therefore, the receiver can align the preamble with incoming symbols, and recompute the correlation for each incoming symbol until the end of packet reception. If a correlation spikes in the middle of packet reception, we know another packet is colliding with the packet being received. The position of the correlation spike is precisely the beginning of the colliding packet. Δ is thus determined. 2.2

More Than Two Interferers

In Section 1, we have brieﬂy explained the process of decoding two colliding duplication packets using onion decoding. In general, onion decoding can decode

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Fig. 3. More than two interferers

collision packets caused by multiple colliding duplication packets. For an illustrative example, consider a collision packet caused by three duplication packets as shown in Figure 3. Δ1 is the oﬀset between duplication packet A and duplication packet B. Δ2 is the oﬀset between duplication packet B and duplication packet C. Initially, onion decoder bootstraps the process with P iece 1, which is a collision free chunk of the original packet. It decodes P iece 1 directly using a standard decoder, and obtains Chunk 1. Then, it re-encodes Chunk 1 according to the characteristics (e.g. attenuation, phase shift) of duplication packets B and C, in order to obtain an estimation of Chunk 1’s symbols in B and C. After subtracting the re-encoded estimation values from P iece 2 of the collision packet, Chunk 2 can be decoded from the remaining symbols of P iece 2. The onion decoder works this way iteratively. During the ith iteration, it re-encodes Chunk i decoded in the (i − 1)th round, according to the characteristics of duplication B and C. Then, it subtracts the re-encoded symbols from P iece (i + 1) of the collision packet to obtain Chunk (i + 1). This iterative decoding process continues until all chunks of the original packet have been successfully decoded.

3

Challenges

In this section, we discuss the challenges of using onion decoding for eﬃcient wireless broadcasting. Before we proceed to the discussion, it is worth pointing out that broadcast throughput maximization and broadcast delay minimization are actually one problem. They all seek to broadcast n packets within a minimum amount of time. Delay minimization is a special case of throughput optimization, where the number of packets to be broadcast is n = 1. That said, in our discussion we shall focus on analyzing the time needed to broadcast a packet. Maximizing throughput and minimizing delay of wireless broadcasting are non-trivial problems. Even in homogeneous single-rate wireless networks, the problem of minimum delay broadcasting is known to be NP-hard [6]. Since minimum delay broadcasting is a special case of maximum throughput broadcasting, the latter is clearly NP-hard as well. Moreover, as single-rate networks are special cases of multi-rate networks, these problems are also NP-hard in multi-rate wireless networks. Here, we brieﬂy discuss a few major challenges of wireless broadcasting in multi-rate networks, with possible support from onion decoding.

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Fig. 4. The same node may need to transmit the same packet more than once, in order to achieve optimal performance

3.1

How Many Times to Transmit?

One of the key questions in multi-rate broadcasting is whether a node should transmit duplication packet just once, or multiple times (even if previous transmissions are successful). We show that, in multi-rate networks, the same node transmitting the same packet multiple times at diﬀerent data rates may reduce the total time needed to broadcast a packet. To demonstrate this interesting fact, consider the network in Figure 4. There are 2n nodes in the network, evenly distributed in two rows. In the sequel, the number beside each link is the number of time slots needed to transmit a packet at the highest allowable data rate on that link. If each node transmits the same packet once only, they have to use the lower data rate, which takes m time slots to transmit a packet. In total, nm time slots are needed. In contrast, if nodes do consider the option of transmitting the same packet multiple times, total broadcast time can be signiﬁcantly reduced. In particular, broadcasting can be scheduled to ﬁnish in two rounds. – In the ﬁrst round, nodes in the top row transmit the packet in turn, at the higher data rate. This process takes n − 1 time slots in total. By the end of this round, every node in the top row shall have received the packet. – In the second round, nodes in the top row transmit the packet at the lower data rate simultaneously, which takes m time slots. Every node in the bottom row will receive the packet. In total, m + n − 1 time slots are enough. When n → ∞, the performance nm → m. ratio is n+m−1 3.2

Using Lower Rates Can Yield Better Performance?

Choosing the right data rate for packet transmission is another key problem. Intuitively, to increase throughput and reduce delay, higher data rates are preferred. Interestingly, with the deployment of onion decoding, using lower data rates can actually yield better performance. To see that, consider the example in Figure 5, where node 1 has a packet to broadcast. If node 2 choose to transmit the packet at a data rate that takes m − 1 time slots, the transmission by node 3 cannot proceed concurrently. Because node 3 is not able to match that data rate. If scheduled to proceed concurrently,

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Fig. 5. Using lower data rates yields much better performance

Fig. 6. Letting node 4 postpone its transmission, waiting for node 5 to transmit ﬁrst yields much better performance

the two transmissions will collide at node 5, which will not be able to decode the heterogeneous collision packet. Therefore, the two transmissions have to be serialized. In total, 2m time slots are needed to broadcast the packet. Interestingly, if node 2 chooses to transmit the packet at the lower data rate that takes m time slots, the transmission by node 3 (at the same data rate) can proceed concurrently with the transmission by node 2. Because node 5 will be able to decode the collision packet using onion decoding. Thus, the total number of time slots needed to broadcast the packet comes down to m + 1. 3.3

Who/When to Transmit?

The other key problem in wireless broadcasting that every node have to ﬁgure out is when should I start transmitting? Or equivalently, we need to answer another question: who is the next guy to transmit? To illustrate the subtlety and importance of these questions, let us consider the network in Figure 6, where node 1 has a packet to broadcast. When node 4 receives the packet by the end of time slot 2, node 5 is in the middle of receiving the packet. As node 4’s transmission will not aﬀect reception at node 5, node 4 seems to have a strong case for transmitting the packet right away. If node 4 chooses to transmit immediately, the optimal solution is for node 5 to wait until node 4’s transmission completes. Then, node 5 and 7 transmit simultaneously. 2m + 2 time slots are needed in total.

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In contrast, if node 4 resists the temptation and waits until node 5 to transmit ﬁrst, nodes 4, 7 and 8 will have all received the packet by the end of time slot 4. Then, these three nodes transmit simultaneously, reaching node 6, 9 and 10. The total time needed is m + 4 time slots only.

4

A Simple Heuristic Algorithm

In this section we describe a simple heuristic algorithm, called Cost-eﬀective Broadcast Scheduling (CBS) algorithm, for eﬃcient wireless broadcasting in multi-rate environments. The algorithm runs iteratively. In each iteration, one node is selected to transmit one packet at a certain rate, starting from a certain moment. In particular, this choice is made according to the following guidelines. Non-conﬂict: When scheduling new packet transmissions, existing packet transmissions should not be interrupted by new transmissions. Speciﬁcally, new transmissions should not collide at any node that is receiving a packet it does not have. Moreover, if a node is in the middle of receiving a packet it does not have, the node will not be scheduled to transmit so that it has to abort the reception. Cost-eﬃciency: Typically, there will be multiple non-conﬂicting packet transmissions that can be launched at a certain moment. For better throughput and delay, we prefer packet transmissions with higher utility-to-cost ratios. If a new packet transmission is scheduled, we shall be able to compute the set of nodes that will be able to correctly receive this transmission, considering possible interference from previously scheduled packet transmissions. Among these nodes, those who would not have received the packet without the new schedule transmission are the utility of the scheduled transmission. We use the number of such nodes as the utility of the new transmission. The cost of a new transmission is the amount of time it takes. We prefer packet transmissions with highest utility-to-cost ratio. Order: If two or more allowable packet transmissions have the same utility-tocost ratio, we prefer transmission of the packet with the smallest packet id or sequence number, in order to preserve their order of transmission as much as possible. Note that the CBS algorithm can be applied with both standard decoders and onion decoder. The only diﬀerence is on determining whether a packet transmission can be correctly received (for calculating cost-eﬃciency) and whether a transmission should be allowed (for collision avoidance).

5

Simulation

We implement the CBS algorithm (with standard decoder and onion decoder) in MATLAB 7.0, and conduct simulations to evaluate the eﬀectiveness of onion decoding on improving throughput and delay. In our simulations, 400 802.11b nodes are deployed in a 2000m × 2000m area. As we know, every 802.11b node has four transmission rates: 1Mbps, 2Mbps,

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(a) CDF of throughput.

(b) CDF of delay.

Fig. 7. Simulation results

5.5Mbps and 11Mbps. The corresponding transmission ranges are 483m, 370m, 351m and 283m, respectively [5]. The ﬁrst node is used as the source node, which sits in the center of the area, while the other 399 nodes are randomly placed. 400 such random topologies are generated. On each topology, we conduct two experiments. In one experiment, 100 packets are broadcast from the source node. We record the total time needed to broadcast, and divide the number of packets by this number to obtain throughput. In another experiment, one single packet is broadcast, the total time needed to broadcast is just the broadcast delay. In our simulations, we schedule packet transmissions for each time slot in a slot by slot manner. In each time slot, we determine a set of packet transmissions that are allowed to start in that time slot. The set of packet transmissions are selected one by one, according to their utility-to-cost ratio. If two transmissions have the same utility-to-cost ratio, packet with smaller id/sequence number is given higher priority. The scheduling process continues until no more transmission can be added to the set. When all nodes have collected all packets, simulation is completed. The CDF of the 100 pairs of observed throughput results and delay results are presented in Figure 7(a) and 7(b), respectively. As we can see, onion decoding has the potential to signiﬁcantly improve throughput and delay of wireless broadcasting. Throughput wise, onion decoding increases broadcast throughput by up to 152%, about 58% for majority of the topologies. Delay wise, onion decoding reduces broadcast delay by up to a factor of 2.22, about 20% for most topologies.

6

Related Work

PHY layer decoding: To combat packet collisions, successive interference cancellation [8] proposes that if k−1 of k colliding packets are previously known, the unknown packet can be decoded by successively cancelling the known packets’ signals from the collision signal. Thus, the unknown packet is correctly received

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as if collision did not happen at all. However, the beneﬁt oﬀered by successive interference cancellation is limited. Because for these schemes to work, the colliding senders have to transmit at a data rate that is signiﬁcantly lower than allowed by their channel quality. Moreover, some of the colliding packets must have been known already. Recently, Gollakota and Katabi proposed ZigZag decoding [7] for decoding collisions. Compared with successive interference cancellation, ZigZag decoding is more eﬀective in that it allows senders to transmit at normal data rates allowed by channel quality, and does not need to know any of the colliding packets. Unfortunately, to be able to decode, ZigZag decoding requires that the colliding packets must collide together multiple times. This requirement may be appropriate for unicast packets, which are typically retransmitted multiple times until correctly received. But for broadcast, this requirement can make ZigZag decoding a poor choice because wireless broadcast packets are typically not acknowledged and not retransmitted, as is the case in IEEE 802.11. In contrast, onion decoding perfectly suits wireless broadcasting, as it does not need know any colliding packet, and is able to decode even if concurrent transmissions of a packet collide once only. Eﬃcient wireless broadcasting: Minimum delay broadcasting in single-rate environment has been extensively studied in the literature, not only in general graphs [4, 3] , but also in unit disk graphs [2, 6, 10]. In both cases, the problem is proven to be NP-hard [4, 6], and a number of approximation algorithms have been proposed [2, 3, 10]. In multi-rate wireless networks, the minimum delay broadcasting problem is much less researched [5,9]. Since single-rate networks are a special type of multirate networks, the problem is clearly NP-hard in multi-rate networks as well [5]. In [9], a localized broadcasting approach based on connected dominating set is proposed by Chou et al. to reduce the worst case broadcasting delay in multirate wireless networks. Among all existing work, the proposal of Chou et al. in [5] is the closest to our work. The scheduling proposed in [5] aims at achieving a low latency broadcasting scheduling in wireless network. It ﬁrst constructs a broadcasting tree using a variation of Prim’s algorithm, then a greedy algorithm is applied to schedule transmissions in order to minimize overall broadcasting delay. However, in [5] each node is allowed to transmit each packet once only. Consequently, every forwarding node can only broadcast at the lowest rate of all the point-to-point links between that node and its children in the broadcasting tree. In contrast, our CBS algorithm does allow each node to transmit the same packet multiple times. Moreover, we introduce onion decoding, a PHY layer technique that can eﬀectively improve broadcast delay in multi-rate environment.

7

Summary

This paper presents onion decoding, a novel PHY layer technique enabling receivers to decoding colliding transmissions of the same packet in wireless broadcasting. Hence, broadcasting transmissions can be parallelized more aggressively,

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leading to improved throughput and delay. Unlike previous proposals, onion decoding does not require colliding transmissions to collide together multiple times, and hence perfectly suits for wireless broadcast, where failed transmissions are not retransmitted. We also propose a simple heuristic algorithm for applying onion decoding in wireless broadcasting. Simulation results demonstrate that onion decoding can signiﬁcantly improve throughput and delay of wireless broadcasting.

References 1. Andrews, J.G.: Interference cancellation for cellular systems: A contemporary overview. IEEE Wireless Communications 12(2), 19–29 (2005) 2. Zhenming, C., Chunming, Q., Jinhui, X., et al.: A constant approximation algorithm for interference aware broadcast in wireless networks. In: Proceedings of INFOCOM, May 2007, pp. 740–748. IEEE Press, Anchorage (2007) 3. Chlamtac, I.: The wave expansion approach to broadcasting in multihop radio networks. IEEE Transactions on Communications 39(3), 426–433 (1991) 4. Chlamtac, I., Kutten, S.: On broadcasting in radio networks–problem analysis and protocol design. IEEE Transactions on Communications [legacy, pre - 1988] 33(12), 1240–1246 (1985) 5. Chun, T.C., Archan, M., Junaid, Q.: Low-latency broadcast in multirate wireless mesh networks. IEEE Journal on Selected Areas in Communications 24(11), 2081–2091 (2006) 6. Rajiv, G., Srinivasan, P., Arunesh, M.: Minimizing broadcast latency and redundancy in ad hoc networks. IEEE/ACM Trans. Netw. 16(4), 840–851 (2008) 7. Shyamnath, G., Dina, K.: ZigZag decoding: Combating hidden terminals in wireless networks. In: Proceedings of ACM SIGCOMM, August 2008, pp. 159–170. ACM Press, Seattle (2008) 8. Halperin, D., Anderson, T., Wetherall, D.: Taking the sting out of carrier sense: interference cancellation for wireless LANs. In: Proceedings of ACM MobiCom, pp. 339–350. ACM Press, San Francisco (2008) 9. Qadir, J., Chou, C.T., Misra, A., et al.: Localized minimum-latency broadcasting in multi-rate wireless mesh networks. In: Proceedings of WOWMOM, pp. 1–8. IEEE Press, Espoo (2007) 10. Scott, C.-H.H., Peng-Jun, W., Chinh, T.V., et al.: Nearly constant approximation for data aggregation scheduling in wireless sensor networks. In: Proceedings of INFOCOM, May 2007, pp. 366–372. IEEE Press, Anchorage (2007)

A Spectrally Eﬃcient Anti-Jamming Technique Based on Message Driven Frequency Hopping Lei Zhang, Jian Ren, and Tongtong Li Department of Electrical & Computer Engineering Michigan State University, East Lansing, MI 48824 {zhangle3,renjian,tongli}@egr.msu.edu

Abstract. This paper considers spectrally eﬃcient anti-jamming system design based on message-driven frequency hopping (MDFH). Unlike conventional FH where the hopping frequencies are determined by a preselected pseudonoise (PN) sequence, in MDFH, part of the message acts as the PN sequence for carrier frequency selection. It is observed that MDFH has high spectral eﬃciency and is particularly robust under strong jamming. However, disguised jamming from sources of similar power strength can cause performance losses. To overcome this drawback, in this paper, we propose an anti-jamming MDFH (AJ-MDFH) system. The main idea is to transmit a secure ID sequence along with the information stream. The ID sequence is generated through a cryptographic algorithm using the shared secret between the transmitter and the receiver. It is then exploited by the receiver for eﬀective signal detection and extraction. It is shown that AJ-MDFH can eﬀectively reduce the performance degradation caused by disguised jamming. Moreover, AJ-MDFH can be extended to a multi-carrier scheme for higher spectral eﬃciency and/or more robust jamming resistance. Simulation example is provided to demonstrate the performance of the proposed approaches. Keywords: jamming resistance, physical layer security, message-driven frequency hopping.

1

Introduction

As a widely used spread spectrum technique, frequency hopping (FH) was originally designed for secure communication under hostile environments [1,2]. In conventional FH, each user hops independently based on its own PN sequence, a collision occurs whenever there are two users transmitting over a same frequency band. Mainly limited by the collision eﬀect, the spectral eﬃciency of the conventional FH is very low [3]. To improve the spectral eﬃciency, FH systems that exploit high-dimensional modulation scheme have been studied in the literature [4,5]. However, the performance of these systems are still limited by the collision or self-jamming eﬀect.

This research is partially supported by NSF under awards CNS-0746811 and CNS0845812.

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To break the low capacity barrier and allow secure high speed communications, a message-driven frequency hopping (MDFH) scheme has been proposed in [3,6]. The main idea of MDFH is that a large portion of the message acts as the PN sequence for carrier frequency selection at the transmitter. That is, selection of carrier frequencies is directly controlled by the encrypted information stream rather than by a pre-selected pseudo-random sequence as in the conventional FH. At the MDFH receiver, the carrier frequencies are captured using a ﬁlter bank which selects the strongest signals from all the frequency bands. Note that transmission through hopping frequency control essentially adds another dimension to the signal space, and the resulting coding gain enables MDFH to increase the system spectral eﬃciency by multiple times. It has been observed that MDFH is very robust under strong jamming scenarios, and outperforms the conventional FH by big margins. The underlying argument is that: strong jamming can enhance the power of the jammed signal and hence increases the correct detection probability. When the system experiences disguised jamming, that is, when the jamming power is close to the signal power, it is diﬃcult for the MDFH receiver to distinguish jamming from true signal, resulting in performance losses. To improve the performance of MDFH under disguised jamming, in this paper, we propose an anti-jamming MDFH (AJ-MDFH) scheme. The basic idea is to insert some signal identiﬁcation (ID) information during the transmission process. This ID information is generated through a cryptographic algorithm using the shared secret between the transmitter and the receiver. Therefore, it can be used by the receiver to locate the true carrier frequency or the desired channel. At the same time, it is computationally infeasible to be recovered by malicious users. Comparing with MDFH, AJ-MDFH can eﬀectively reduce the performance degradation caused by disguised jamming and deliver signiﬁcantly better results when the jamming power is close to that of the signal power. At the same time, it is robust under strong jamming just as MDFH. Moreover, AJ-MDFH can be extended to a multi-carrier scheme for higher spectral eﬃciency and/or more robust jamming resistance. Simulation example is provided to demonstrate the eﬀectiveness of the proposed approaches.

2

Anti-Jamming MDFH (AJ-MDFH) System

To enhance the jamming resistance of MDFH under disguised jamming, in this section, we will introduce the anti-jamming MDFH system, named AJ-MDFH. 2.1

Transmitter Design

The transmitter structure of AJ-MDFH is illustrated in Figure 1. The encrypted information sequence is transmitted through carrier frequency selection, and each user is assigned a secure ID sequence. This ID information is shared between the transmitter and the receiver, therefore, it can be used by the receiver to locate the true carrier frequency. Our design goal is to reinforce jamming resistance without sacriﬁcing too much on spectral eﬃciency.

A Spectrally Eﬃcient Anti-Jamming Technique

Encrypted Information

Initial Vector, Key

Channel Coding

PN Sequence Generation

Interleaving

Encryption

Yn

Xn

Carrier Frequency Selection

Symbol Mapper

237

f Xn

sn Baseband Signal

Modulation

s (t )

Generation

Secure ID Generation

Fig. 1. AJ-MDFH transmitter structure

Let Nc be the total number of available channels, with {f1 , f2 , · · · , fNc } being the set of all available carrier frequencies. Without loss of generality, here we assume that Nc = 2Bc for an integer Bc . Let Ω be the selected constellation that consists of M symbols, each symbol in the constellation represents Bs = log2 M bits. Let Nh be the number of hops in one ordinary symbol period. At each symbol period, MDFH transmits a block of length L Nh Bc + Bs bits. Each block contains Nh Bc carrier bits and Bs ordinary bits. The carrier bits are used to determine the hopping frequencies, and the ordinary bits are mapped to a symbol which is transmitted through the selected channels successively. Comparing with MDFH, in AJ-MDFH, the spectral eﬃciency is only reduced 1 . by a factor Nh BBcs+Bs . Take Nh = 5, Bc = 8, Bs = 4, for example, Nh BBcs+Bs = 11 It should be noted that, in order to prevent impersonation attack, each user’s ID sequence needs to be kept secret from the malicious jammer. Therefore we generate the ID sequence through a reliable cryptographic algorithm, such as the Advanced Encryption Standard (AES) [7], so that it is computationally infeasible for the malicious user to recover the ID sequence. That is, we ﬁrst generate a pseudo-random sequence using a linear shift feedback register, encrypt it using AES, and then take the AES output as our ID sequence. 2.2

Receiver Design

The receiver structure for AJ-MDFH is shown in Figure 2. The receiver regenerates the secure ID through the shared secret (including the initial vector, the LFSR information and the key). For each hop, the received signal is ﬁrst fed into the bandpass ﬁlter bank. The output of the ﬁlter bank is then demodulated, and used for carrier bits (i.e., the information bits) detection. Demodulation. Let s(t), J(t) and n(t) denote the ID signal, the jamming interference and the noise, respectively. For AWGN channels, the received signal can be represented as r(t) = s(t) + J(t) + n(t). We assume that s(t), J(t) and n(t) are independent of each other. If the spectrum of J(t) overlaps with the frequency band of s(t), then the signal

BPF, f1(t) BPF, f2(t) r (t )

Signal Detection & Extraction

Demodulation

Recovered Information

BPF, fNc(t)

sn Initial Vector, Key

Secure ID Generation

Yn

Symbol Mapper

Fig. 2. AJ-MDFH receiver structure

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is jammed ; otherwise, the signal is jamming-free. If J(t) spreads over multiple channels, we have multi-band jamming; otherwise, we have single band jamming. Note that the true information is embedded in the index of the active carrier over which the ID signal s(t) is transmitted. For i = 1, 2, · · · , Nc , the output of the ith ideal bandpass ﬁlter fi (t) is ri (t) = fi (t) ∗ r(t) = αi (t)s(t) + Ji (t) + ni (t).

(1)

Here αi (t) ∈ {0, 1} is a binary indicator for the presence of signal in channel i at time instant t. At each hopping period, αi (t) is a constant: αi (t) = 1 if and only if s(t) is transmitted over the ith channel during the mth hopping period; otherwise, αi (t) = 0. Ji (t) = fi (t) ∗ J(t) and ni (t) = fi (t) ∗ n(t). When there is no jamming presented in the ith channel, Ji (t) = 0. For demodulation, ri (t) is ﬁrst shifted back to the baseband, and then passed through a matched ﬁlter. At the mth hopping period, for i = 1, · · · , Nc , the sampled matched ﬁlter output corresponds to channel i can be expressed as ri,m = αi,m sm + βi,m Ji,m + ni,m ,

(2)

where sm , Ji,m and ni,m correspond to the ID symbol, the jamming interference and the noise, respectively; αi,m , βi,m ∈ {0, 1} are binary indicators for the presence of ID signal and jamming, respectively. Note that the true information is carried in αi,m . Signal Detection and Extraction. Signal detection and extraction is performed at each hopping period. For notation simplicity, without loss of generality, we omit the subscript m in (2). That is, for a particular hopping period, (2) is reduced to: ri = αi s + βi Ji + ni , for i = 1, · · · , Nc . (3) Deﬁne r = (r1 , · · · , rNc ), α = (α1 , · · · , αNc ), β = (β1 , · · · , βNc ), J = (J1 , · · · , JNc ) and n = (n1 , · · · , nNc ), then (3) can be rewritten in vector form as: r = sα + β · J + n. For single carrier AJ-MDFH, at each hopping period, one and only one item in α is nonzero. That is, there are Nc possible information vectors: α1 = (1, 0, · · · , 0),· · · , αNc = (0, 0, · · · , 1). If αk is selected, and the binary expression of k is b0 b1 · · · bBc −1 , where Bc = log2 Nc , then estimated information sequence is b0 b1 · · · bBc −1 . So at each hopping period, the information symbol α, or equivalently, the hopping frequency index k, needs to be estimated based on the received signal and the ID information which is shared between the transmitter and the receiver. Here we use the maximum likelihood (ML) detector. If the input information is equiprobable, that is, p(αi ) = N1c for i = 1, 2, · · · , Nc , then MAP detector is reduced to the ML detector. For the ML detector, the estimated hopping frequency index kˆ is given by kˆ = arg max p{r|αi }. 1≤i≤Nc

(4)

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Recall that the information signal (including α and s), the jamming interference and the noise are independent to each other. Assume both the noise and the jamming interference are totally random, that is, n1 , · · · , nNc , J1 , · · · , JNc are all statistically independent, then r1 , · · · , rNc are also independent. In this case, the joint ML detector in (4) can be decomposed as: kˆ = arg max

Nc

1≤i≤Nc

Nc

p{rj |αi } = arg max

1≤i≤Nc

j=1

p{rj |αj = 0} · p{ri |αi = 1}(5)

j=1,j =i

Nc Since j=1 p{rj |αj = 0} is independent of i, (5) can be further simpliﬁed as i |αi =1} kˆ = arg max1≤i≤Nc p{r βi p(ri |αi = 1, βi )p(βi ) p{ri |αi =0} , where p{ri |αi = 1} = and p{ri |αi = 0} = βi p(ri |αi = 0, βi )p(βi ), with βi ∈ {0, 1}. Deﬁne Λi p{ri |αi =1} p{ri |αi =0}

be the likelihood ratio for channel i, then (5) can be rewritten as: kˆ = arg max Λi .

(6)

1≤i≤Nc

If we further assume that n1 , · · · , nNc are i.i.d. circularly symmetric Gaussian random variables of zero mean and variance σn2 , and J1 , · · · , JNc are i.i.d. circularly symmetric Gaussian random variables of zero mean and variance σJ2i , then it follows from (3) and (6) that: kˆ = arg max

1≤i≤Nc

P {βi =0} 2 πσn

2

exp{− riσ−s }+ 2

P {βi =0} 2 πσn

n

2

exp{− rσi2 } + n

P {βi =1} 2 +σ 2 ) π(σn J i

P {βi =1} 2 +σ 2 ) π(σn J i

2

i −s exp{− r } σ2 +σ2 n

Ji

2

i exp{− σr 2 +σ 2 } n

.

(7)

Ji

In the ideal case when the jammed channel indices are known, or equivalently, βi is known for i = 1, · · · , Nc , then the ML detector above can be further simpliﬁed. Deﬁne n ˜ i = βi Ji + ni , then n ˜ i is circularly symmetric Gaussian with zero mean and variance σi2 = βi σJ2i + σn2 . It follows from (3) and (6) that kˆ = 2

2

i −s arg max1≤i≤Nc R1 (ri ), where R1 (ri ) = ri −r . σi2 However, in reality, jamming side information is generally unknown. Here we develop the following two suboptimal detectors. Note that σi2 is generally unknown. If we replace the overall interference power σi2 in R1 (ri ) with the instantaneous received signal power ri 2 , then we can obtain another detector 2 kˆ = arg min1≤i≤Nc R2 (ri ), where R2 (ri ) = rri −s 2 . i For more tractable theoretical analysis, we can replace the instantaneous received signal power ri 2 in R2 (ri ) with average signal power observed in channel i, Pi = E{ri 2 }, then we obtain an alternative detector

kˆ = arg min R3 (ri ), 1≤i≤Nc

where R3 (ri ) =

ri −s2 . Pi

(8)

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ID Constellation Design and Its Impact on System Performance

For AJ-MDFH, ID signals are introduced to distinguish the true information channel from disguised channels invoked by jamming interference. The general design criterion of the ID constellation is to minimize the carrier detection error probability under a given signal power. Under this criterion, there are two questions need to be answered: (1) How does the size of the constellation impact the system performance? (2) How does the type or shape of the constellation inﬂuence the detection error and which type should we use for optimal performance? In this section, we will try to address these questions under diﬀerent jamming scenarios. Recall that for AJ-MDFH, the message signal is embedded in the index of the hopping frequency or channel. In the worst case if the ID is known to the jammers, or can be easily guessed by the jammers, then the jammers can disguise itself by sending the same ID symbol over a diﬀerent or fake channel. In this case, it would be diﬃcult for the receiver to distinguish the true channel from the disguised channel, leading to high detection error probability. We deﬁne this kind of jamming as ID jamming or ID attack. In literature, jamming has generally been modeled as Gaussian noise. We refer this kind of jamming as noise jamming. Here we consider the constellation design problem under these two jamming scenarios separately: – Constellation Design under Noise Jamming: Without loss of generality, we assume that the ID symbol is transmitted through channel 1. When detection metric in (8) is used, it can be shown that the carrier detection error probability Pe is upper bounded by ⎡ Nc −1 ⎤ s2 (s2 +σ2 ) 2 1 − 2 1 σ 2 2 1 ⎣1 − 1 − ⎦, Pe ≤ PeU = e σm (s +2σ1 ) (9) |Ω| s2 + 2σ12 s∈Ω

where m = arg max{σl2 } for 2 ≤ l ≤ Nc and |Ω| is the size of the ID constellation Ω. Further mathematical analysis shows that: when SNR is high 2 enough, i.e., s 1, the upper bound of the detection error probabilσ2 1

ity, PeU , is minimized when the constellation is constant modulus, i.e., when s2 = Ps for all s ∈ Ω. An intuitive explanation for this result is that the signal power in constant modulus constellations always equals to the maximal signal power available. – Constellation Design under ID Jamming: In this case, the entropy or uncertainty of the ID symbol needs to be maximized. Under the assumption that all the symbol in a constellation Ω of size M are all equally probable, 1 the entropy H(s) = − log2 |Ω| = log2 |Ω| = log M. In the ideal case when the channel is noise-free, the optimal constellation size would be M = ∞. However, when noise is present, larger M also implies that there is a larger probability for an ID symbol to be mistaken for its neighboring symbols.

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More speciﬁcally, it can be shown that: for a given SNR and assuming PSK constellation is used, the carrier detection error probability Pe will converges to a limit P¯e as constellation size M increases. That is, for any > 0, there always exists an Mt such that for all M > Mt , |Pe − P¯e | < .

4

Multi-carrier AJ-MDFH

For more eﬃcient spectrum usage and more robust jamming resistance, in this section, we extend the concept of AJ-MDFH to multi-carrier AJ-MDFH (MCAJ-MDFH). The transmitter and receiver structure of the MC-AJ-MDFH system are illustrated in Figure 3. The idea is to split all the Nc channels into Ng non-overlapping groups, and each subcarrier hops within the assigned group based on the AJ-MDFH scheme. To ensure hopping randomness of all the subcarriers, the groups need to be reorganized or regenerated securely after a prespeciﬁed period, named group period. In the following, we will ﬁrst describe the secure group generation algorithm, and then discuss the design of MC-AJ-MDFH with and without additional frequency diversity. X 1,n

S/P

Encrypted Information Sequence

Channel Coding and Interleaving

X Ng ,n

Channel Coding and Interleaving Initial Vector, Key Initial Vector, Key Secure ID

Generation

Y1,n , , YNg ,n

Subcarrier Selection Over Group Gn1

Subcarrier Selection Over Group GnNg

f1,n

Group Gn1 Subcarrier Detection

BPF, f1

f Ng ,n

s (t )

r (t )

BPF, f2

Modulation

Gn1 , , GnNg Secure Group Generation

s1 (t ), , sNg (t ) Symbol Mapping and Baseband Signal Generation

Secure Group De-assignment

Initial Vector, Key

(a) Transmitter structure

Recovered Information Message Reconstruction

Group GnNg Subcarrier Detection

BPF, fNc

Initial Vector, Key

Secure ID Generation

Y1,n , , YNg ,n

s1,n , , sNg ,n Symbol Mapper

(b) Receiver structure

Fig. 3. Transmitter and receiver structure of MC-AJ-MDFH

4.1

Secure Group Generation

In this section, we propose a secure group generation algorithm to ensure that: (i) Each subcarrier hops over a new group of channels during each group period, so that it eventually hops over all the available channels in a pseudo-random manner; (ii) Only the legitimate receiver can recover the transmitted information correctly. Secure group generation is synchronized at the transmitter and the receiver. At the receiver, the received signal is fed to a bank of single-carrier AJ-MDFH receivers for signal extraction and recovery. Recall that we assume there are a total of Nc available channels and there are Ng subcarriers in the system. For l = 0, · · · , Ng − 1, the number of channels assigned to subcarrier i is denoted as Ngi . As diﬀerent subcarriers transmit over Ng non-overlapping set of channels, we have Ngi = Nc . i=0

For secure group generation, ﬁrst, generate a pseudo-random binary sequence using a 32-bit linear feedback shift register (LFSR) as in Section 2, which is initialized by a secret sequence shared between the transmitter and receiver.

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Encrypt the generated sequence into a ciphertext using the AES algorithm and a secure key. Pick an integer L ∈ [ N2c , Nc ] and let q = L log2 Nc = LBc . Take q bits from the ciphertext and put them as a q-bit vector e = [e1 , e2 , · · · , eq ]. Second, partition the ciphertext sequence e into L groups, such that each group contains Bc bits. For k = 1, 2, · · · , L, the partition of the ciphertext is represented as pk = [e(k−1)∗Bc +1 , e(k−1)∗Bc +2 , · · · , e(k−1)∗Bc +Bc ], where pk corresponds to the kth Bc -bit vector. For k = 1, 2, · · · , L, denote Pk as the decimal number corresponding to pk . And denote P = [P1 , P2 , · · · , PL ] as the permutation index vector. For k = 0, 1, 2, · · · , L, denote Ik = [Ik (0), Ik (1), · · · , Ik (Nc − 1)] as the index vector at the kth step. The secure permutation scheme of the index vector is achieved through the following steps: 1. Initially, the index vector is I0 = [I0 (0), I0 (1), · · · , I0 (Nc − 1)] and the permutation index is P = [P1 , P2 , · · · , PL ]. We start with I0 = [0, 1, · · · , Nc −1]. 2. For k = 1, switch I0 (0) and I0 (P1 ) in index vector I0 to obtain I1 . In other words, I1 = [I1 (0), I1 (1), · · · , I1 (Nc − 1)], where I1 (0) = I0 (P1 ), I1 (P1 ) = = 0, P1 . I0 (0), and I1 (m) = I0 (m) for m

3. Repeat the previous step for k = 2, 3, · · · , L. In general, if we already have Ik−1 = [Ik−1 (0), Ik−1 (1), · · · , Ik−1 (Nc − 1)], then we can obtain Ik = [Ik (0), Ik (1),· · · , Ik (Nc − 1)] through the permutation deﬁned as Ik (k − 1) = Ik−1 (Pk ), Ik (Pk ) = Ik−1 (k − 1), and Ik (m) = Ik−1 (m) for m

= k − 1, Pk . 4. After L steps, we obtain the channel center frequency vector as FL = [fIL (0) , fIL (1) , · · · , fIL (Nc −1) ]. 5. Vector FL is used to assign the channels to Ng groups. We assign channels {fIL (0) , fIL (1) , · · · , fIL (Ng0 −1) } to the ﬁrst group; Assign {fIL (Ng0 ) , fIL (Ng0 +1) , · · · , fIL (Ng0 +Ng1 −1) } to the second group, and so on. Because each frequency index appears in FL once and only once, the proposed algorithm ensures that all the subcarriers are transmitting on non-overlapping sets of channels. 4.2

Multi-Carrier AJ-MDFH without Diversity

In this case, each subcarrier transmits independent bit stream. The spectral eﬃciency of the AJ-MDFH system can be increased signiﬁcantly. Let Bc = log2 Nc and Bg = log2 Ng , then the number of bits transmitted by the MC-AJMDFH within each hopping period is BMC = (Bc − Bg )Ng = (Bc − log2 Ng )Ng . BMC is maximized when Bg = Bc − 1 or Bg = Bc − 2, which results in BMC = 2Bc −1 . Note that the number of bits transmitted by the AJ-MDFH within each hopping period is Bc , it can be seen that BMC > Bc as long as Bc > 2. Take Nc = 256 for example, then the transmission eﬃciency of AJ-MDFH can be Bc −1 increased by BBMcC = 2 Bc = 16 times. We assume that jamming is random and equally distributed among all the groups. Then the overall carrier detection error probability Pe of MC-AJ-MDFH is equal to that corresponding to each subcarrier fk for k = 1, · · · , Ng . Let Pe,k

A Spectrally Eﬃcient Anti-Jamming Technique

243

denote the carrier detection error probability corresponding to the kth subcarrier or the kth group, then we have Pe = Pe,k , and Pe,k = P0,k · P {incorrect carrier detection|signal not jammed} + P1,k · P {incorrect carrier detection|signal jammed},

(10)

where P0,k , P1,k denote the probability that the kth subcarrier is jamming-free or jammed, respectively. 4.3

Multi-carrier AJ-MDFH with Diversity

Under the multi-band jamming, diversity needs to be introduced to the AJMDFH system for robust jamming resistance especially. A natural solution to achieve frequency diversity is to transmit the same or correlated information through multiple subcarriers. The number of subcarriers needed to convey the same information diﬀers in diﬀerent jamming scenarios. Ideally, the number of correlated signal subcarriers should not be less than the number of jammed bands. At the receiver, the received signals from diﬀerent diversity branches are often combined for joint signal detection. To achieve better performance, appropriate diversity combination schemes need to be selected for diﬀerent metrics used. In this paper, we propose to use the equal gain combination scheme [8] while choosing R3 (ri ) be the detection metric, since it can also be regarded as a normalized square-law metric. Assume that the same infor10 MDFH mation is transmitted through conventional FH AJ−MDFH the hopping frequency index of MC−AJ−MDFH: without diversity MC−AJ−MDFH: with diversity 10 Nd subcarriers over Nd groups {Gn1 , Gn2 , · · · , GnNd } simultaneously, each group has the 10 same number of channels, denoted as Ngc . Note that the se10 cure group generation algorithm ensures that the channel index 10 −20 −15 −10 −5 0 5 10 15 20 in each group is random and JSR(dB) does not necessarily come in ascending or descending order. Let Fig. 4. The performance comparison under 2band ID jamming. R3 (rinl ) denote detection metric value for ith channel in group Gnl , then the active hopping frequency index can be estimated as kˆ = d nl arg min1≤i≤Ngc N l=1 R3 (ri ). The diversity order Nd can be dynamic in diﬀerent jamming scenarios to achieve tradeoﬀ between performance and eﬃciency. 2

BER

0

−2

−4

−6

5

Simulation Results

In this section, we illustrate the performance of the proposed AJ-MDFH and MCAJ-MDFH through a Monte Carlo simulation example using Matlab. We assume

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that the signal is transmitted through AWGN channels and experiences 2-band ID jamming. For conventional FH, BFSK modulation is used; For MDFH, 8-PSK is adopted for ordinary bits modulation; For AJ-MDFH and MC-AJ-MDFH, 32PSK is adopted for ID constellation to combat possible ID jamming. The SNR is taken as Eb /N0 = 10dB. JSR is deﬁned as the ratio of the jamming power to signal power per channel. The jamming bands of the multi-band ID jamming are independently and randomly selected, and each ID jamming randomly transmits a symbol from the constellation of corresponding modulation scheme. The number of available channels is Nc = 64 (Bc = 6). Choose Ng = Nd = 4 and Ng = 32 for MC-AJ-MDFH with and without diversity, respectively. From Figure 4, it can be seen that AJ-MDFH and MC-AJ-MDFH deliver much better performance than MDFH and conventional FH.

6

Conclusion

In this paper, we proposed a highly eﬃcient anti-jamming scheme AJ-MDFH based on message-driven frequency hopping. It was shown that AJ-MDFH is robust under strong jamming and can eﬀectively reduce the performance degradation caused by disguised jamming. Moreover, AJ-MDFH can be extended to multi-carrier AJ-MDFH for higher spectral eﬃciency and more robust jamming resistance. The proposed approaches can be applied to both civilian and military applications for reliable communication under jamming interference.

References 1. Viterbi, A.: A processing satellite transponder for multlple access by low rate mobile users. In: Proc. Digital Satellite Commun. Conf., Montreal, Canada (October 1978) 2. Pickholtz, R., Schilling, D., Milstein, L.: Theory of spread-spectrum communications: A tutorial. IEEE Trans. Commun. 30(5), 855–884 (1982) 3. Ling, Q., Ren, J., Li, T.: Spectrally eﬃcient spread spectrum system design: Message-driven frequency hopping. In: Proc. IEEE Intl. Conf. Commun., May 2008, pp. 4775–4779 (2008) 4. Cho, J., Kim, Y., Cheun, K.: A novel frequency-hopping spread-spectrum multipleaccess network using M-ary orthogonal Walsh sequence keying. IEEE Trans. Commun. 51(11), 1885–1896 (2003) 5. Choi, K., Cheun, K.: Maximum throughput of FHSS multiple-access networks using MFSK modulation. IEEE Trans. Commun. 52, 426–434 (2004) 6. Ling, Q., Li, T.: Message-driven frequency hopping: Design and analysis. IEEE Trans. Wireless Commun. 8(4), 1773–1782 (2009) 7. Advanced Encryption Standard, FIPS-197, National Institute of Standards and Technology Std. (November 2001) 8. Miller, L., Lee, J., Kadrichu, A.: Probability of error analyses of a BFSK frequencyhopping system with diversity under partial-band jamming interference–part III: Performance of a square-law self-normalizing soft decision receiver. IEEE Trans. Commun. 34(7), 669–675 (1986)

Secure RFID Application Data Management Using All-Or-Nothing Transform Encryption* Namje Park1 and Youjin Song2,** 1

Computer Science and Engineering, Arizona State University, 699 S. Mill Avenue, Tempe, Arizona, AZ 85281, USA [email protected] 2 Department of Information Management, Dongguk Univertsity, 707 Seokjang-dong, Gyeongju, Gyeongsangbuk-do, 780-714, Korea [email protected]

Abstract. Ensuring the security of RFID's large-capacity database system by depending only on existing encryption schemes is unrealistic. Therefore, data sharing for security management to supplement it is drawing attention as an extremely secure scheme. However, applying the existing secret sharing scheme to this method makes the size of the share equal to that of the original data. Thus, it is not suitable for application to large-scale database. This paper proposes secret sharing algorithms that enable efficient data sharing security management based on the characteristics of the All-Or-Nothing Transform encryption mode. The proposed algorithms enable fast sharing and reconstruction in terms of processing speed and allow the sum of shares to be equal to that of the plaintext, thereby making them suitable for large-capacity database storage.

1 Introduction While common RFID technologies are used in B2B (Business to Business) models like supply channels, distribution, logistics management, networked mobile RFID technologies are used in the RFID reader attached to an individual owner's cellular phone through which the owner can collect and use information of objects by reading their RFID tags; in case of corporations, it has been applied mainly for B2C (Business to Customer) models for marketing [1]. Though most current RFID application services are used in fields like the search of movie posters and provision of information in galleries where less security is required, they will be expanded to and used more frequently in such fields as purchase, medical care, electrical drafts, and so on where security and privacy protection are indispensable. With the question of secure storage and management of mobile RFID’s personal data becoming an increasingly urgent matter, data is encrypted to eliminate threats to security. However, the encryption scheme requires plenty of time and memory for *

**

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(No. 2009-0087849)

Corresponding author.

G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 245–252, 2010. © Springer-Verlag Berlin Heidelberg 2010

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encoding and decoding. Since the encryption scheme such as Advanced Encryption Standard (AES) encodes confidential information in its entirety, there is a risk of entire outflow of said information once the secret key is decoded; hence the difficulty of key distribution and management. Unlike the encryption scheme, there is no such thing as life of a key under the secret sharing scheme, which presents no problem in renewing the public certificate of authentication and helps reduce considerable operational expenses. In contrast, under the existing secret sharing scheme [2], if the original data are separately stored, the size of share (i.e., distributed information) becomes equal to that of the original data; thus increasing the data volume to be stored. Recently, security has been found to be improvable without reducing the efficiency of the existing encryption scheme through the appropriate transformation of plaintext (in encryption mode) before encoding it. This paper proposes secret sharing algorithms having the characteristics of All-OrNothing Transform (AONT), aiming for long-term, secure, and efficient storage of large-capacity data. In other words, algorithms for the sharing and reconstruction of large-capacity data whose security is improved using the AONT encryption mode and to which XOR operations are applied to boost efficiency are suggested. The research results of this paper are expected to be utilized as algorithms that ensure availability and manage the efficient distribution of large-capacity data including highly confidential data and secrets involving customers' personal information, even though some of the data are leaked to the outside.

2 Secure Data Management in Networked Mobile RFID Network mobile RFID middleware applications may access application tag data in three ways. The left below is a diagram of a tag being read by a mobile RFID reader to get a key [5]. This is used by an application to access database storage to get additional data. Note that key and storage contain application data, such as personal data, etc. This is at risk of theft and abuse. In the middle, tag key and data are both read by a mobile RFID reader to get the personal data, etc. In this case, no additional data is required. Note that both key and data may contain sensitive data that is at risk. On the right, RFID tag key and data are read by a mobile RFID reader to get the personal data, etc. Key also used by an application to access storage to get additional data, such as whether this person is on a specific watch list. This is the most challenging combination because key, data, and storage are all at risk.

Fig. 1. Secure Data Management in Mobile RFID Environment

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3 AONT-Based Data Sharing Security Management Scheme This chapter proposes a new algorithm based on AONT transformations [3] and Exclusive OR (XOR) operations. The proposed algorithm is shown in Figure 2.

Fig. 2. Proposed secret sharing algorithm

against The proposed scheme generates the share the size of the plaintext P of size . The sum of all share sizes is equal to the size of the plaintext, which can be described as follows: z1+z2+,...,+zn = The scheme has greater advantage compared with the existing Shamir scheme or the secret sharing scheme [2] proposed by Kurihara [6,10]. Specifically, the existing secret sharing scheme is such that the sum of share sizes is equal to the plaintext size x n, requiring the same storage space as the plaintext for each database at the time of database storage. In contrast, the proposed algorithm is such that the sum of share sizes is equal to that of the plaintext, requiring average storage space of as much as n/ per database. Such characteristics are quite suitable for large-capacity database. 3.1 Detailed Composition The composition of the data sharing system as proposed in this paper is as follows: 1) Data distribution stage - Segment plaintext by randomly generating random numbers. Align the segmented plaintext in ascending or descending order of small size and perform XOR operations for each segmented plaintext. In the process, invert the resulting value from descending order in even number rounds. - With regard to the resulting value produced from the performance of such repetition 16 times, make the segmented fragments into shares and store each share separately on DB for management. 2) Data reconstruction stage - Aggregate each share that is separately stored on DB. - For each share, repeat the XOR operations and ascending/descending order alignments in reverse order and differently from the data distribution stage. In this process, plaintext can be reconstructed in odd number rounds by reverting 16 times the resulting value from descending order.

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3.2 Details of the Proposed Methods The terms used in this paper's proposed methods are as follows: : Exclusive-OR (XOR) operation , : Bit connection : Plaintext, : Size of plaintext, : Number of blocks in the plaintext : Segmented plaintext, : Random number generated by BBS generator

・・・

・

・

・・

(1) Data distribution stage With regard to -sized plaintext m, generate as many uniform random numbers less than as appropriate using the pseudo random number generator. Here, use the BBS random number generator. Based on random number , segment plaintext into and rearrange them in ascending order in proportion to the segmented size. At this time, the size is equal to the plaintext despite the rearrangement.

① ②

③ Plaintext

arranged in ascending order generates the resulting value through XOR operation. The XOR operation involves getting the segmented value at the very last as it is, (i.e., at the very last) the segmented value of the plaintext right before it and the XOR-operated resulting value are placed before . If the calculation includes the segmented value of plaintext at the very front, the value of can be derived.

Fig. 3. Exercise 1

At this time, if the size between the segmented plaintext at the back and that at the front are different, the XOR operation is performed by adjusting the size of the segmented plaintext value at the back starting from the front and subsequently erasing the last part because said operation is only possible if the segmented plaintext size at the back is equal to that at the front. Next, from the resulting value , segment into based on the random number and rearrange them in descending order as follows:

④

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Afterward, invert the value rearranged in descending order and create the resulting value of by performing the same XOR operation as in as follows:

③

Invert

⑤ Repeat the aforementioned procedure 16 times until the resulting value

, is created and compose the shares based on the fragmented shares of the finally created value, storing them on k databases.

Fig. 4. Exercise 2

(2) Data reconstruction stage First, aggregate the data that are separately stored on k databases. Afterward, perform XOR operations on the segmented values. Here, the XOR operation is performed in the same manner as in the distribution stage in . For the XOR operation, in the case wherein it is different from the preceding size, it does not involve erasing the last part to adjust the size of the segmented value to be operated as in the distribution stage; here, in the process of lowering the segmented value since it is at the very last and performing an XOR operation of the value that is lowered as it is, it involves performing the XOR operation again for the immediately preceding part and the resulting value derived from the XOR operation, not for the immediately preceding or succeeding part. At this time, if the preceding part is bigger, the succeeding small part should be used repeatedly. Segment again the resulting value produced into random numbers and revert it. Here, the random value within the same seed range is known.

①

③

②

Fig. 5. Restoration status (Exercise 2)

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③ The final plaintext value can be obtained by repeating ①, ② above. The proposed secret sharing algorithm is summarized in Table 1. And, the secret reconstruction algorithm is summarized as follows (see Table 2): Table 1. The proposed algorithm INPUT : P ଲ OUPTPUT : 1. Generate ri(i=1,...,t) by BBS Generator 2. Calculate set s ଶ {s0,...,st-1} which LENGTH(si)=ri andd si=SUBSTRING(P)riri+1 3. for Rĸ0 to 16 do 4. for iĸ0 to n-1 do 5. if R mod 1 = 0 then 6. bĸMAX(s) 7. aiĸb 8. discard MAX(s) 9. else 10. bĸMIN(s) 11. aiĸInvert(b) 12. discard MIN(s) 13. end if 14. end for i 15. for jĸ0 to n-1 do 16. if j=n-1 then 17. sjĸsj||(aj) 18. else ) 19. sjĸ(ajْPADD 20. end if 21. end for j 22. end for R ) 23. return (

Table 2. Secret reconstruction algorithm INPUT : s1,...,st OUPTPUT : Plaintext P 1. Generate ri(i=1,...,t) by BBS Generator 2. Calculate set s ଶ {s0,...,st-1} which LENGTH(si)=ri andd si=SUBSTRING 3. for Rĸ0 to 16 do 4. for iĸ0 to n-1 do 5. if R mod 1 = 1 then 6. bĸMAX(s) 7. aiĸb 8. discard MAX(s) 9. else 10. bĸMIN(s) 11. aiĸInvert(b) 12. discard MIN(s) 13. end if 14. end for i 15. for jĸn-1 to 0 do 16. if j=n-1 then 17. sjĸsj||(aj) 18. else ) 19. sjĸ(ajْPADD 20. end if 21. end for j 22. end for R 23. return (s1,...,st)

4 Analysis 4.1 Advantages of the Proposed Scheme The proposed algorithms have the following advantages compared with the existing secret sharing algorithms: 1) Minimization of Redundancy The sum of secret sharing data is the same as the plaintext in size, thereby making it radically different from the existing secret sharing schemes [3]. In other words, there is redundancy in the existing threshold value-based secret sharing scheme such that the problem of the total size of secret sharing increasing in proportion to the number of shares remains. In contrast, the proposed algorithms allow for efficient processing because the total size of secret sharing is equal to the plaintext. 2) Variability in Size The existing secret sharing scheme is unable to process share size with variability. In other words, share size is outputted in the same manner as plaintext, which in reality does not ensure diverse handling by share. In the proposed algorithms, share size is variable, and each share size can be freely adjusted; thus offering the advantage of enabling the easy handling of shares.

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3) Cryptographic Characteristics The proposed algorithms have cryptographic characteristics since the plaintext and the end product are equal in size. The difference lies in the fact that such characteristics make them take on characteristics similar to those of the existing encryption scheme, but they enable separate data sharing and storage that is extensive enough to memorize or store vast plaintext; decryption does not depend on whether or not the key is owned. This aspect has the huge advantage of making key management hassle-free. 4) Variability in the Number of Shares The existing threshold value-based secret sharing algorithms determine the number of shares in the secret sharing stage; once the shares are segmented, readjusting their number is impossible. However, the proposed algorithms have the characteristics of being able to merge the shares. Thus, the number of shares can be freely adjusted after all the shares have been created, thereby facilitating their handling. 5) Fast Secret Distribution/Reconstruction Since the proposed algorithms are activated based on XOR and pseudo random numbers, they are efficient compared with Shamir's algorithms. Moreover, they have the advantages of being able to process at a time the entire length that can be operated by a memory device without being limited by block units during the operation process. 4.2 Suitability for Large-Capacity DB The following can be cited as requirements for handling large-capacity database: - Fast processing should be enabled. - Input of data should be reduced as much as possible. - The larger the DB capacity is, the more data that should be inputted. The aforementioned principal characteristics of the proposed algorithms fulfill the three requirements above since they enable fast processing based on XOR and pseudo random numbers along with the characteristics of nonexistent redundancy and capability of variable adjustment of the secret distribution size. In short, the proposed algorithms are suitable for large-capacity database. The proposed scheme and existing algorithms are summarized as follows: Table 3. Comparison of algorithms

Shamir

Kurihara

Average share size

l

l

Verifiability of security

○

Capability of fast processing

×

Suitability for large-capacity DBMS

×

Variability in share size Variability in share number

× × (○ : Superior,

△

○

△

○

○

△

Proposed Algorithm

○

× ○ × ○ : Moderate, × : Unsatisfactory)

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5 Conclusion This research paper has proposed deriving the security requirements for the safe storage and management of large-capacity data and AONT-based secret sharing and reconstruction structures. First, it has defined the major functions required for largecapacity data management, database attack, and security threats and proposed the security requirements for the storage and management of large-capacity data. Based on this, the proposed AONT-based structures of secret sharing and reconstruction enhance security using the AONT encryption mode wherein data quantity is invariable and rely on the XOR operation-based secret sharing scheme to ensure efficiency. The proposed scheme has realized the variability of the AONT encryption mode and fast processing and redundancy at the same time and enabled privilege management after the weight is given. Moreover, the proxy re-encryption technique (which enables other users to decode encrypted data through delegated decryption privileges) has been reviewed as a way of managing large capacity data. At present, such delegation of privileges is likely to be applied to the question of managing safe data sharing when it comes to large-capacity data management. The research results of this paper are expected to be used in the cloud services environment since they are structured to ensure the safe, efficient distribution management of large-capacity data such as highly confidential medical data and trade secrets including customers' personal information.

References 1. Park, N., Kwak, J., Kim, S., Won, D., Kim, H.: WIPI Mobile Platform with Secure Service for Mobile RFID Network Environment. In: Shen, H.T., Li, J., Li, M., Ni, J., Wang, W. (eds.) APWeb Workshops 2006. LNCS, vol. 3842, pp. 741–748. Springer, Heidelberg (2006) 2. Shamir, A.: How to Share a Secret. Communication of the ACM 22(11), 612–613 (1979) 3. Rivest, R.L.: All-or-nothing encryption and the package transform. In: Biham, E. (ed.) FSE 1997. LNCS, vol. 1267, pp. 210–218. Springer, Heidelberg (1997) 4. Dean, J.: Handling Large Datasets at Google: Current Systems and Future Directions. In: Data-Intensive Computing Symposium (2008) 5. Park, N., Kim, S., Won, D., Kim, H.: Security Analysis and Implementation leveraging Globally Networked Mobile RFIDs. In: Cuenca, P., Orozco-Barbosa, L. (eds.) PWC 2006. LNCS, vol. 4217, pp. 494–505. Springer, Heidelberg (2006) 6. Kurihara, J., Kiyomoto, S., Fukushima, K., Tanaka, T.: A fast (3, n)-threshold secret sharing scheme using exclusive-or operations. IEICE Trans. Fundamentals E91-A(1), 127–138 (2008) 7. Fujii, Y., Tada, M., Hosaka, N., Tochikubo, K., Kato, T.: fast (2, n)-threshold scheme and its application. In: Proc. CSS 2005, pp. 631–636 (2005) 8. Tada, M., Fujii, Y., Hosaka, N., Tochikubo, K., Kato, T.: A secret sharing scheme with threshold 3. In: Proc. CSS 2005, pp. 637–642 (2005) 9. Kuwakado, H., Tanaka, H.: Strongly non-separable encryption mode for throwing a media away. Technical Report of IEICE 103(417), 15–18 (2003) 10. Kurihara, J., Kiyomoto, S., Fukushima, K., Tanaka, T.: A New (k, n)-Threshold Secret Sharing Scheme and Its Extension. In: Wu, T.-C., Lei, C.-L., Rijmen, V., Lee, D.-T. (eds.) ISC 2008. LNCS, vol. 5222, pp. 455–470. Springer, Heidelberg (2008)

Prevention of Wormhole Attacks in Mobile Ad Hoc Networks by Intrusion Detection Nodes Ming-Yang Su and Kun-Lin Chiang Department of Computer Science and Information Engineering, Ming Chuan University, 5 Teh-Ming Road, Gwei Shan District, Taoyuan 333, Taiwan [email protected], [email protected]

Abstract. This study proposes an approach to deploy a node implementing IDS (Intrusion Detection System) in MANETs for detecting and isolating wormhole nodes, if they exist. This study uses ns2 to evaluate the proposed IDS nodes, and the results show that these IDSs can rapidly and correctly block the wormhole nodes under the circumstances of low false positives. Keywords: MANETs (Mobile ad hoc networks), wormhole attacks, IDS (Intrusion Detection System), ns2.

1 Introduction Wormhole attacks are often referred to as attack activities jointly launched by two malicious nodes at different locations, one of which would transmit the routing message to the other through a secret tunnel. Despite considerable distance between them, these two malicious nodes appear to be adjacent to each other. Therefore, there is a high possibility that the hop count passing the malicious nodes will be shorter than that passing the normal nodes so as to increase the chances of grabbing the route for data transmission, thereby eavesdropping or dropping the data packets passing the malicious nodes. The secret tunnel in wormhole attacks can be represented by a packet encapsulated channel and out-of-band channel [1], as shown in Figs. 1(a) and (b), respectively. A packet encapsulated channel is also called an in-band channel, which means a malicious node puts the received routing messages into the payload of a data packet and uses regular nodes that constitute a tunnel, to transmit to the other malicious node. In Figure 1(a), a route is created between m1 and m2, of which the node s is the source and the node d is the destination. When RREQ (Route Request) is initiated and broadcast by s and received by m1; m1 will encapsulate it in a data packet and transmit it via the route between m1 and m2. Destination d will receive three RREQs from different routes. So destination node d will choose to send an RREP (Route Reply) in response to m2 because this path is the shortest. The main difference between the out-of-band channel and the encapsulated packet channel is the type of tunnel. It is possible that a special channel is connected between two malicious nodes through a wired cable or implemented through a long-distance signal with high power transmission, so as to form a secret tunnel between two nodes, as shown in Figure 1(b). G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 253–260, 2010. © Springer-Verlag Berlin Heidelberg 2010

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(a) Packet encapsulated channel (in-band channel)

(b) Out-of-band channel

Fig. 1. Two methods of wormhole attacks

In this paper, m1 and m2 of Figure 1 are called a tunnel ingress node and a tunnel egress node, respectively. Conversely, if a route is from node d to node s, then m2 is the tunnel ingress node and m1 is the tunnel egress node. A malicious node can be a tunnel ingress node and/or an egress node. The paper discusses how to deploy some nodes, executing an intrusion detection system (IDS) into a MANET so as to detect and isolate wormhole nodes. The defense mechanism executed on an IDS node is called an Anti-Wormhole Mechanism (AWM), which is used to sniff routing messages of regular nodes within their transmission range. IDS nodes can determine if there is a tunnel ingress node according to the pairing of RREQ and RREP in the AODV routing protocol, and also determine if there is a tunnel egress node, according to RREP, being normally processed in a valid time. When the abnormal score exceeds the threshold, IDS can broadcast a block message to all nodes so as to jointly isolate the malicious nodes that possibly execute wormhole attacks. The remainder of this paper is organized as follows: Section 2 reviews the related works on prevention of wormhole attacks in MANETs; Section 3 presents the function of AWM executed in IDS; Section 4 shows some results of ns2; and Section 5 presents the conclusions.

2 Related Works The following is a briefing on related researches concerning the three different approaches in defending against wormhole attacks. The first is merely to modify the routing protocol to avoid wormhole nodes in path discovery, such as [2], and [3]. The second is to equip extra hardware for a node, like a positioning system, a time synchronization mechanism, or a directed antenna, in addition to modifying the routing protocol. Some of them are [1], [4], and [5]. Finally, the third is to deploy some special nodes executing Intrusion Detection Systems (IDSs) with or without hardware support, such as [6] and [7].

3 The Proposed Intrusion Detection System Three prerequisites are assumed in this paper as follows: 1. Two neighboring IDS nodes are in the transmission range of each other so as to transmit block messages to each other.

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2. Authentication mechanisms exist in MANETs. The identity of each node cannot be falsified and a block message transmitted by an IDS node cannot be modified or falsified. 3. Each IDS is set as a promiscuous mode so as to sniff all routing messages within the range of transmission. Authentication mechanisms in MANETs is not detailed in this paper. The network topology in this study has four types of nodes executing four algorithms, respectively. • Wormhole node: executes a WAODV (Wormhole AODV) routing algorithm to conduct wormhole attacks. • Tunnel node: executes a TAODV (Tunnel AODV) routing algorithm and cooperates with a wormhole node to quickly transmit RREQ and RREP messages to another wormhole node at the other endpoint, provided the hop count of the message is not changed. • Regular node: executes a slightly modified AODV, which is also called MAODV (modified AODV), so as to carry out normal routing and cooperate with IDS nodes to block wormhole nodes if necessary. • IDS node: executes the AWM (Anti-Wormhole Mechanism) algorithm to detect wormhole nodes and broadcast corresponding block messages. When an RREQ/RREP is received by a wormhole node, it is converted into a format called WRREQ/WRREP that can be recognized only by tunnel nodes. It rapidly passes through the tunnel nodes without increasing the hop count. After being received by the wormhole node at the other endpoint, it is converted back to a common RREQ/RREP. When an IDS detects a wormhole node within its transmission range, one block message, mainly containing the identity of the wormhole node, is broadcast to all nodes in the MANET. Precisely, a block message includes: the IDS node issued, the wormhole node detected, and a timestamp. After being received by a regular node, the message can be added into the Block Table, as shown in Table 1. Table 1 keeps the information that malicious Node 1 was issued by IDS_A and Node 6 was issued by IDS_C; the timestamps are also recorded in the table. The block message transmitted by IDS will be authenticated by regular nodes before updating their Block Tables. Therefore, nodes other than the IDS node cannot broadcast a valid block message. Table 1. Block Table IDS

Malicious node

Timestamp

IDS_A IDS_C

1 6

2009/02/19 12:51 2009/02/19 12:55

MAODV, executed by a regular node, is basically identical to AODV, except that the intermediate node is not allowed to reply RREP. Besides, the differences are: 1) when a node receives a block message broadcast by an IDS, the malicious node in the block message should be added to the Block Table; i.e., in addition to the routing table, a Block Table is needed to record the malicious nodes; and 2) when a node receives an RREP forwarded by a malicious node residing in the Block Table, the RREP will be dropped immediately.

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The following paragraphs will explain the algorithm of AWM executed on IDS nodes. AWM maintains four tables: the RQ Table, RP Table, SN Table, and Nr Table, as shown in Table 2. The RQ table records RREQs sniffed by an IDS node within its transmission range. The RP Table records RREPs sniffed by an IDS node within its transmission range. This implies that for the RREP, the latest forwarded node is Node 5 and is destined to Node 1. If Node 1 continues to be within the IDS’s transmission range and is not the end of the RREP (the source node of the path), and does not forward the RREP within a specific period, then the suspicious value of Node 1 will be added with 1 by the IDS. The Nr Table records Hello Messages sniffed by an IDS node within its transmission range; namely, it records the list of neighboring nodes. As shown in Table 2 (c), Nodes 3, 4, and 7 exist within the range of transmission of the IDS node. The SN (suspicious node) Table, as shown in Table 2 (d), records the suspicious values of neighboring nodes within the IDS’s transmission range. The suspicious value is an important basis for a IDS, determining whether a neighboring node is a malicious node. At present, the suspicious value of Node 3 in Table 2 (d) is 1. Assuming it is less than the threshold, the status is “inactive”; whereas the suspicious value of Node 4 is 6; assuming that it reaches the threshold, the status is “active” and should be blocked. Table 2. RQ, RP, Nr, and SN Tables (a) RQ Table Route

Src 1 3

Maximal hop count

Broadcasting nodes

Expiration time

3001

2

2, 4, 5

02:41:12

5012

4

1, 6

02:44:34

Dest

Src_seq

6 5

(b) RP Table Route Src

Dest

Dest_seq

2 5

5 1

2101 4006

Forwarding nodes

Expected node

Expiration time

3, 4, 5 1, 6

1 4

05:21:10 06:23:36

(c) Nr Table Node ID

Expiration time

3 4

02:40:12 02:42:34

7

02:48:34

(d) SN Table Node ID

Suspicious Value

Status

3

1

inactive

4

6

active

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The AWM algorithm executed on all IDS nodes mainly includes four procedures: Procedures A, B, C, and D. Procedure_A is for IDS sniffing an RREQ to update the RQ Table. Procedure_B is for IDS sniffing an RREP to update the RP Table and find the tunnel ingress wormhole node. Procedure_C is for IDS periodically checking the RP Table to find the tunnel egress wormhole node. Finally, Procedure_D is used to communicate between the IDSs, mainly to process the block message. The approaches for judging the tunnel ingress wormhole node and tunnel egress wormhole node are different. The former can be carried out immediately by checking the RQ Table when sniffing an RREP, while the latter can only be conducted by periodically checking the RP Table.

4 Experimental Data and Analysis In this study, ns2 is used to verify performance of the proposed AWM mechanism that is executed by all IDS nodes. Fifty movable common (regular) nodes are randomly distributed in an area of 2000 m × 2000 m, and MAODV is executed for normal routing. Two kinds of wormhole nodes are considered: (1) two wormhole nodes (a pair) are located diagonally in a 500 m × 500 m area, as shown in Figure 2(a), and (2) four wormhole nodes are located diagonally as shown in Figure 2(b), in which (W1, W2) are a pair and (W3, W4) are another pair. In Figures 2(a) and 13(b), T denotes a tunnel node. The wormhole nodes execute WAODV to conduct wormhole attacks, while tunnel nodes execute TAODV to assist wormhole nodes in rapidly forwarding encapsulated routing packets. Furthermore, some IDS nodes execute AWM to detect and broadcast block messages, if necessary. The threshold in AWM is set as 2, that is, when RREP is processed abnormally twice by a common node and is detected, it is considered a wormhole node.

(a) Two (a pair) wormhole nodes

(b) Four (two pairs) wormhole nodes

Fig. 2. Deployments of wormhole nodes in the simulations

In terms of total packet loss rate, Figure 3 first shows the comparison of the original AODV and the proposed MAODV, and the influence of wormhole attacks in the cases of one pair and two pairs. Different pause times will affect the packet loss rate, but on average, when no attack exists, the total packet loss rate of the original AODV is 10.98%, and that of MAODV provided for normal node routing in this study is 10.61%.

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60%

total packet loss rate

50% 40%

AODV, no wormholes

30%

M AODV, no wormholes

20%

M AODV+2 Wormholes (one p air)

10%

M AODV+4 Wormholes (two pairs)

0% 0

5

10

15

pause time (second)

Fig. 3. Total packet loss rates without IDS

When one pair of wormholes exists, the packet loss rate is increased to 47.66%. If two pairs of wormholes exist, the packet loss rate will continue to increase to 54.11%. Figure 3 shows that wormhole nodes can successfully grab the routes and drop packets to conduct a high packet loss rate. Considering the deployment of IDS nodes, if IDS nodes need to cover the entire 2000 m × 2000 m area, 90 IDS nodes are required as shown in Figure 4(a). However, it is difficult to deploy 90 IDS nodes to realize such a full coverage of IDS in a real MANET. In this study, we let four neighboring IDSs form a local coverage. It is assumed that for the case of two wormhole nodes, a local coverage can cover one wormhole node, as shown in Figure 4(b), and for the case of four wormhole nodes, two local coverages can cover two wormhole nodes of the two pairs, as shown in Figure 4(c).

(a) Full coverage

(b) A 4-IDS covers

(c) Two 4-IDS

Fig. 4. Full coverage, one local coverage, and two local coverages of IDS

When two (one pair) wormhole nodes exist in a MANET, improvement is shown by the decrease in the total packet loss rate by the full coverage and local coverage as shown in Figure 5(a). As for the full coverage of IDS (90 IDS nodes), the performance is best when the pause time is 10, with the packet loss rate down from 45.07% to 12.55%. This is close to the packet loss rate of AODV and MAODV when there is no attack, representing the fact that wormhole nodes can be detected rapidly and blocked.

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The performance is the worst when the pause time is 0, with the packet loss rate only down from 50.63% to 25.93%. As for one local coverage (4 IDSs), the performance is best when the pause time is 10, with the packet loss rate down from 45.07% to 29.24%; the performance is worst when the pause time is 5, with the packet loss rate only down from 45.10% to 35.86%. In the case of four (two pairs) wormhole nodes, the improvement of the total packet loss rate is shown in Figure 5(b). As for the full coverage (90 IDSs), the performance is best when the pause time is 5, with a packet loss rate of 27.64%, reduced from 51.01%. As for two local coverage (8 IDSs), the performance is best when pause time is 0, with a packet loss rate of 40.63%, reduced from 56.83%.

60%

total packet los rate

50%

AODV, no wormholes

40%

MAODV, no wormholes MAODV, 2 Wormholes + 0 IDS

30%

MAODV, 2 Wormholes + 90 IDSs

20%

MAODV, 2 Wormholes + 4 IDSs

10% 0% 0

5

10

15

pause time (second)

(a) Two wormholes (one pair)

total packet loss rate

60% 50% 40%

AODV MAODV

30%

MAODV, 4 Wormholes + 0 IDS MAODV, 4 Wormholes + 90 IDSs

20%

MAODV, 4 Wormholes + 8 IDSs

10% 0% 0

5

10

15

pause time (second)

(b) Four wormholes (two pairs) Fig. 5. Improvements of total packet loss rate by IDSs

5 Conclusions This paper proposes the deployment of IDS nodes in MANETs so as to detect and isolate malicious nodes that can selectively execute wormhole attacks. The average packet loss rate is 10.61% in the whole network when there is no wormhole attack. When there are two (one pair) wormhole nodes, the average rate of total packets lost is increased to 47.66%. Through deployment of IDS nodes and based on the assumption of full coverage (90 IDSs), the packet loss rate can be successfully decreased to 12.55%

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(pause time set as 10 seconds). If there is only local coverage (4 IDSs), the packet loss rate can be decreased to 29.24% (pause time set as 10 seconds). When there are four (two pairs) wormhole nodes in the MANET, the average of total packet loss rate is 54.11%. In the case of full coverage (90 IDSs), the packet loss rate can be successfully decreased to 27.64% (pause time set as 5 seconds). If there are two local coverage (8 IDSs), the packet loss rate can also be decreased to 40.63% (pause time set as 0). Acknowledgments. This work was partially supported by the National Science Council with Contracts NSC-98-2221-E-130-007.

References [1] Khalil, I., et al.: LITEWORP: A Lightweight Countermeasure for the Wormhole Attack in Multihop Wireless Networks. In: DSN 2005 (2005) [2] Naıt-Abdesselam, F., Bensaou, B., Yoo, J.: Detecting and Avoiding Wormhole Attacks in Optimized Link State Routing Protocol. In: The Proc. of the IEEE Conference on Wireless Communications and Networking (2007) [3] Su, M.-Y.: WARP: A Wormhole Avoidance Protocol by Anomaly Detection in Mobile Ad Hoc Networks. Computers & Security 29(2), 208–224 (2010) [4] Lazos, L., et al.: Preventing Wormhole Attacks on Wireless Ad Hoc Networks: A Graph Theoretic Approach. In: WCN 2005 (2005) [5] Wang, X.: Intrusion Detection Techniques in Wireless Ad Hoc Networks. In: The Proc. of the IEEE International Computer Software and Applications Conference (2006) [6] Azer, M.A., et al.: Intrusion Detection for Wormhole Attacks in Ad hoc Networks: A Survey and a Proposed Decentralized Scheme. In: The Proc. of the IEEE 3rd International Conference on Availability, Reliability and Security (2008) [7] Gorlatova, M.A., et al.: Detecting Wormhole Attacks in Mobile Ad Hoc Networks through Protocol Breaking and Packet Timing Analysis. In: Proc. of the IEEE Conference on Military Communications (2006)

A Publicly Verifiable Encryption Scheme with Short Public/Private Keys Yujun Liu1 , Yonggang Cui2 , and Limin Liu3 1

Department of Information Engineering, The Academy of Armored Forces Engineering, Beijing 100072, China 2 State Key Laboratory of Information Security, Graduate School of the Chinese Academy of Sciences, Beijing 100049, China 3 Department of Optical and Electronic Engineering, PLA Ordnance Engineering College, Shijiazhuang 050003, China

Abstract. A publicly veriﬁable public key encryption scheme has the property that the validity of ciphertexts can be veriﬁed without knowledge of private key, which facilitate its non-interactive threshold decryption. Based on a selective identity secure IBE by Boneh et. al, a new publicly veriﬁable encryption scheme is presented using two hash functions. We prove the IND-CCA2 security of the new scheme under decisional BDHI assumption and target collision resistant property of two hash functions. Furthermore, our scheme has very short public and private keys that are independent of the security parameter, witch signiﬁcantly simpliﬁes public key certiﬁcate management and reduces the computation cost for encryption or decryption. Keywords: chosen ciphertext security, identity-based encryption, public veriﬁability.

1

Introduction

A publicly veriﬁable public key encryption scheme has the property that the validity of ciphertexts can be veriﬁed without knowledge of private key, which facilitate its non-interactive threshold decryption. Besides security, shorter public key and private key are also desired characters of threshold encryption schemes since they can simplify public key certiﬁcate management and reduce the computation cost for encryption or decryption. Indistinguishability against chosen ciphertext attack (IND-CCA2) [8] is by now the accepted standard security deﬁnition for public key encryption (PKE) schemes. While there have been several eﬃcient encryption schemes shown to be heuristically secure in random oracle model, it wasn’t until recently that Cramer and Shoup [6] designed an encryption scheme that was both eﬃcient and provably IND-CCA2 secure in the standard model. Somewhat surprisingly, Canetti et. al. [5] gave a generic transformation (CHKtransformation) from a selective identity secure IBE scheme to a IND-CCA2 secure PKE scheme. The main advantage of CHK-transformation over CramerShoup scheme or its variants is that schemes constructed by this way are publicly G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 261–265, 2010. c Springer-Verlag Berlin Heidelberg 2010

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veriﬁable, i.e. the validity of ciphertexts can be veriﬁed without knowledge of private key, which facilitate its non-interactive threshold decryption. Boneh and Katz [3] further improved the eﬃciency of CHK-transformation by replacing the strong one-time signature with a message authentication code, at the cost of losing public veriﬁability. In [4], Boyen et.al. considered to use non-black box transformation from IBE schemes to PKE schemes. Especially, they present a eﬃcient publicly veriﬁable PKE scheme, based on Waters IBE scheme [7], that has very short ciphertext (consisting of 3 group elements), while having long public (private) key, consisting O(λ) (λ is security parameter) group elements. A natural question is whether we can construct publicly veriﬁable PKE schemes with short ciphertext and public (private) key that is independent of the security parameter. We answer this question in the aﬃrmative. In this paper, a publicly veriﬁable PKE scheme is present, based on the second IBE in [1], by two target collision resistant (TCR) hash functions. The ciphertext of our scheme consists of 4 group elements, while public and private key consisting of 3 group elements plus two TCR-Hash functions. We also give a simple proof of the IND-CCA2 security of our scheme in the standard model using a new security model for IBE schemes.

2

The New Encryption Scheme

Let G be a bilinear group of prime order p and g be a generator of G (the security parameter determines the size of G). Assume e : G × G → GT is a bilinear map and denotes (G, GT , g, q, e) by Γ . Let H1 : G × GT → Zp and H2 : G → Z∗p be target collision resistant hash family. Deﬁne the following encryption scheme: Gen(1k ): Select random elements x, y, z ∈ Z∗p and deﬁne X = g x , Y = g y and Z = g z . Choose uniformly H1 ←R H1 and H2 ←R H2 and outputs a public key pk and a secret key sk, where pk = (Γ, X, Y, Z, H1 , H2 ) and sk = (x, y, z). Enc: To encrypt a message M ∈ GT , pick two random s, t ∈ Z∗p and output CT = (g s·α X s , Y s , e(g, g)s · M, t), where α ← H2 (Y β Z t ) and β ← H1 (Y s , e(g, g)s · M ). Dec: To decrypt a ciphertext CT = (A, B, C, t) ∈ G2 × GT × Z∗p , compute β = H1 (B, C), α = H2 (Y β Z t ) and verify A = B (α+x)/y and reject if not. Then, Dec output M = C · e(B 1/y , g)−1 . Note that the validity of ciphertexts can also be veriﬁed by testing if e(A, Y ) = e(B, g α X). Thus, our scheme is publicly veriﬁable since e(B, g α X) and e(A, Y ) can be computed without private key. Theorem 1. Suppose the (t, q, ε)- Decisional BDHI assumption holds in G and H1 is (t1 , ε1 )-TCR, H2 is (t2 , ε2 )-TCR Hash family, then the above defined en cryption scheme is (t , q , ε )-IND-CCA secure for any q ≤ q and ε ≤ 2 + 1 + 2 + q /p.

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3 3.1

263

Security Analysis The Security of BB-IBE Scheme

To prove the Theorem 1, we ﬁrstly present the IBE scheme by Bnoeh et al.[1],which is called BB-IBE scheme in the following. Let Γ = (G, GT , g, q, e) be deﬁned as above and assume the identity space is Z∗p . The BB-IBE scheme works as follows: Setup(1k ): Select random elements x, y ∈ Z∗p and deﬁne X = g x and Y = g y . The public parameters are params = (Γ, X, Y ) and the secret master-key is master-key=(x, y). Encrypt(params, ID, M ): To encrypt a message M ∈ GT under public key ID ∈ Z∗p , pick a random s ∈ Z∗p and output the ciphertext CT = (g s·ID X s , Y s , e(g, g)s · M ). We omit the description of algorithms KeyGen and Decrypt which are not needed for our proof. Boneh and Boyen proved the IND-sID-CPA security of BB-IBE based on decisional BDHI assumption. In fact, BB-IBE scheme can be proved secure, based on the same assumption, in a more strong security model, which we call IND-IDs-CPA model. The IND-IDs-CPA model is as same as IND-sID-CPA model except that the adversary can adaptively choose the challenge identity ID∗ after receiving partial public parameters params1, while not known the other part of params i.e. params2. Let the public parameters of BB-IBE params=(params1, params2) with params1=(Γ, Y ) and params2 = X, then BB-IBE can be proved IND-IDsCPA secure since the params1 part is absolutely independent of ID∗ during the construction of simulator in the security proof of BB-IBE in [1]. This is very crucial for our security proof and details is refer to [1]. 3.2

Proof of Theorem 1

Assuming A is IND-CCA2 adversary against our new scheme AE, we build an IND-IDs-CPA adversary B of BB-IBE scheme using A. When receiving partial parameters params1=(Γ, Y ) of BB-IBE, B selects random r, l ∈ Z∗p and computes Z ← Y r , D∗ ← Y l . Then, B chooses H1 ←R H1 and H2 ←R H2 , where H1 and H2 are target resistant family, and commits the challenge identity ID∗ = H2 (D∗ ). After receiving the other part of public parameters params2 = X, B gives pk=(Γ, X, Y, Z, H1 , H2 ) to A. When receiving a ciphertext CT = (A, B, C, t)(w.l.g. we always assume CT is valid since AE is publicly veriﬁed), B computes β = H1 (B, C), α = H2 (Y β Z t ) and aborts if α = ID∗ . B makes a query to obtain the private key dα correspond = (A, B, C) using dα . ing to identity α and decrypts the BB-IBE ciphertext CT Then, B sends the resulting plaintext to adversary A. When A submits a pair of plaintexts, B submits the same pair to its own ∗ = (A∗ , B ∗ , C ∗ ). Then, B encryption oracle and gets challenge ciphertext CT responds A with CT ∗ = (A∗ , B ∗ , C ∗ , t∗ ), where t∗ = (l − β ∗ )/r and β ∗ = ∗ ∗ H1 (B ∗ , C ∗ ) (so D∗ = Y β Z t ).

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Public Key Length Private Key Length Ciphertext Length

CHK-BBES

4|G|

2|G|

7|G|

BMW-Scheme [1]

O(λ)|G|

O(λ)|G|

3|G|

Our Scheme

3|G|

3|G|

4|G|

| · | denots the bit length of an element in group G. λ is security parameter.

At last, B outputs the same bit as A. That completes the construction of B. For any valid ciphertext of AE scheme CT = (A, B, C, t), deﬁne β(CT ) = H1 (B, C), D(CT ) = Y β Z t and α(CT ) = H2 (D(CT )). Let F denote the event that A queries a valid ciphertext CT such that α(CT ) = ID∗ during an attack, then B simulates perfectly the decryption oracle IDs−CPA of adversary A unless F occurs, so we have AdvCCA2 A,AE ≤ AdvB,BB−IBE + Pr[F ]. Then, to prove Theorem 1, it is enough to give an upper bound of Pr[F ]. Let F1 denote the event that A queries a valid ciphertext CT such that D(CT ) = D∗ and α(CT ) = ID∗ during the attack. Then, Pr[F1 ] ≤ Advtcr B,H2 ≤ 2 because D∗ is randomly selected in G. Let F2 denote the event that A queries CT such that D(CT ) = D∗ before obtaining CT ∗ , we have Pr[F2 ] ≤ q /p due to the same reason as above. Let F3 denote the event that A queries CT such that D(CT ) = D∗ after obtaining CT ∗ , we can prove that Pr[F3 ] ≤ + 1 and omits it due to page limit. It is easy to see that F = F1 ∨ F2 ∨ F3 , so Pr[F ] ≤ + 1 + 2 + q /p.

4

Discussion and Comparisons

In this section, we make a comparison between our scheme with some public veriﬁable encryption schemes in reference. We ﬁrstly consider a scheme, called CHK-BBES, built by applying CHK-transform to BB-IBE using a strong onetime signature with very high eﬃciency present in [2] over pairing group. Then, the publicly veriﬁable scheme present by Boyen et. al. in [4], called BMW-scheme, is also considered. The comparison is assembled in Table 1. Compared with CHK-BBES scheme, the ciphertext length of our scheme is much shorter, although private key is a slightly longer. BMW-scheme inherits the drawback of the original Waters IBE scheme [7], i.e. the public and private key are both consisted of O(λ) group elements for security parameter λ, while our scheme has short public and private key which are independent of λ, witch signiﬁcantly simpliﬁes public key certiﬁcate management and reduces the computation cost for encryption or decryption.

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References 1. Boneh, D., Boyen, X.: Eﬃcient Selective-ID Secure Identity-Based Encryption without Random Oracles. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 223–238. Springer, Heidelberg (2004) 2. Boneh, D., Boyen, X.: Short Signatures without Random Oracles. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 56–73. Springer, Heidelberg (2004) 3. Boneh, D., Katz, J.: Improved Eﬃciency for CCA-Secure Cryptosystems Built Using Identity-Based Encryption. In: Menezes, A. (ed.) CT-RSA 2005. LNCS, vol. 3376, pp. 87–103. Springer, Heidelberg (2005) 4. Boyen, X., Mei, Q., Waters, B.: Direct chosen ciphertext security from identitybased techniques. In: ACM Conference on Computer and Communications Security 2005, pp. 320–329 (2005) 5. Canetti, R., Halevi, S., Katz, J.: Chosen-Ciphertext Security from Identity-Based Encryption. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 207–222. Springer, Heidelberg (2004) 6. Cramer, R., Shoup, V.: A Practical Public Key Cryptosystem Provably Secure Against Adaptive Chosen Ciphertext Attack. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 13–25. Springer, Heidelberg (1998) 7. Waters, B.: Eﬃcient Identity Based Encryption without Random Oracle Model. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 114–127. Springer, Heidelberg (2005) 8. Rackoﬀ, C., Simon, D.R.: Non-Interactive Zero-Knowledge Proof of Knowledge and Chosen Ciphertext Attack. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 433–444. Springer, Heidelberg (1992)

Algorithm on Self-organization of Wireless or Connectionless Clustering Zhen Shen, Sheng Qiang, and Dong-yun Yi School of Science, National University of Defense Technology Changsha, Hunan, 410073, China

Abstract. The astronautics oriented computer simulation has two remarkable characteristics: one is the high dynamics of the simulation scenario; the other is the huge quantity of computing. To meet the above two requirements in complex scenario simulating computing (such as constellation optimization, resource scheduling and space-based early warning, et al), we developed an astronautics simulation and cluster computing environment. We employed distributed clusters as the computing environment to increase it’s numerical calculation ability and controllability, and we developed it based on High Level Architecture to enhance it’s dynamic simulation adaptation. The organization algorithm of the distributed clusters is mainly discussed in this paper. We provided an self-organization algorithm based on wireless or connectionless communication protocol. Keywords: Self-organization, Astronautics Simulation, Agent Technology, Cluster Computing, High Level Architecture, Parallel Virtual Machine.

1 Introduction The researcher in astronautics simulation field should know the famous Satellite Tool Kit (STK), but as a general simulation platform, STK has the following disadvantages: 1.

2. 3.

STK is not a complete simulation framework, while it is mainly a program majored in satellite related computing and demonstrating; It just provide the utility to interact with other simulation framework, but the integration needs more programming works. STK is not designed for high performance cluster system, and has no extra performance advantage in huge quantity computing tasks; STK is expansive and not free, although it has many function in astronautics field, there are problems it can not do easily, such as: anti-missile and midcourse maneuvering, small satellite formation, special sensor imaging and volume covering, et al [1, 2].

Of course, there are more VO models and physical models which STK provides for us to learn from, and we are doing this. The original intention of this paper is to build an open source STK based on open source software, and become an astronautics simulation platform and cluster calculation environment [3-6]. We set up a simulation environment, with its main function to make up the above-mentioned three disadvantages. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 266–273, 2010. © Springer-Verlag Berlin Heidelberg 2010

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The organization algorithm of the distributed clusters is mainly discussed in this paper. We provided an self-organization algorithm based on wireless or connectionless communication protocol.

2 Parallel and Management Parallel virtual machine based distributed cluster system is the foundation of ASPCCE’s parallel computing and management layer, its main functions are: 1.

2.

parallel computing task management, such as computer node organization, computing resource distribution configuration, computing task process management, computing task inter-process communication, et al; Simulation management, such as: simulation federation’s configuration and start, runtime control and sudden event simulation, simulation monitoring and data collection, simulation result demonstration, et al.

The horizontal structure of the simulation platform is divided into the following three layers: the interactive layer, the logic supporting layer, the data storage layer. The interactive layer is mainly responsible for interaction between users and systems. In the interactive layer, the system will use double-agent mechanism, the logical support layer and interface interaction layer loosely coupled links. The interactive layer invokes the logic supporting layer functions through the logical proxy, the logic supporting layer feedback through the interface proxy. The logical layer is mainly responsible for the main business logic calculations. On one hand it receives commands coming from interaction layer interface information, on the other hand it read or store information to the data storage layer through the data agent when needed. The data storage layer is mainly responsible for data storage capabilities. The data storage layer manage a variety of data including: data files, parameter configuration, components, scripts, scene script, simulation state, simulation results and so on. From the vertical the simulation platform can be divided into three kinds of running status: simulation design state, simulation run state, simulation evaluation state. Simulation design first appeared as a running state is primarily aimed at the development of component configuration and updates, scene script development and update, components and scripts management. This work will be the basis of simulation, should be completed first time and most of its major components and typical scenarios should be designed in the development phase first. However, as the system needs change after the completion of this part, this is the most vulnerable part, and should change after the completion of secondary development according to the needs. Simulation run state mainly refers to the entire period from the start of the simulation until the end of it. In the simulation run state, the system mainly do the following tasks: parsing scene script, building simulation federal, starting passive simulation components, managing initiatives component, driving simulation forward, accepting and executing real-time control command, recording simulation state, saving the simulation results and so on. Simulation evaluation status is mainly refers to the simulation state about the analysis and evaluation about simulation result data after the end of the simulation. Mainly including: various forms of simulation demonstration, statistical analysis, report generation, etc.

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The computation node can compose an effective colony computing system through wired or the wireless networking way, at least below need some premises: 1. 2. 3. 4.

The computation node must be able to carry on the effective discrimination. The network system must have the broadcast ability. The network system must have p2p communication ability. The computation node itself must have the multi-advancement concurrent ability.

In the computation node set which element satisfies these conditions, the so-called computation node organization's final outcome is an only node relational tree which depth is 1. The computation node's organization function is the core and foundation of other function, after each computation node has formed the computation cluster, the computation task management and the node load equalization can easily realized: The computation duty's management may realize with the help of the point-to-point communication ability and the multi-advancement ability of each node; The computation loading balance may realize through the task process count’s equal distribution principle between various nodes. Below mainly discuss how to organize the computation node, this article attempts to establish one kind of self organization algorithm to maintain computation node. Certainly speaking for the cluster system which based on the wired confined network connections' personal computer, because it nearly does not have the system structure change which is unpridictable, definitely may use the more explicit system organization strategy. But for based on wireless sensor's network cluster system (e.g.: the wireless local area network, the 3G network, the satellite network and so on) system own topology structure possibly momentarily changes, it needs the self organizing ability to realize the biggest flexibility. Computation node's multi-advancement ability is very important, because each node needs to execute a cluster proxy process, this process represents this node to interact with the cluster system, and it is responsible for all cluster calculate tasks’ analysis and execution. The cluster proxy process needs to dispose or preserve following cluster calculation related information: 1. 2.

Disposition information: Node ID, polling cycle, connection overtime. Node relational tree information: Father node ID or subnode ID tabulation.

The information about node ID, father node ID and subnode ID tabulation are node relational tree's key information for differentiation and indication, node ID and the network system communication address must have unique corresponding relations. The computing node can divide into two kinds: One kind is a root node, one kind is a sub-node. In a cluster system there’s only one root node, but for the self organization's cluster system, the only root node is not fixed, but has the real-time change along with the cluster node's change. The parallel computing duty is the same between each subnode as well as the only root node, the only difference is the entire cluster related information is maintains by the root node. The root node may be called the isolated root node when its sub-node tabulation is empty, its essence has not distinguished with the non-cluster's computation node itself. The polling cycle and the connection overtime are parameters which help to realize cluster's dynamic, the cluster proxy process will defer to the polling cycle to carry on the node relational tree information’s refreshment,

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in the refresh process, it will defer to the connection overtime to decide whether the father node or the sub-node is alive and will carry on corresponding processing. In order to better describe the self-organization algorithm for computing nodes, we state from each node’s join, maintain and withdrawal progress. When the computation node want to join the cluster system, it's cluster proxy process's primary mission is:

Fig. 1. Cluster Join Process

After passing through the join progress of the cluster system, the computation node enters the cluster maintain state. Because the network reason or the computation node’s join time fortuitousness, above join progress has crack, it will has the possibility to form certain isolated root nodes which can not connect with each other, there two kind of reasons: First, between the computing nodes do not have network connections, in this kind of situation it is unable to form the cluster computing system; Second, between the computing nodes which are unable to connect with each other temporarily, this needs the node carries on unceasingly joins the colony the attempt in the respective polling cycle, until connects finally successfully, this part of work can be done in the computation node’s maintain state. Here this article believed that the cluster proxy process starts successfully expressed that the node’s join progress finish successfully. When the computation node enter its maintain state, this node's cluster proxy process mainly carries on the following task: When computation node prepared to withdrawal cluster system, this node's cluster proxy process’s execution duty is quite simple, so long as the node broadcast quitting message and then terminated, other nodes may obtain the response in a polling cycle and organize again. Based on the high performance cluster computing system in high speed local network, The self organization function and the core cluster proxy process are developed with the help of wxWidgets’s wxConnection framework.

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Fig. 2. Cluster Maintenance Process

3 Simulation Engine We develop the ASPCCE’s simulation engine according to the high level architecture (HLA) specification. To realize dynamic adaptation, the global unique federation object model (FOM) is separated and distributed into every federate, and it will be aggregated by the simulation engine just before the normal HLA simulation starts. Every federate is modeled as agent object pattern with the internal handling threads and autonomous execution willingness based on the specific simulation logic. The simulation engine processes simulation execution according to HLA’s time advance mechanism. Every federate agent must interact with the engine through the communication mechanism that is provided strictly by the engine. The functional difference between simulation engine and management layer can be summarized as: the simulation engine is the core communication mechanism and the driving engine for the simulation’s autonomous execution; The management layer is responsible for the preparation works before the simulation starts, the collection works after the simulation stops, the human machine interface when simulation is running, and the background parallel computing tasks. The simulation engine is developed based on the open source RTI implementation, CERTI.

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Fig. 3. Simulation runtime structure

High Level Architecture provides a well-established simulation specification includes six main aspects of services: the federal management, time management, declaration management, object management, ownership management and data distribution management. The first four management services is the core of the simulation support services, specify the forms of participation, the promoting process and the interactive rules of the HLA simulation object. Strictly in accordance with HLA interface requirements specification and reference the implementation of open source RTI library CERTI’s design, we have a choice Realization of a number of core services, including: z Time management: Regulating, Constrained. z Declaration Management: Object, Interaction. z Object Management: Register, Discover, Delete, Remove, Update, Reflect, Send, And Receive. As a kind of parallel computing tasks, the simulation federation’s basic federal management service is implemented based on the parallel task management. The above services constitute a subset of HLA specifications, and can be designed in relatively simple and efficient ways to meet the needs of the astronavigation simulation. Scripts used to control the process and guide the implementation of information processing is a major feature of the simulation system. The script is a plain text stored procedures, in general, a computer script is a combination of computing, in which the logic can be achieved with certain branches. Script development, it is rather the general procedure is relatively close to natural language, may not be interpreted by the compiler, which will do some help to quickly develop or some light-weight control. There are many scripting language now, the general implementation of a scripting language interpreted only with a specific interpreter related, so long as the computer has the appropriate language interpreter it can be cross-platform. In choice of script language specification, there are two options: One is to build a custom syntax for a simple scripting language to achieve the basic functions; one is using the existing

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standard scripting language. For the first approach, requires a certain amount of work, the resulting language skills are limited, but it does not require additional development kits support, can maintain platform-independent features. The latter is necessary to use some existing scripting language development kit, which kit may not be platform-independent, and will bring the system constraints.

Fig. 4. System usage demonstration

4 Conclusion We take the full integrated existing achievement's way as a foundation, elaborated the construction plan about general simulation platform face the astronavigation simulation domain. We use our knowledge and mastery about existing aerospace-related open-source software, and integrate the appropriate library at the appropriate entry point, but it’s not a simple integration, it’s the selective re-creation on the existing software’s ideas (for example: the owner designed 3D engine, the high-level architecture and cluster system’s integration, etc.). The structured and functional simulation platform is able to fill in the field of aerospace simulation of open-source generalpurpose simulation platform gaps. We believe that the completion of the platform can play an important role for scientific research in the field of aerospace simulation. Some astronautics field simulation tests showed that ASPCCE can fulfill complex astronautics simulation and computing works. We hope it could play an important role in the decision support and the system optimization in the astronautics field. Meanwhile, as open source software, we hope it is valuable in the scientific research field for the astronautics.

References 1. Wang, W.-G., Sun, X.-Y., Xu, Y.-P., et al.: Satellite orbit maneuver simulation application based on HLA. Journal of System Simulation 20(1), 89–93 (2008) 2. Li, K.-X., Cong, M.-Y., Zhang, W.: Infrared sequence image generation of point target in deep space. Optics and Precision Engineering 17(12), 3062–3068 (2009)

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3. Fengrui, Mengxin, Panzhongshi: The design and implementation of STK-pRTI middleware in optical detection real time simulation system. In: Proceedings of the 2nd International Conference on Modeling and Simulation, ICMS 2009, vol. 6, pp. 414–418 (2009) 4. Liu, G.-M., Wen, Y.-L., Liao, Y.: Research on dynamic simulation of satellite based on multi software platform. Journal of System Simulation 19(2), 308–311 (2007) 5. Wang, D., Qiu, X.-G., Huang, K.-D.: Study on STK-RTI middleware based modeling and simulation of space-ground integrated combat. Journal of System Simulation 17(2), 501–503 (2005) 6. Zhang, J., Cao, X.: An integrated system for satellite orbit design and mission analysis. In: Collection of Technical Papers - AIAA Modeling and Simulation Technologies Conference, vol. 1, pp. 26–31 (2004)

A Strongly Partitioned Operating System Model for Data Link Networks Xiaoming Tang1, Yuting Zhao1, Yinjuan Li2, and Yujun Liu3 1

School of Automation, Northwestern Polytechnical University, 710072 Xi’an, China 2 Flight Automatic Control Research Institute, 710065 Xi’an, China 3 Academy of Armored Forces Engineering, 100084 Beijing, China [email protected]

Abstract. A strongly partitioned operating system model for data link (DL) networks is proposed. Every partition is completely independent and isolated in this model. A fault in one partition only affects the partition itself rather than other partitions. The operating system kernel, called DL-OS, is discussed in sub-topics including system architecture, partition scheduling, task model, inter-partition communication and memory management. DL-OS is analyzed and evaluated. DL-OS is applied in a DL network, in which the traditional three systems were integrated into only one platform. The size, weight and power (SWAP) of the DL network terminal is dramatically decreased. Keywords: Data Link Networks, SWAP, Strongly partitioned, OS.

1 Introduction Data link (DL) networks play an important role in the evolution toward network centric warfare (NCW). Many types of intelligent bombs and missiles were developed by virtue of DL networks, such as JASSM from Lockheed Martin. One of the most important keys to the success of DL networks is the embedded real-time operating system widely used in tactical communication systems nowadays. In the federation architecture, which is common used in today’s tactical communication systems, each subsystem is a self-contained and isolated system. The trend of next generation systems is integrated modular, in which the different subsystem with different critical level can be integrated in to only one platform. DL-OS presented in the paper guarantees that every partition of the operating system is completely independent and isolated. A fault in one partition will only affect the partition itself rather than other partitions. DL-OS is very suitable for intelligent systems like data link terminals mentioned in this paper. The rest of the paper is organized as follows: some related works are in Section 2; Section 3 presents the philosophy and model of DL-OS; the implementation of DL-OS about spatial partition, time partition, task management is described in Section 4, followed by the evaluation of DL-OS in section 5; Section 6 gives a case study in the development of a data link network and the conclusion is presented in Section 7. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 274–281, 2010. © Springer-Verlag Berlin Heidelberg 2010

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2 Related Works Currently, there are two popular operating systems with spatial and temporal partitioning, VxWorks 653 by Windriver Inc. and Integrity-178B by Greenhill Inc. The DEOS developed by Honeywell only has the property of temporal partitioning. VxWorks 653 is derived from the traditional operating system VxWorks and it has been used in mission critical systems. Due to many defects rooted from VxWorks, there are no reports about the deployment of VxWorks 653 in safety critical systems. DEOS employs the slack stealing which efficiently allocates unscheduled and reclaimed CPU time for aperiodic task execution. DL-OS uses the same technique to schedule the aperiodic tasks. Besides, a novel multi-server is added in DL-OS to schedule the debugging information. The multi-server allocates the idle slot to debugging agent to guarantee the deterministic behavior even in software development process. Integrity-178B uses the resource partitioning to guarantee the high reliability and determination. DL-OS employs not only the spatial, temporal and resource partitioning, but also the cache partitioning. The resource partitioning make it deterministic for the access and control to I/O, and the cache partitioning can dramatically decrease the cache jitter for safety critical partition.

3 Model Description The new architecture of operating system kernel, DL-OS, as shown in Fig. 1 (left), adopts the two-level scheduling system model, rather than traditional scheduling model commonly used in commercial real-time operating systems. A partition represents an independent domain in time and space. Within a domain, DL-OS schedules and manages the tasks of one specific application. An application may consist of many tasks, such as periodic, non-periodic or sporadic tasks. Different partition can employ different scheduling strategy and run on different kernel. Each partition is represented in the form of tuple (x, y), in which x means the worst case execute time (WCET) of the partition and y means the period of the partition. DL-OS adopts cyclic schedule to dispatch partitions as shown in Fig. 1 (right). There are three partitions named P1(2,8), P2(8,16) and P3(6,32) in Fig. 1 (right). According to [6], we get the minor frame, mf = 8ms, main frame, MF = 32ms. In the first minor frame, P1 and the first part of P2, named P2.1, get a chance to execute. In the second minor frame, P1, P2.2 and P3.1 are scheduled. P1, P2.2 and P3 execute in the 4th minor frame. At the end of the MF, There is a 2ms idle slot. In the DL-OS task

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Fig. 1. Strongly partitioned OS model (left). Partition Scheduling (right).

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Fig. 2. Inter-partition communication Model (left). Task Model (right).

model, it is assumed that all tasks arrive periodically. The task set is represented by (Ci, Mi Di ,Ti), where Ci, Di, Ti, Mi means the WCET, the deadline, the period and the size of transmitted message, respectively. Here we decomposed Di into CDi and MDi, as indicated in Fig. 2 (right), where CDi means the computing deadline and MDi means the message deadline. Computing deadline affects the allocation of the bandwidth. The minimum required bandwidth for task i is Bimin = Mi/MDi. For instance, if Mi is 100KB, and MDi is 10ms, the required bandwidth should be 10MBps at least.

，

4 Model Implementation 4.1 Architecture DL-OS is designed to meet the demand of the temporal and spatial partitioning in the DL. The three-layer architecture of DL-OS is illustrated in Fig. 3. The architecture includes the modular support layer (MSL), core operating system (COS) and partition operating system (POS).

Fig. 3. DL-OS Architecture

MSL is similar with the board support package (BSP) in commercial operating system like VxWorks, but MSL is an independent layer in DL-OS and has its own image. Based on this three-layer philosophy, the MSL, COS and different POSs are totally isolated and can be designed, linked and debugged independently. They are like components in Joint Tactical Radio System (JTRS) Software Communication Architecture (SCA) [9]. MSL masks the difference of the specific hardware implementation and provide the universal interface for upper layer. MSL also manage the hardware resources, like basic

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memory management, I/O resources and bus management. From the aspect of COS, MSL acts as an agent to access the lower hardware. Thus, MSL is independent with COS and flight application, just relied on the specific hardware. COS is consist of the partition management, inter-partition communication, interrupt and exception management, health monitor, system synchronization for redundant hardware. In addition, the COS provides an optional task management component as well. COS is not relevant with hardware and flight application. POS, resided on the COS, provides the run-time container for specific flight application. The applications with different critical levels can run on the COS within different POSs. Through the generic interface APEX [1], POS can interact with COS. POS is related with flight application, rather than COS and MSL. 4.2 Spatial Partition The partition is the resource domain for a run-time application. It has its exclusive environment, context, configuration information and memory map. DL-OS uses the MMU provided by most CPU to implement the spatial partition. COS runs in the privilege mode, and all the POSs in the user mode. POS accesses the resources in privilege mode by the system call. Like other real-time operating systems, DL-OS doesn’t use the virtual memory, and the map from logical address to physical address is one to one directly. The physical memory spaces of MSL, COS and various POSs are independent and not overlapped. In addition, to guarantee the protection and isolation of different application systems, the health management plays a very important role. The health monitor modular identifies the exception and error, and then classifies and submits to the fault management modular. The duty of fault management is to recovery from the fault, or at least confine the fault in trigged application system. Except the fetal error, like power invalidation, the core operating system must not shutdown. 4.3 Time Partition In the time domain, DL-OS should guarantee that every partition has been allocated some time slots to run in its own slots and no interference from other partitions. For example, the deadlock or deadline missing in partition 1 just affect itself, after its time slot, the cyclic scheduler for partitions will switch partition 1 out whatever. As requirement in DO-297[2] for incremental development and deployment, DL-OS can add and delete the partition at run-time, which need cyclic scheduler can adjust the slot windows within the main frame. The characteristics of partition schedule are as follows: 1) The scheduling unit is partition only. 2) There is no priority for partition. 3) The main frame and minor frame are allocated off-line by system integrator. Only if there is a partition need to be added or delete, the slot’s offset in main frame may be changed. DL-OS will schedule the virtual partition called IDLE partition if there is no partition to run. Besides, there are some tasks in core level, like health monitor task. They can be executed at any time, yet, the time those tasks consumed should account in the running partition, which should be considered in schedule analysis.

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4.4 Task Management Task management is an optional component for partition operating systems. There is only one execution loop in some partitions and no task management or partition operating system at all. The basic scheduling method is priority preempted scheduling. The difference between the periodic and non-periodic scheduling lies on the opportunity of schedule.

5 Performance Evaluation We measured the four key performance metrics of DL-OS based on the Rhealstone [7]. The hardware platform is featured by PowerPC755 CPU@210MHz and no external cache. Each test case executed 100 times, and then the maximum, minimum and average value is calculated. Table 1. DL-OS time measurement Time Performance

Context Switch Time Interrupt Delay Time Task Create Time Partition Switch Time

Max 2.720us 0.520us 13.500us 10.250us

Min 1.345us 0.345us 4.050us 5.000us

Avg 1.504us 0.404us 6.900us 5.250us

The time performance is illustrated in Table 1. Context switch time is the time taken by two independent tasks in ready queue with equal priority to switch from the one context to another. Interrupt delay time is the time interval from the interrupt granted to the first instruction application related. Interrupt delay time doesn’t include the delay caused by bus, peripheral devices. Task create time is the time to create the task. This time depends on the time to allocate the resources for task and the numbers of tasks have been created. Partition switch time, the index dedicated to DL-OS, is used to evaluate the performance of switching the partition. At the time of switching partition, the partition scheduler saves the context of the current partition, load the context of the next partition, and then run this partition. Furthermore, we have optimized the mechanism of memory management of DL-OS according to the applications. There are three modes for memory management in PowerPC architecture. Real mode doesn’t provide the privilege check and memory protection. BAT mode manages and protects the memory through the eight registers’ pairs of IBAT and DBAT. Segment & Page (S&P) mode manages and protects the memory with the page table. We measured the RAM accessing time 10,000 times. The result is shown in Fig. 4 (left). In BAT mode, the average time is 102.7us, but is 258.3us in S&P mode, one time larger than the former. The performance of accessing I/O 10,000 times under BAT, S&P mode, and no memory management enabled is showed in the Fig. 4 (right). According to this figure, the average time is 756.5us in BAT mode, 20,204.5us in S&P mode and 9,032us in MMU disabled mode. Why the performance in no memory management is better than that in S&P mode for accessing the I/O? The reason is no matter in what mode DL-OS always disables the cache for I/O. In S&P mode, the performance of accessing I/O is affected not only by address translation but also by cache disabled.

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Fig. 4. Performance of Accessing RAM in BAT and PAGE (left); Performance of accessing IO in BAT, PAGE and no MMU (right)

Based on the measure and analysis above, we optimize the spatial management of DL-OS. Combining the BAT and S&P mode, we deploy the core operating system in BAT mode, and partition operating system in Segment & Page mode.

6 Case Study: DL Networks DL-OS has been successfully applied to DL network terminals. CPU is PowePC-755@210MHz. Memory consists of 128MB RAM and 8M Flash. The main board in DL network terminals is displayed in Fig. 5 (left).

Fig. 5. Main Board in DL Network Terminals (left). Partition Scheduling Table for a DL Network (right).

There are three partitions in the DL-OS implementation. For the real-time response and deployment convenience, no other but timer interrupt is scheduled. Only two FIFOs are used for polling mechanism. The measured metrics are listed in Table 2. Table 2. DL-OS Partition Switch Time Partitions Metrics

Task Measures Tasks Period Task 1 5ms Partition 1 Task 2 1s Task 1 10ms Partition 2 Task 2 10ms Task 3 10ms Partition 3 Task 1 10ms

WCET 230us 920us 560us 422us 104us 53us

Partition WCET 1.521ms 1.596ms 132us

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In particular, we use the complex health management and the Periodic I/O model mentioned before for safety and reliability. The execution time of those tasks should be added to the specific partition at run-time. Based on the analysis above, we adopt the main frame which includes four minor frames. The sequence is partition 1, 2 and 3, as shown in Fig. 5 (right).

Fig. 6. AADL Model of a DL Network

According to the standard AADL [8], we map the partition in DL-OS into the process in AADL, the task into the thread, and the message channel into the connection. The corresponding AADL model of this system is showed in Fig. 6. In this model, the messages sent and received are via different connection and the inter-partition communication is implemented by the model Periodic I/O. Table 3. End-to-end latency analysis [5] Source Partition GD FK FK BA

Destination Partition FK GD BA FK

Delay 4ms 5ms 6ms 7ms

The constraint of data transmission between the different partitions should be no more than the corresponding latency analyzed above; otherwise, we should adjust the partition scheduling table and optimize the application system to meet the constraint.

7 Conclusions To meet the requirement of IMA, DL-OS can integrate the traditional systems into only one platform. Three systems were integrated into one system with DL-OS, the size, weight and power (SWAP) of DL network terminals were dramatically decreased with DL-OS. Although the design and application of DL-OS have got very success, the robust design, support to redundancy architecture should be considered in detail further. Moreover, the architecture analysis and model under DL-OS, and formal verification of DL-OS should be extended in our next research.

References 1. Aeronautical Radio, Inc. Avionics application software standard interface, required services (December 2005) 2. RTCA, Inc. Integrated modular avionics (IMA) development guidance and certification considerations (November 2005)

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3. RTCA. Software considerations in airborne systems and equipment certification (December 1992) 4. Aeronautical Radio, Avionics Application Software Standard Interface (January 2007) 5. Feiler, P.H., Gluch, D.P., Hudak, J.J., Lewis, B.A.: Embedded System Architecture Analysis Using SAE AADL. Technical Note,CMU/SEI (June 2004) 6. Kim, D., Lee, Y.-H.: DC2 Scheduling of Aperiodic Tasks for Strongly Partitioned Real Time Systems. IEEE Real Time Computing Systems and Applications (December 2000) 7. Kar, R.: Rhealstone—a Real-time Benchmarking Proposal. Dr. Dobbs Journal (February 1989) 8. SAE International. Achitecture analysis & design language(AADL) (August 2004) 9. JPEO, Software Communications Architecture Specification V3.0 (August 2004)

Twin Hybrid ElGamal Encryption over Signed Quadratic Residue Groups Yonggang Cui1 and Yujun Liu2 1

State Key Laboratory of Information Security, Graduate School of the Chinese Academy of Sciences, Beijing 100049, China 2 Department of Information Engineering, The Academy of Armored Forces Engineering, Beijing 100072, China

Abstract. In this paper, we consider the security of twin Hybrid ElGamal (THEG) scheme when instantiated over the group of signed quadratic residues. In random oracle model, the scheme is proved INDCCA secure under composite computation Diﬃe-Hellman assumption, which is at least as hard as factoring. In the standard model, we give a more tight security reduction for THEG, using a new hash proof system, than that of Hybrid ElGamal (HEG) in [4]. Therefore, THEG can be instantiated over smaller RSA modulus than HEG, which results in shorter ciphertexts for the same bit security and hence reduces the communication complexity of encrypted data transmitted over public communication lines. Keywords: chosen ciphertext security, Hybrid ElGamal encryption, signed quadratic residue group.

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Introduction

A fundamental task of cryptography is to protect the secrecy of data transmitted over public communication lines. For this purpose we use encryption schemes which use some secret information (a key) to encode data in a way that an adversary cannot decoded it. Besides security, shorter ciphertext is also a preferable character of encryption scheme since it can reduce the communication complexity of the same amount of encrypted data. The Hybrid ElGamal encryption scheme (HEG) combines the original ElGamal encryption scheme with a hash function for key derivation and a symmetric cipher. As ”Diﬃe-Hellman integrated encryption scheme” (DHIES) [1], it is contained in several standards bodies for public key encryption, e.g., in IEEE P1363a, SECG, and ISO 18033-2. Hofheinz and Kiltz [4] consider the security of HEG when instantiated over the + group of signed quadratic residues (QN+ N ). They proved that HEG over QNN is IND-CCA [5] secure in the random oracle model under the factoring assumption. Under a very diﬀerent assumption, the higher residuosity assumption, they also proved that HEG over QN+ N is IND-CCA secure in the standard model. This is the ﬁrst security result for HEG in the standard model from a non-interactive computational assumption. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 282–286, 2010. c Springer-Verlag Berlin Heidelberg 2010

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In this paper, we consider the security of a twin version of HEG, called THEG, when instantiated over QN+ N . In the random oracle model, the scheme is proved IND-CCA secure under composite computational Diﬃe-Hellman assumption, which is at least as hard as factoring. In the standard model, we give a more tight security reduction for THEG, using a new hash proof system, than that of HEG in [4]. Therefore, THEG can be instantiated over smaller RSA modulus than HEG, which results in shorter ciphertexts for the same bit security and hence reduces the communication complexity of encrypted data.

2 2.1

Preliminaries Signed Quadratic Residue Groups

Let N = P Q be a n-bit Blum number, i.e. P and Q are both n/2-bit primes congruent 3 modulo 4. By QNN we denote the group of quadratic residues moduleo N . For x ∈ ZN , we deﬁne |x| as the absolute value of x, where x is represent as a signed integer in the set {(1−N )/2, · · · , (N −1)/2}. Let QN+ N = {|x| : x ∈ QNN }, + then QN+ N is a group with the following operation. Namely, for g, h ∈ QNN we deﬁne g ◦ h = |g · h mod N |. The reader is refer to [4] for details. We also deﬁne g x := g ◦ g ◦ · · · ◦ g = |g x mod N | for an integer x. x times

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Hash Proof Systems

A hash proof system HPS = (Params, Pub, Priv) consists of three algorithms. The randomized algorithm Params(1k ) generates parameterized instances of params = (G, C, K, V, PK, SK, Λ, μ : SK → PK), where G is a group, C, K are sets and Λ = (Λsk )sk∈SK with each Λsk a hash function from C to K. We require that V ⊂ C and for random C0 ∈ V and random C1 ∈ C\V the two elements C0 , C1 are computationally indistinguishable. The deterministic public evaluation algorithm Pub inputs the projection key pk = μ(sk), C ∈ V and a witness r of the fact C ∈ V and returns K = Λsk (C). The deterministic private evaluation algorithm Priv inputs sk ∈ SK, C ∈ C and returns Λsk (C), without knowing a witness. HPS is called κ-entropic if for all C ∈ C\V, the min-entropy H∞ (Λsk (C)|pk) ≥ κ where in the above pk = μ(sk) for sk ←R SK. Details is refer to [3].

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Twin Hybrid ElGamal over Signed Quadratic Residues

Let 0 < δ ≤ 1/4 be a constant and n(k) a function. RSAgen is an algorithm that generates elements (N, P, Q, S) such that N = P Q is a n-bit Blum number and S > 1 is a divisor of φ(N )/4 with 1 < gcd(S, (P − 1)/2) < (P − 1)/2 and 1 < gcd(S, (Q − 1)/2) < (Q − 1)/2. Furthermore, the prime factors of φ(N )/4 are pairwise distinct and at least δn-bit integers.

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The Encryption Scheme

Let SE=(E,D) be a symmetric cipher with key-space {0, 1}l(k) and H = (Hk )k∈N be a family of hash functions with H : {0, 1}3n(k) → {0, 1}l(k) for each H ∈ Hk . Deﬁne the following encryption scheme THEG=(Kg,Enc,Dec): KeyGen. Kg(1k ) chooses randomly an RSA modulus N = P Q generated with RSAgen and a hash function H ∈ Hk . Let g ∈ QR+ N be random generator and 0 < x, z < N/4 two integers. Kg then sets X = g x , Z = g z and outputs the public key pk = (N, g, X, Z, H) and secret key sk = (N, x, z, H). Encryption. Enc(pk, m) chooses uniformly integer 0 < y < N/4, sets Y = gy

K = H(Y, X y , Z y )

ψ = EK (m)

and outputs the ciphertext (Y, ψ) ∈ × {0, 1}∗. Decryption. Dec(sk, (Y, ψ)) veriﬁes that Y ∈ QR+ N and rejects if not. Then, Dec computes K = H(Y, Y x , Y z ) and outputs DK (ψ). QR+ N

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Security of THEG

We will prove that the same scheme THEG is secure in the standard model and in the random oracle model under diﬀerent assumptions. Theorem 1. Assume the factoring or computation diﬃe-hellman assumption holds for RSAgenn(k),δ , H is modeled as a random oracle, and SE is one-time IND-CCA secure. Then THEG is IND-CCA secure. When factoring assumption holding for RSAgenn(k),δ , Theorem 1 follows by [[4], Theorem 3]. On the other hand, Theorem 1 follows by [[2], Theorem 4] when computation diﬃe-hellman assumption holds for RSAgenn(k),δ . In the standard model, the following theorem will be proved in Section 4. Theorem 2. Assume the Higher Residuosity assumption holds for RSAgenn(k),δ , H is a family of 4-wise independent hash functions, and SE is AE-OT secure with l-bit 4l keys. If |QR+ N |/S > 2 , then THEG is IND-CCA secure.

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A Security Proof in the Standard Model

Let (N, P, Q, S) be generated by RSAgen and GS the unique subgroup of order S of Z∗N . To prove Theorem 2, we will consider a slightly diﬀerent scheme, THEG =(Kg , Enc,Dec). It is deﬁned as THEG, with the only diﬀerence that in Kg , the element g from key generation is a uniform element from G+ S (instead of an ). uniform element from QN+ N The higher residuosity (HR) assumption for RSAgenn(k),δ states that distinguishing a random element from GS and a random element from QRN is computational infeasible. It is easy to see that, under the HR assumption, THEG is IND-CCA if and if only THEG is IND-CCA. So, the rest of this section is devoted to prove that THEG is IND-CCA secure.

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IND-CCA Secure Encryption via Randomness Extraction

Let HPS = (Params, Pub, Priv) be a hash proof system and H a family of hash function with H : K → {0, 1}l(k) . Let SE = (E, D) be an AE-OT secure symmetric encryption scheme with key-space KSE = {0, 1}l(k) . A public key encryption scheme PKGHPS = (Kg, Enc, Dec) is deﬁned as follows. Key generation. Kg(1k ) picks par ←R Par(1k ), sk ←R SK and deﬁne pk = μ(sk) ∈ PK. Next, it picks a random hash function H ←R H. The public key is (par, H, pk) and the secret key is (par, H, sk). Encryption. Enc(pk, m) picks C ←R V together with its witness r that C ∈ V. Session key K ← H(Λsk (C)) ∈ {0, 1}l is computed as K ← H(Pub(pk, C, r)). The symmetric ciphertext is ψ = EK (m). The ciphertext is (C, ψ). Decryption. Omit. Theorem 3. [3] Assume HPS is κ(k)-entropic with hard subset membership problem and negligible collision probability, H is a family of 4-wise independent hash functions with H : K → {0, 1}l(k) , and SE is AE-OT secure. If κ(k) ≥ 2(l(k) + k) then PKEHPS is secure in the sense IND-CCA. 4.2

A Hash Proof System for THEG

We now give a hash proof system HPS that yields the scheme THEG via the transformation given in the above subsection. Let (N, P, Q, S) ←R RSAgen(1k ) + and deﬁne G = Z∗N , C = QR+ N and V = GS . Assume g is a uniform generator + r of GS , then V = {g : r ∈ ZS }. A value r ∈ Z is a witness of C = g r ∈ V. Note that it is possible to sample an almost uniform element from V together with a witness by ﬁrst picking r ∈ Zλ (λ = (N − 1)/4) and deﬁning C = g r . Furthermore, membership in C can be eﬃciently checked by Lemma 1 in [4]. + 3 2 2 Deﬁne SK = (Zλ )2 , PK = (G+ S ) and K = (QRN ) . For sk = (x, z) ∈ (Zλ ) , + 2 x z k deﬁne μ(sk) = (X, Z) = (g , g ) ∈ (GS ) . This deﬁnes the output of Par(1 ). Let h be a random generator of QN+ N and deﬁne C for C ∈ C, such that ◦ C = h. For C ∈ C deﬁne C if C ∈ V, (C, C x , C z ), Λsk (C) = x+z x+z (C, C ,C ), if C ∈ C\V. This deﬁnes Priv(sk, C), which uses the trapdoor of HPS, i.e. the order of group GS . Given pk = μ(sk), C ∈ V and a witness r ∈ Z such that C = g r , Pub(pk, C, r) computes K = Λsk (C) as K = (g r , X r , Z r ).

This complete the description of HPS. Note that PKEHPS is exactly THEG . Therefor the IND-CCA security of THEG can be proved by combining Theorem 3 with the following lemma.

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Lemma 1. Under the HR assumption, the subset member problem is hard in HPS. Furthermore, HPS is (log2 T )-entropic with collision probability δ = 0, where T = |QR+ N |/S. Proof. The subset membership is hard in HP S by the deﬁnition of HR assumption. The collision probability δ is zero since Λsk (C) = (C, C x , C z ) contains the element C. To show HPS is (log2 T )-entropic we can assume SK = (ZT S )2 , instead of SK = (Zλ )2 since random element of (ZT S )2 is computationally in+ distinguishable from a random element of (Zλ )2 . Let C ∈ C\V = QR+ N \GS and 2 sk, sk ∈ (ZT S ) . If μ(sk) = μ(sk ), then x ≡ x mod S, z ≡ z mod S. Further x+z = C x +z C x +z = hx +z more, if Λsk (C) = Λsk (C), then hx+y = C x+z C and so x + z ≡ x + z mod T S. Therefor H∞ (Λsk (C) | pk) ≥ log2 T .

5

Discussion and Comparisons

Compared to the Hybrid ElGmamal in [4], THEG over QR+ N can be proved INDCCA secure, in random oracle model, based on the computational DH assumption which is at least as hard as factoring N [6]. Furthermore, in random oracle model, THEG can be secure instantiated over a more abroad class of RSA modulus than TEG, whenever CDH assumption holds for RSAgen. In the standard model, we construct a hash proof system, where Priv using the trapdoor of the underlying sunset membership problem, to provide a more tight security reduction for THEG than that of TEG. For example, for k = 80-bit security one may chooses n(k) = 1024 and δ = 1/8. Then N can be sampled as N = P Q for P = 2P1 P2 + 1 and Q = 2P3 P4 + 1 for primes Pi such that Pi ≥ 2δn and S = P1 P3 . In this case, 48-bit key can be extracted by Lemma 1 and Theorem 3, while the proof in [[4], Lemma 8] has no sense for this case.

References 1. Abdalla, M., Bellare, M., Rogaway, P.: The oracle Diﬃe-Hellman assumptions and an analysis of DHIES. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 143–158. Springer, Heidelberg (2001) 2. Cash, D.M., Kiltz, E., Shoup, V.: The twin diﬃe-hellman problem and applications. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 127–145. Springer, Heidelberg (2008) 3. Kiltz, E., Pietrzak, K., Stam, M., Yung. M.: A New Randomness Extraction Paradigm for Hybrid Encryption. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 589–608. Springer, Heidelberg (2010) 4. Hofheinz, D., Kiltz, E.: The group of signed quadratic residues and applications. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 637–653. Springer, Heidelberg (2009) 5. Rackoﬀ, C., Simon, R.: Non-interactive zero-knowledge proof of knowledge and chosen ciphertext attack. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 433–444. Springer, Heidelberg (1992) 6. Shmuely, Z.: Composite diﬃe-hellman public-key generating systems are hard to break. Technical Report 356, Computer Science Department, Technion, Israel (1985)

Extra Slot Allocation for Fair Data Collection in the Slot-Based Grid Network Junghoon Lee and Gyung-Leen Park Dept. of Computer Science and Statistics, Jeju National University, 690-756, Jeju Do, Republic of Korea {jhlee,glpark}@jejunu.ac.kr

Abstract. This paper designs a fair message collection scheme on the grid-style wireless network, aiming at accelerating the fast and stable deployment of smart grid systems. Based on the slot-based medium access protocol, the proposed scheme allocates extra slots on each control round for the nodes along the fringe of the grid, as they unavoidably have poor delivery ratio due to long hops from the coordinator. Along the fringe path, the farther away from the coordinator, the higher priority a node has for message loss compensation. The simulation results, obtained from a discrete event scheduler, show that we can achieve up to 15.6 % improvement of delivery ratio for the farmost node and 12.3% reduction of standard deviation, with 14.3 % extension of control loop length.

1

Introduction

The smart grid is expected to provide power quality for 21st century needs, having self-healing, consumer-friendly, and attack resistant features [1]. To this end, it essentially incorporates information technology, communication infrastructure, and modern sensors at large scales for both on-line and back-oﬃce services to facilitate the operation and management of assets [2]. Based on this infrastructure, new monitoring, control, and protection applications can be developed and run to achieve the objectives of the smart grid. Not just the electric power devices but many diﬀerent components can be a part of this system, extending the target to be monitored. For example, PHEV (Plug-In Hybrid Electric Vehicles), carrying internal or external sensors, create new applications such as a ﬂeet-based data collection and analysis. Monitoring, processing, and controlling functions are performed at diﬀerent places and the data network must deliver the message between each function entity as accurately and timely as possible. This paper considers a grid-style network where each node communicates with its neighbors according to the preassigned schedule based on the time-slotted access scheme [3]. The example of grid topology can be found in a traﬃc light network in the urban area, while

This research was supported by the MKE, Korea, under the ITRC support program supervised by the NIPA. (NIPA-2010-(C1090-1011-0009)). Corresponding author.

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many factory area can install wireless nodes at regularly-spaced coordinates. Figure 1 plots a 4 × 4 grid network, where N0,0 is the coordinator node. In a control round, every monitoring message is reported from each node to the coordinator and the corresponding control action is delivered to each node oppositely. The details on this network can be found in [4], which designs an eﬃcient real-time message delivery scheme based on channel sensing and switching. Even if such a scheme can signiﬁcantly enhance the system-wide delivery ratio, the node closer to the coordinator show a much better delivery ratio than that far away from the coordinator. This problem makes it diﬃcult for the coordinator to have the global view for the current status on the whole control area, possibly leading to a misdecision. To solve this problem, we are developing a compensation scheme that allocates more slots for the node having poorer message delivery ratio.

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Compensation Policy

Figure 1 also marks the message delivery ratios for the respective nodes in the 4 × 4 grid network. As shown in this ﬁgure, every node has a diﬀerent delivery ratio, even if each wireless link has the same channel error rate, namely, 0.2. The success ratios of 1-hop nodes such as N1,0 and N0,1 are much higher than that of the 6-hop node, N3,3 by 34.7 %. Furthermore, for the nodes the same hops away from the coordinator, their success ratios are diﬀerent as the node on the fringe of the grid cannot beneﬁt from the split-merge operation which adapts the transmission path according to the current channel condition [4]. In a control loop, time slots are assigned to each node based on the routing protocol and a speciﬁc collection policy, both for upstream and downstream. Basically, the BFS (Bread First Search) scheme allocates 48 slots for the upstream transmission to each node one by one [3]. The loop length can be optimized by

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overlapping those transmissions which can be done in a single time slot without interfering each other [5]. In addition to the regular slot allocation, this paper proposes an additional slot allocation for the fringe nodes, namely, N3,x and Nx,3 in the 4 × 4 grid. At this stage, we allocate 6 slots, which corresponds to the hop length from the coordinator to the farmost node, N3,3 . The 6 slots are allocated one by one either for N3,3 → N3,2 , N3,2 → N3,1 , N3,1 → N3,0 , N3,0 → N2,0 , N2,0 → N1,0 , and N1,0 → N0,0 , or N3,3 → N2,3 , N2,3 → N1,3 , N1,3 → N0,3 , N0,3 → N0,2 , N0,2 → N0,1 , and N0,1 → N0,0 . For the odd-numbered rounds, the ﬁrst allocation works, while for the even-numbered rounds, the second allocation works. At the beginning of this extra procedure, N3,3 sends it message if it has failed to transmit in this control round. Otherwise, the slot goes idle. At the next slot, N3,2 , in the odd-numbered round, if a message has arrived from N3,3 at the previous slot, it relays the message. Otherwise, it can retransmit its message, which failed to be transmitted. In this way, the route proceed along the fringe nodes to give them one more chance to recover the message transmission failure.

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Performance Measurement

We evaluate the performance of our scheme via simulation using SMPL, which provides a simple and robust discrete event trace library [6]. The 4 × 4 grid is selected for the experiment, which assumes that every wireless link is directional and has the same packet error rate to focus on the performance of the compensation scheme. Here, the overhead ratio is 6/42, namely, 14.3 %. First, Figure 2 plots the improvement of delivery ratio for the farmost node. The poorest ratio is important for the fair message collection among the sensor nodes. While the noncompensated case shows almost linear decrease along with the increase of the error rate, the compensation scheme can enhance the delivery ratio by up to 15.6 % when the error rate is 0.15. Beyond this point, the eﬀect of compensation gradually diminishes, as the recovery process also needs a multi-hop transmission along the error-prone links. 1

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Next, Figure 3 shows the standard deviation of delivery ratios for all nodes in the 4 ×4 grid. The smaller the standard deviation, the fairer the collection step, making the coordinator can much better catch the current status of the target system. Be it compensated or noncompensated, we cannot avoid the increase of standard deviation in the delivery ratio stemmed from the increase of the error rate. A high error rate more aﬀects the node many hops away from the coordinator. Anyway, the compensation scheme can reduce the standard deviation by up to 0.123 when the error rate is 0.2. As a result, with just a simple addition of extra slot allocation along the error-prone fringe nodes in the grid network, we can achieve up to 15.6 % improvement of delivery ratio for the farmost node and 12.3% reduction of standard deviation.

4

Concluding Remarks

This paper has designed a fair message collection scheme on the slot-based wireless network, allocating extra slots for the node along the fringe of the grid, as they inherently suﬀer from poor message delivery ratio. The simulation result shows that the proposed scheme can achieve 12.3% reduction of standard deviation, with 14.3 % extension of control loop length. This work is important in deploying the smart grid system which is necessarily based on monitoring and controlling facilities located accross the wide area. We are currently pursuing many research goals for the smart grid system in Jeju area, especially focusing on the eﬃcient integration of information technology and prospective interaction with the vehicular network.

References 1. Pipattanasomporn, M., Feroze, H., Rahman, S.: Multi-agent systems in a distributed smart grid: Design and implementation. In: IEEE PES Power Systems Conference and Expositions (2009) 2. Power Systems Engineering Research Center, U.S. Energy Infrastructure Investment: Large-Scale Integrated Smart Grid Solutions with High Penetration of Renewable Resources, Dispersed Generation, and Customer Participation, White paper (2009) 3. Han, S., Song, J., Zhu, X., Mok, A.K., Chen, D., Nixon, M., Pratt. W., Gondhalekar, V.: Wi-HTest: Compliance test suite for diagnosing devices in realtime WirelessHART network. In: The 15th IEEE Real-Time and Embedded Technology and Applications Symposium, pp. 327–336 (2009) 4. Lee, J., Song, H., Mok, A.K.: Design of a reliable communication system for gridstyle traﬃc control networks. Accepted at the 16th IEEE Real-Time and Embedded Technology and Applications Symposium (2010) 5. Lee, J., Park, G., Shin, I., Kim, C., Kim, S.: A control loop reduction scheme for wireless process control on traﬃc light networks. In: Gervasi, O. (ed.) ICCSA 2010, Part III. LNCS, vol. 6018, pp. 1–10. Springer, Heidelberg (2010) 6. MacDougall, M.: Simulating Computer Systems: Techniques and Tools. MIT Press, Cambridge (1987)

An Efficient Multipath Existence Checking Scheme for Wireless Sensor Networks Feng Wei1 , Yingchang Xiang2 , and Bowu Zhang3 1

Department of Computer Engineering, Heilongjiang Infotech Profession College Harbin, Heilongjiang, 150025, China 2 Department of Basic Courses, Rizhao Polytechnic College Rizhao, Shandong, 276826, China 3 Department of Computer Science, The George Washington Universiy Washington, DC, 20052, USA [email protected], [email protected], [email protected]

Abstract. Multipath routing has been employed to improve the network throughput, to provide load balance and to achieve reliability for many years in both wired and wireless networks. In wireless networks, the communication cost of the multipath construction scheme is at least polynomial. In this paper, we focus on the problem of how to chose a right position to deploy a wireless sensor node such that the existence of K multipaths, which connect a wireless sensor node and the data sink, is guaranteed. Traditionally, at a given position, the communication cost of the multipath existence checking is at least polynomial through using a multipath construction scheme. To reduce the communication overhead, we propose a simple and efficient multipath existence checking scheme, which incurs only a constant communication overhead. Keywords: Multipath Routing, Existence Checking, Wireless Networks, Topology Control.

1 Introduction Wireless Sensor Networks (WSN) have been extensively investgated for both military and civilian applications. A WSN consists of spatially distributed sensors to cooperatively monitor physical and environmental conditions, such as temperature, target movement, and sound. In a typical WSN’s deployment, there exist a large number of sensor nodes and a small number of data sinks. The sensor nodes report their readings to the sinks through multihop communications. Multipath routing was employed in a WSN to improve the network throughput and to decease the message delay. In wireless sensor networks, all the current multipath construction schemes are based on the message broadcasting, such as the one proposed in [2]. As a result of broadcasting, the communication cost for mutipath construction is lower bounded by O(N ), where N is the number of nodes in the network. Assuming that we are going to deploy a new sensor in a network as shown in Fig. 1, where the black rectangle indicates the sink, the black circles indicate the sensor nodes and the white circles indicate the G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 291–294, 2010. c Springer-Verlag Berlin Heidelberg 2010

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candidate positions at which the new sensor could be deployed. The requirement of the deployment is that the new deployed sensor can construct at least K multipaths to the sink. If there exist M number of candidate positions, then the communication cost for finding out the right position to deploy the senor is at least O(N × M ) if applying the multipath construction schemes at each candidate position in the worst case.

Fig. 1. Network model for sensor deployment

It is well known that one of the important issues in WSN is energy conservation. As the wireless transmission is the major source of energy consumption, we focus on reduce the communication cost for deploying a new sensor at a right K multipath enabled position. Our methodology is to design a light-weight K multipath existence checking scheme with communication cost O(P ), where P N , such that the overall communication cost for finding out the right position and constructing K multipath is reduced to O(P × M + N ). In the rest of this paper, we first introduce the most related work in Sec. 2. Then, sufficient conditions for K multipath Existence are presented in Sec. 3. Based on the sufficient conditions, we formally introduce our K multipath existence checking scheme in Sec. 4. Finally, the paper is concluded in Sec. 5.

2 Preliminary Result In [1], Cheng etc. presented the sufficient conditions on which there exist one more node-disjoint path in many to one wireless sensor networks. This result is summarized as the following theorem. Theorem 1. Given a sensor node S, which already has K − 1 multipaths to the sink, a sensor node P , which has H multipaths to the sink, such that P is not in any of S’s paths and S is not in any of P ’s paths, and S and P are direct neighbors. If H > |PS | or |SP | < |PS |, where |PS | is the number of P ’s paths that joint at least one of S’s paths and |SP | is the number of S’s paths that joint at least one of P ’s paths , then there must exist K multipaths from S to the sink. Based on the above sufficient conditions, [1] presented the sufficient conditions on which K multipaths exist for newly deployed sensor nodes as shown in Theorem 2.

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Theorem 2. The existence of K vertex-disjoint paths to the sink is guaranteed if the newly-deployed sensor node has K neighbors that have at least K vertex-disjoint paths to the sink. This result is beautiful. However, we notice that there might exist alternative sufficient conditions as the result presented in Theorem 2 is not necessary. We introduce our sufficient conditions for checking the existence of K multipaths in the next section.

3 Sufficient Conditions for K Multipath Existence In Theorem 2, to guarantee the K-multipath existence, the newly-deployed sensor has to find K number of neighbors that have at least K multipaths. We notice that it is not necessary to require all the K neighbors to have K multipaths. Without loss of generality, we assume that the sensor’s jth neighbor has pj number of multipath to the sink, where 1 ≤ j ≤ K, and pj ≤ pi when 1 ≤ j ≤ i ≤ K. Then, we obtain the sufficient conditions for K-multipath existence as shown in the following theorem. Theorem 3. The existence of K vertex-disjoint paths is guaranteed if the sensor node has K neighbors, and the jth neighbor has at least j vertex-disjoint paths that disjoint any ith neighbor, where 1 ≤ j < i ≤ K. Proof. Let S be newly-deployed sensor node that needs to construct its K vertexdisjoint paths to the sink. Let v1 , v2 , · · · , vj , · · · , vK be the neighbors of S, where vj has at least j vertex-disjoint paths. Then, for any vj , there must exist j vertex-disjoint paths that do not pass S because S is the newly-deployed sensor. In the following, we construct K vertex-disjoint paths for S step by step. – Pick one of v1 ’s paths, which does not pass any of v2 , · · · , vi , · · · , vK . The concatenation of this path and the edge between S and v1 forms the first path for S. – ··· – Pick j of vj ’s paths, which do not pass any of vj+1 , · · · , vK . Based on Theorem 1, there exist j vertex-disjoint paths from S to the sink. – ··· – Pick K paths from vK ’s paths. Based on Theorem 1, there exist K vertex-disjoint paths from S to the sink. Therefore, the claim is true.

It can be verified that a sensor node must satisfy all the conditions of Theorem 3 if it can satisfy all the conditions of Theorem 2. This means that Theorem 2 is a special case of our proposed Theorem 3.

4 K Multipath Existence Checking Scheme Based on the K multipath existence conditions studied above, we propose our K multipath existence checking scheme in this section. As stated in Theorem 3, the required

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information that is essential for checking the K-multipaht existence is only associated with the sensor node’s direct neighbors. Therefore, the sensor node can collect all those information through one-hop broadcasting. At each candidate position, the following K-multipath existence checking scheme is employed before launching the multipath construction scheme. Scheme K Multipath Existence Checking (1) The sensor node broadcasts a request to its direct neighbors. (2) All the sensor’s direct neighbors respond the request by sending their multipath information back. (3) The sensor node collects all its direct neighbor’s multipath information. (4) The sensor node checks whether all the conditions in Theorem 3 are satisfied. (5) The sensor node confirms the K multipath existence at a candidate position if (4) returns a positive result. In the above scheme, the communications only occur between the sensor node and its direct neighbors. Thus, the communication cost for the proposed multipath existence checking scheme is O(D), where D is the node degree. At a candidate position, we start the multipath construction only after the checking scheme confirms the multipath existence. As a result, the communication cost for K multipath construction with our existence checking scheme is bounded by O(D × M + N ).

5 Conclusion In this paper, we study the problem of multipath existence checking in wireless sensor networks. We propose sufficient conditions for K-multipath existence, which are more general than the existing results. Based on the proposed theoretical results, we design an efficient K multipath existence checking scheme. By using our existence checking scheme, the communication cost for K multipath construction is reduced from O(M × N ) to O(D × M + N ), where M is number of candidate positions, D is the average node degree, and N is the number of nodes in the network.

Acknowledgments The research was partially supported by the US National Science Foundation under grants CNS-0347674 and CNS-0721669.

References 1. Cheng, W., Xing, K., Cheng, X., Lu, X., Lu, Z., Su, J., Wang, B., Liu, Y.: Route recovery in vertex-disjoint multipath routing for many-to-one sensor networks. In: Proceedings of the 9th ACM international symposium on Mobile ad hoc networking and computing (Mobihoc 2008), Hongkong, China, May 2008, pp. 209–219 (2008) 2. Lou, W., Kwon, Y.: H-spread: hybrid multipath scheme for secure and reliable data collection in wireless sensor networks. IEEE Transactions on Vehicular Technology (May 2006)

Data Collection Scheme for Two-Tier Vehicular Sensor Networks Junghoon Lee1 and Mikyung Kang2, 1

Dept. of Computer Science and Statistics, Jeju National University, 690-756, Jeju Do, Republic of Korea 2 University of Southern California - Information Sciences Institute, VA22203, USA [email protected], [email protected]

Abstract. This paper proposes a data collection scheme for the twotier vehicle sensor network, which is comprised of high-speed WLANs surrounded by the ubiquitous cellular network, aiming at improving the accuracy and speed of event detection out of bunch of sensor data. Each vehicle reports periodically summary data including its location, timestamp, and alarm data via the cellular network, while it sends whole set, entering the WLAN range. For event detection, the information server locates the vehicles whose data must be further investigated according to the legacy spatio-temporal query or the trajectory comparison method. The WLAN can give precedence to the vehicle which might have the important data by operating two diﬀerent frequencies. We are currently reﬁning this framework and planning to test in Jeju area.

1

Introduction

With the development of vehicular networks, vehicles can possibly play a role of the sensor carrier. That is, a lot of sensors can be installed in the car, collecting the data and then reporting to an information server for further processing [1]. In this scenario, the telematics device, or the in-vehicle computer, controls sensor activities and communicates with the wireless network. Basically, vehicle sensors include speedometers, GPS locators, electronic device gauges, and the like. The record of those data, just as the vehicle black box, makes it possible to develop diverse kinds of new services, for example, traﬃc accident analysis, environment monitor, and so on. In addition, other sensors for the external environment can be installed in the vehicle to detect air pollution or accident. Moreover, PHEV (Plug-In Hybrid Electric Vehicles) is expected to accelerate new applications on the vehicle sensor network, carrying even video cameras. The amount of sensor data grows greatly and it is not possible to gather all of sensor data from all vehicles in the whole surveillance area due to limited network connectivity and unsatisfactory network bandwidth. These days, several networks can connect vehicles. First, the cellular network provides ubiquitous

This research was supported by the MKE, Korea, under the ITRC support program supervised by the NIPA. (NIPA-2010-(C1090-1011-0009)). Corresponding author.

G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 295–298, 2010. c Springer-Verlag Berlin Heidelberg 2010

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connectivity, but it’s not freely available and has limited bandwidth, as this network is originally optimized for the voice traﬃc. On the other hand, WLAN provides suﬃcient bandwidth, but it cannot connect all vehicles over the wide area. Like other mobile devices, vehicular telematics devices can install both two network interfaces and we can assume a two-tier network, comprised of WLANs surrounded by a ubiquitous cellular network as shown in Figure 1 [2]. Within the WLAN range, the device sends whole records using the WLAN interface. Outside the WLAN range, it sends just a summary or basic data set.

Fig. 1. 2-tier network architecture

In this network, raw sensor data are kept in the vehicles, while only the subset of these data can be reported to the information server. The information server estimates the current status of a vehicle and speciﬁc area with this subset. For better accuracy, it is necessary to extend the subset. However, most sensor data have routine values, just containing normal status readings. Hence, what data the subset includes is more important. Moreover, it needs the sensor data from multiple vehicles to exactly catch the event. So, when the server ﬁnds something suspicious from the report from a vehicle, say, V , it needs the sensor data from the vehicles which have been around V , as they commonly witnessed the same event, even with diﬀerent sensors. Hence, even though the sensor data might be considered to be normal from a single vehicle’s viewpoint, the global server can decide to investigate the sensor value the vehicle has collected. How to pick vehicles possibly having such relevant data is very important to the eﬃcient and correct system operation. In this regard, this paper functionally designs how the vehicle reports the sensor data on the two-tier network, how the server locates the vehicle that is likely to have critical data, and how the network can give precedence to the message carrying critical data.

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Each vehicle gathers sensor data with a relatively short period. Some sensor value can be classiﬁed as alarm-level from the beginning. The vehicle must report

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those data as soon as possible. Others may be critical or normal. Suppose that a temperature sensor catches just a little bit high value originated from ﬁre and the car having the sensor moves away from the ﬁre place. The vehicle cannot know what actually happens. Only the information server having an integrative view can identify the ﬁre after collecting sensor values from cars around the ﬁre at that time. Here, some might capture high temperature, others may capture smoke or gas. When the server synthesizes all of them, it can detect the occurrence of ﬁre. Each vehicle is connected to the information server at least via the cellular network. Our design makes each vehicle periodically report messages consisting of the current location, the current time, and just the alarm level events. Hence, the information server can keep track of every vehicle. Entering the WLAN range, each vehicle tries to send its message according to the MAC and its priority using the abundant WLAN bandwidth. The message covers the current status record and sometimes the previous one, which couldn’t be transmitted to the server due to network disconnection and are still kept in the vehicle’s buﬀer. The information server performs the preliminary analysis using available data to identify the vehicle that must be further investigated. In response to the periodic report message from the vehicle of interest, the information server notiﬁes the priority level of the vehicle’s messages. In the cellular network, the server cannot initiate the message transmission, but can just respond to the report message from the vehicle. 2.2

Locating the Vehicles

If the server catches an alarm or suspicious data item from vehicle V , it must conduct secondary analysis. The server ﬁrst searches the vehicles which have been near V during the interval embracing the event. Most spatial databases provide such spatio-temporal query processing. Sometimes, it is necessary to ﬁnd the vehicles on the same road segment during the speciﬁc time interval. In this case, the information server performs a trajectory-based search by the trajectory of V . For the matched trajectories, the time-stamps are compared to ﬁnally pick the vehicles to investigate. In this way, the vehicles of interest are selected and the information server raises their communication priority in the WLAN range, provided that the necessary record is not stored in the server database. Our previous work has developed the method capable of analyzing the similarity of two trajectories [3]. Each time the server initiates this data collection procedure, one thread is associated. It is necessary to give higher priority to the vehicle having the data many threads need. 2.3

WLAN Operation

Within a WLAN range, an AP (access point) collects messages from each vehicle. Basically, AP runs on the IEEE 802.11 MAC protocol. With a slight modiﬁcation, the cell can implement a prioritized access. AP can tune the frequency channel along the time axis as shown in Figure 2, namely, AP alternates Frequency 1 and Frequency 2 with the speciﬁc interval. Each vehicle, entering the range, ﬁrst listens to the beacon from the AP before its network activity. During CP-H (Contention

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Period-High), where AP tunes its channel to Frequency 1, only the vehicle having the high priority can transmit its message at Frequency 1. Remind that the priority is notiﬁed to each vehicle through the report response message via the cellular network. In the CP-L (Contention Period-Low) interval, where AP operates on Frequency 2, any vehicle can access the network. Other vehicles not obeying this access scheme cannot receive ACK from the AP during CP-H.

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Currently, we are developing the simulation environment for the proposed vehicle sensor network, based on the location history data collected from the Jeju taxi telematics system, which keeps track of each taxi and stores the location report for an eﬃcient dispatch and various analyses [4]. Jeju area, nominated as the telematics model city by the Korean national government, hosts a variety of vehicular network projects to design, develop, test, and deploy a new challenging vehicular telematics service. Many vehicles, especially taxis and rent-a-cars have installed the telematics devices to which many sensors can be attached [5]. In addition, our research team is investigating the characteristics of vehicle and eco sensors installable in the vehicle, focusing on their spatio-temporal stream behavior as well as considering the environmental need for the location-speciﬁc monitoring target.

References 1. Wang, Z., Kulik, L., Ramamohanarao, K.: Proactive traﬃc merging strategies for sensor-enabled cars (2007) 2. Liang, B., Drew, S., Wang, D.: Performance of multiuser network-aware prefetching in heterogeneous wireless systems. Wireless Networks 15, 99–110 (2009) 3. Won, J., Kim, S., Baek, L., Lee, J.: Trajectory clustering in road network environment. In: Proc. IEEE Symposium on Computational Intelligence and Data Mining, pp. 299–305 (2009) 4. Lee, J., Park, G., Kim, H., Yang, Y., Kim, P.: A telematics service system based on the Linux cluster. In: Shi, Y., van Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds.) ICCS 2007. LNCS, vol. 4490, pp. 660–667. Springer, Heidelberg (2007) 5. Wang, Y., Zhao, Q., Annavaram, M.: Markov-Optimal Sensing Policy for User State Estimation in Mobile Devices. In: The 9th International Conference on Information Processing in Sensor Networks (2010)

Energy Efficient Water Filling Ultra Wideband Waveform Shaping Based on Radius Basis Function Neural Networks Weixia Zou1, Bin Li1, Zheng Zhou1, and Shubin Wang1,2 1

Wireless Network Lab, Beijing University of Posts and Telecommunications Key Lab of Universal Wireless Communications, MOE 2 Inner Mongolia University Inner Box.96, BUPT, Beijing 100876, China [email protected] U

UHT

Abstract. In the emerging energy efficient framework, power allocation for ultra wide band (UWB) is much significant given its extremely large bandwidth. For multi-band UWB, this area has been extensively researched in the context of OFDM resources allocation. For pulse-based UWB, however, there is still an urgent need for efficient waveform design technique to embody arbitrary power allocation strategy. In this paper, we present a UWB waveform design method based on the radius basis network neural networks (RBF). The power density spectrum of emitted waveform is firstly abstract to a general mathematic function. Then based on the interpolation theory, RBF network is adopted to generate UWB waveforms given any spectrum shape. Numerical simulations validate our algorithms through the water filling (WF) waveforms shaping. Keywords: Energy efficient; water filling; UWB; radius basis network (RBF).

1 Introduction As large-scale distributed systems, such as Grids, Clouds and P2P networks, gather and share more and more computing nodes and storage resources, their energy consumption is exponentially increasing. Much effort is nowadays put into hardware specific solutions as well as protocol design, aiming to making contributions to the currently advocated energy efficient communications as well as the low carbon campaign [1]. Ultra wideband impulse radio (UWB-IR) is one promising technique in short-range high data rate communication scenes, such as wireless personal area network (WPAN) [2]. Meanwhile, UWB-IR has also been employed in military applications owing to its exceptional multipath resolution and penetration capability [3]. These advantages of UWB-IR are mainly attributed to the enormous bandwidth of its transmitted pulses, which may even occupy several gigahertzes (GHz). However, UWB has also been being confronted with the rigorous restrictions of applications from radio regulatory bodies because of its potential interference to other existing vulnerable wireless systems [4]. The first UWB emission mask was set out by U.S. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 299–306, 2010. © Springer-Verlag Berlin Heidelberg 2010

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Federal Communications Commission (FCC) in 2002, accompanying the authorizing of its unlicensed use in 3.1–10.6GHz [5]. Since then, UWB technology has attracted extensive interest for its wide applications. Due to the unique intermittent emission manner in which the information bit is carried by the short duration pulses, UWB may conduct data transmission at a extremely low power (<-41.3dBm). This is greatly meaningful to the energy efficient communication techniques. When it comes to the realistic applications, there exist interferences with different power from the neighborhood non-cooperative networks that may also occupy in different frequency. According to the classical information theory, the power allocation should be performed to obtain optimal transmission performance given the total UWB emission power. To multi-band OFDM system, this problem has been attached extensive investigations [6]. To impulse radio UWB systems, the WF waveform designing can be employed as is presented in [7], which is based on Hermite-Gaussian functions (HGF). Nevertheless, to the best knowledge, it is still greatly difficult to design an ideal hardware adaptation that produces UWB waveforms completely complying with the optimal power allocation, and simultaneously matching arbitrary regulated emission mask. In this paper, we develop a novel water-filling waveform generator for UWB impulse radio in the context of energy efficient communications. We firstly define the optimal design target as the power density spectrum UWB waveform which has taken the regulatory emission limitation and the in-band interference power into considerations. Then, we abstract this waveform designing as a classical twodimensional mathematic function approximation problem. By resorting to RBF network, UWB waveforms with arbitrary spectrum shape can be adaptively generated. After the short updating process, the obtained WF waveform can entirely match the target spectrum. Therefore, our proposed waveform shaper can essentially make the energy efficient UWB-IR networks possible. The reminder of this paper is structured as follows. Sections 2 gives the problem formulation in energy efficient UWB-IR waveform designing. Section 3 is devoted to establish a novel design algorithm based on RBF neural network. Numerical simulations and performance evaluations are conducted in section 4. We finally draw the conclusion in section 5.

2 Waveform Shaping for UWB Motivated by the trend of high data rates transmissions, the appealing techniques such as OFDM and direct-sequence spread spectrum (DSSS) are generally adopted in the wireless services, such as in LTE, 802.11n, WiMAX. As a result, to UWB pulse systems occupying fB=[0 12.5]GHz, the interference power I(f) from these narrowband systems can be reasonably viewed as a band-limited Guassian noise. We assume the total in-band interference power is σ2. B

B

P

∫

fB

I ( f )df = σ 2

P

(1)

On the other hand, restricted by the regulatory emission mask, the total power of UWB waveforms during the authorized band is also limited by P.

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∫

fB

S 2 ( f )df ≤ P, S ( f ) ≤ M ( f )

301

(2)

where M(f) denotes the UWB emission mask regulated by the radio management department, and P=∫M(f)df. According to the classical Shannon theory [8], the total UWB channel capacity can be written as:

⎤ df CUWB = max ∫ log ⎡1 + G( f ) I ( f )⎦⎥ G ( f ) fB ⎣⎢

(3)

S2( f ) ≤ M ( f )

(4)

under the constraint:

Then by resorting to the well known Lagrange multiplier technique [8], we have: S( f ) = M ( f ) − I ( f )

(5)

Up to now, we have developed the optimal spectrum of the transmitted UWB waveforms given the total emission power and the nearby networks interference. Correspondingly, our objective is to produce the UWB pulse fully matching the target spectrum S(f). At the same time, the designed spectrum should keep below M(f) for coexistence purposes.

3 Waveform Shaping for UWB In fact, we may abstracts the UWB pulse designing problem to a classic function approximation problem in mathematics and then utilize the radial basis function neural networks (RBF) to get a novel pulse designing scheme. This scheme has obtained the perfect spectrum utilization efficiency over other designed pulse in [9], and less hardware implementation complexity. More importantly, without any modifications on hardware structure, the implementation algorithm based on RBF can adaptively adjust the out spectrum, dynamically operating as the optimal energy efficient fashion according to the external interference power. Consider that the UWB pulse generator have to alter its output a according to the current interference information, which exactly corresponds to the spectrum of the UWB-IR pulse, the neural network would adopt the supervised learning algorithm [10]. The target signal t is the optimum objective spectrum determined by (5), in which the interference power can be estimated using spectrum estimation techniques [11]. Then, by adjusting the network weight w according to specific learning rules, the actual output a can trace the target signal t. Each part of the neural network is elaborated as follows: 3.1 Network Target

From the energy efficiency aspect, the UWB transmitted pulse is supposed to make full use of the limited spectral energy to maximize communications capacity. Hence,

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the design target of the neural network output can be directly chosen to be the representative samples of this optimal UWB spectrum in (5). Accordingly, the target output t of the neural network can be given by:

t = S( f )

k = 0,1," N -1

f = kf s

(6)

where fs represents the sample interval, and N is the length of the target vector t. 3.2 Network Input

The input matrix P of the neural net is given by: PN ×( n2 +1) = [p0 p1 p2 "pn2 −1 pn2 ]

(7)

where pi represents a column vector with N elements, which can be easily constructed from the basis function p(n). This basis function p(n) is supposed to meet the following two challenges: 1) The basis function is supposed to be even symmetry around its center; 2) The energy of the basis function must be finite. Base on above restrictions, the most familiar candidates for the basis waveforms is the Guassian function [9]. p(k ) =

⎛ f2 ⎞ 1 exp ⎜ − s 2 k 2 ⎟ , 2πδ ⎝ 2δ ⎠

k = 0,1,"N -1

(8)

Then, each column of the input matrix P can be obtained from: ⎧⎪p 0 = 1N ×1 ⎨ ⎪⎩p i = p ( k − i ⎢⎣ N n2 ⎥⎦ )

(

)

N

k = 1, 2," N − 1, i ≠ 0

(9)

where 1N×1 denotes a column vector whose components are all 1; x((k-i))N represents the cyclic shift process on the sample sequence x(n) with a length of N [12], and ⎣N/n2⎦ is the shift factor. It is noted that p0 represents a constant signal which corresponds to the bias of the neural network w0. The parameter δ is employed to modify the shape of the basis function p(n), whose feasible solution can be obtained independent of the network weight vector w for the purpose of the uncomplicated implementation of neural network [9]. 3.3 Learning Algorithm

It is obvious, from (10), that pi can be viewed as a shift basis waveform with its center located at different frequency bands. If an appropriate weight vector w has been obtained, then the weight sum of these shift waveforms may totally match the spectral masks. Hence, the network output a, which also represents the spectrum of the UWB pulse, can be written as: a = purelin(w T P) where purelin(.) is the transform function of the neural network [10].

(10)

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The mean square error (MSE) between the target vector t and the actual vector a can be defined as: E = a−t

2

N −1 ⎛ n2 ⎞ = ∑ ⎜ ∑ w (i ) p i ( k ) − t ( k ) ⎟ k =0 ⎝ i=0 ⎠

2

(11)

In order to minimize the MSE between the actual output a and the target output t, the network weight vector w should be adaptively updated in the learning stage. In view of the simple realization of the UWB-IR pulse generator, Windrow-Hoff algorithm can be adopted in the iteratively updating process of the network weight vector w [10]: w ( n ) = w ( n −1) + 2η Pe ( m −1)

(12)

where η is the learning step of the iterative algorithm; w (n) represents the weight vector of the nth iteration; P is the input matrix. e(n-1) is the error vector between the actual output a and the target output t after (n-1)th iteration. 3.4 UWB Pulse Generation The residual error between the output a, and the target output t was minimized after the network weight vector w had converged to its optimum value wopt. Considering that the output a is exactly the representation of the UWB-IR pulse in frequency domain, and the UWB waveform s(n) can be easily derived from a: s ( n ) = IDFT (aa b)

(13)

where x denotes the even component constructed from x. If the cyclic shift property and the odd-even property of DFT is applied [12], then the UWB-IR pulse can be written as: ⎧1 str ( n) = Re ⎨ ⎩N

n2

⎛

∑ w(i) p (n) exp ⎜⎝ − j '

i =0

2π ⎞⎫ nil ⎟ ⎬ N ⎠⎭

(14)

where l represents the shift factor which is equivalent to ⎣N/n2⎦; p’(n) is the IDFT of the basis function p(k), which can written as: p ' (n) = IDFT [ p (k )]

(15)

3.5 Implementation TheT Tstructure of the suggest UWB waveform generator is illustrated in Fig.1. Firstly, an impulse sequence with a period of N samples is generated at the sample frequency of 25GHz. The basis waveform p(k) is formed after the impulse signal has passed a Guassian shape filter with the specific BT which corresponds to the beforehand determined δ. In consideration of the periodicity of the p(k), pi can be easily obtained after the i×l samples delay has been performed on p(k). The obtained input matrix P is composed of n2+1 shift basis sequence pi(i=0,1,2,…,n2). Then the

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network weight vector w is updated according to (12) during the learning step. After the network weight has achieved its convergence wopt, and the learning process will be terminated. Finally, the UWB-IR transmit pulse is derived from the network output a [9]. The target signal t may be dynamically determined by the estimated interference power I(f).

Fig. 1. Implementation structure of the presented UWB waveform shaping network. Notice that the module “Environment Cognitive” is used to get the interference power distribution I(f).

4 Numerical Simulations and Evaluations In this part, we evaluate the performance of our design UWB waveform from both aspects of the algorithm convergence and the optimal WF spectrum matching errors. In our simulation, the length of the basis waveform N is 256; the shift factor l is 8, and the corresponding number of the shift sequence is 64, including the network bias p0. The UWB emission mask regulated by FCC is adopted in our analysis [5]. We also assume the interference systems mainly appear at 3.1-10.6GHz (e.g., the interference from 0-1GHz is basically ignored). From Figure 2-(a), it can be clearly seen that, the convergence of our algorithm updating can be basically achieved after 50 iterations. Therefore, this short switching time of our optimal waveform generation is applicable when the interference power I(f) changes dynamically. Figure 2-(b) plot the matching errors of our designed UWB waveforms given the optimal water-filling emission spectrum. It can be obviously noted that the spectrum mismatching is rather remarkable during the abrupt edges of target spectrum, while the complete matching can be observed during the plat spectral range. We find that the maximum mismatching happens near 0.9GHz which rightly corresponds to the sharpest changes in FCC mask. We also note that our designed UWB unfavorably surpasses the emission mask in this range. Figure 3(a) and (b) show the designed WF UWB signal both in the frequency domain and the time domain. From the spectral matching aspect, our designed UWB waveform can indeed efficiently embody the optimal power allocation under arbitrary B

B

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interference power distribution. The numerical derived spectrum efficiency is about 97.6%, which is much superior to the HGF based method in which the achieved spectrum efficiency is only about 65%. So, the optimal channel capacity can be achieved given the total emission power. As a result, energy efficient communications can be further realized in UWB-IR system by our presented waveform design algorithm. -3

7

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Fig. 2. (a) Convergences performance. (b) Mismatching errors of the designed WF spectrum. 1

WF UWB pulse FCC UWB pulse

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FCC Emission Mask Expected Optimal WF PSD Proposed WF UWB Proposed FCC UWB 0

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8

-0.4 10

12

1.5

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2.5 Time /sec

3

3.5 x 10

-9

(b)

Fig. 3. The proposed UWB waveform. (a) Frequency domain. (b) Time domain.

5 Conclusions We address the UWB waveform designing problem in the context of energy efficient communication networks. Given the total emission power of UWB and the channel interference power, we derive the optimal transmitted water-filling spectrum to maximize the channel capacity. Based on the formulation of mathematics function approximation, UWB waveform design under arbitrary spectrum shape is realized by RBF networks. The designed UWB pulse can fully match the expected WF spectrum after a short switching time. So, our presented method makes the UWB-IR based energy efficient networks into practice.

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Acknowledgments. This work was supported by NSFC (60772021, 60972079, 60902046), the Research Fund for the Doctoral Program of Higher Education (20060013008, 20070013029) and the National High-tech Research and Development Program (863 Program) (2009AA01Z262) and Important National Science & Technology Specific Projects(2009ZX03006-006/-009). This research was also partly supported by the Ministry of Knowledge Economy, Korea, under the ITRC support program supervised by the Institute for Information Technology Advancement (IITA2009-C1090-0902-0019). T

References 1. Younisand, O., Fahmy, S.: Distributed clustering in ad-hoc sensor networks: A hybrid, energy efficient approach. In: Proceedings of the Twenty-Third Annual Joint Conference of the IEEE Computer and Communications Societies, INFOCOM (2004) 2. Siep, T., et al.: Paving the way for personal area network standards: An overview of the IEEE P802.15 working group for wireless personal area networks. IEEE Pers. Commun. 7, 37–43 (2000) 3. Withington, P., Fluhler, H., Nag, S.: Enhancing homeland security with advanced UWB sensors. IEEE Microwave Magazine (September 2003) 4. Hämäläinen,, M., et al.: On the UWB System Coexistence with GSM900, UMTS/WCDMA, and GPS. IEEE Journal on Selected areas in Communication 20, 1712–1721 (2002) 5. Federal Communications Commission (FCC). Revision of part 15 of the commission’s rules regarding ultra-wideband transmission systems. First Report and Order (Released: April 22, 2002) 6. Xu, Z., Liu, L.: Power allocation for multi-band OFDM UWB communication networks. In: 2004 IEEE 60th Vehicular Technology Conference, VTC 2004-Fall, vol. 1, pp. 368–372 (2004) 7. Yang, R.: Hermite Gaussian Orthogonal Functions, Orthogonal Basis: Their Applications in Ultra Wideband and Ultra Narrowband Cognitive Radio. Ph.D. dissertation, Beijing University of Posts and Telecommunications (June 2007) 8. Proakis, J.: Digital Communications, 4th edn. McGraw-Hill, New York (February 2001) 9. Li, B., Zhou, Z., Zou, W.: A novel spectrum adaptive UWB Pulse: Application in cognitive radios. In: IEEE VTC 2009-fall, Anchorage, USA, pp. 1–5 (2009) 10. Hagan, M.T., Demuth, H.B., Beale, M.H.: Neural Networks Design. PWS Pub. Co. (1995) 11. Kay, S.M.: Modern Spectral Estimation: Theory and Application. Prentice Hall, Englewood Cliffs (January 1988) 12. Oppenheim, A.V., et al.: Discrete-Time Signal Processing, 2nd edn. Prentice Hall, Englewood Cliffs (1999)

An Introduction to Bayesian Techniques for Sensor Networks Bin Liu Department of Electrical and Computer Engineering, Duke University Durham, NC, 27708, U.S. [email protected]

Abstract. The purpose of this paper is threefold. First, it brieﬂy introduces basic Bayesian techniques with emphasis on present applications in sensor networks. Second, it reviews modern Bayesian simulation methods, thereby providing an introduction to the main building blocks of the advanced Markov chain Monte Carlo and Sequential Monte Carlo methods. Lastly, it discusses new interesting research horizons. Keywords: Bayes procedures, Monte Carlo methods, Sensor networks.

1

Introduction

Dramatic advances in computational devices, such as MEMS IC design, have made possible the use of Bayesian techniques for a variety of sensor network applications. Bayesian statistics is one of the two main branches of statistical science (the other branch is commonly termed ’frequentist’ approach), which provides us a system for describing epistemological uncertainty using the language of probability. In principle, any problems involving uncertain beliefs in states of nature can be considered by Bayesian thoughts and handled by Bayesian techniques [1]. Since 1990s Bayesian methods have seen applications in really diverse areas, e.g., computer science (under the name of statistical machine learning) [2,3,4], biology (under the name of bio-informatics)[3], economics [5,6], astrophysics (under the name of astro-informatics) [7], and engineering problems, such as radar/sonar target tracking [8,9,10], signal processing in wireless communications [11], Robotics [12,13], etc. In this paper, I focus on describing theory and algorithms that I feel are the main building blocks of modern Bayesian method, which could be used to handle challenging sensor networks problems. I should emphasize Bayesian Monte Carlo simulation algorithms, which have been witnessed, in the last two decades, to be powerful alternatives for scientiﬁc computation [14]. The remainder of this paper is organized as follows. In Part 2, I present the basic idea of Bayesian modeling and inference, along with a brief introduction of their applications in sensor networks. Part 3 provides the introduction of Markov Chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC) algorithms. In Part 4, I describe some important research frontiers. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 307–313, 2010. c Springer-Verlag Berlin Heidelberg 2010

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Bayesian Inference

In this section I review the Bayesian inference theory and introduce some applications of it in sensor network problems. Bayesian method provides a rigorous basis for quantifying uncertainties in model parameters θ, through considering the posterior probability distribution: p(θ|y, M) =

p(θ|M)p(y|θ, M) dθp(θ|M)p(y|θ, M)

(1)

where y denotes the observations, M the hypothesis, i.e., the model under consideration, p(θ|M) the prior distribution of θ, and p(y|θ, M) is the likelihood function. Note that the prior is involved to do inference for θ, leading to a posterior distribution of θ, rather than a point estimate. The prior can come from previous experiments, theory, even intuition. Given the joint posterior of (θ1 , θ2 ) ∈ Θ1 × Θ2 , one may also interested in the posterior p(θ1 |y, M), which can be obtained via the following marginalization procedure p(θ1 |y, M)dθ2 . (2) p(θ1 |y, M) = Θ2

We now consider another problem, namely Bayesian model comparison. Let subscript i index the model components being involved. Then one can take into account each model component by computing its posterior probability as follows: p(Mi )L(Mi |y) p(Mi |y) = M j=1 p(Mj )L(Mj |y) where M is the total number of model components, and L(Mi |y) = dθi p(θi |Mi )p(y|θi , Mi )

(3)

(4)

is called the marginal likelihood (or normalizing constant). The Bayes factor of (Mi ) to (Mj ) is given by p(Mi |y) BFij = , (5) p(Mj |y) which is usually interpreted as ”the odds” provided by the data y for Mi versus Mj . To deal with the problem of accounting for model uncertainty, one can resort to a strategy termed Bayesian model averaging (BMA), which calculates the posterior distribution of θ given data y as follows p(θ|y) =

M

p(θ|y, Mi )p(Mi |y),

(6)

i=1

which is shown to be an average of the posterior distributions under each of the models considered, weighted by their posterior model probability. It’s reported

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that this BMA method provides better average predictive ability than using any single model, conditional on the set of all models being considered [15,16]. Now I summarize some sensor networks problems, which has seen Bayesian solutions based on the aforementioned inference framework. [17] addresses suppression and failures in sensor networks via Bayesian method, in which the knowledge of the suppression scheme and the application-level redundancy is incorporated by Bayesian inference. [18] utilizes a Bayesian method termed nonparametric belief propagation, for both estimating sensor locations and representing location uncertainties. [19] uses a Bayesian network to model the sensor network, proposing a methodology to answer queries with limited and stochastic information. In next section, I’ll focus on Bayesian simulation techniques, with a look at their current applications for sensor networks.

3

Bayesian Computation

For practical problems, it requires simulation techniques to handle highdimensional complex integrals involved in Equations 1-6, among which MCMC and SMC are the most commonly used methods in the literature. The Metropolis algorithm, an instance of MCMC, is recognized among the ten algorithms that have had the greatest inﬂuence on the development and practice of science and engineering in the 20th century [20]. The idea of Monte Carlo simulation is to draw an independent and identicallydistributed (i.i.d.) set of random draws {θn }N n=1 from a target distribution deﬁned on a high-dimensional space Θ, e.g. the space on which the posterior is deﬁned. These N draws can then be used to approximate the target distribution, denoted π(θ) here, with the following empirical point-mass function π(θ) =

N 1 δθn (θ), N n=1

(7)

where δθn (θ) denotes the delta-Dirac mass located at θn . Consequently, one can approximate the complex integrals by tractable sums, and this approximation converges to the real answer by the strong law of large numbers. See details in [14]. The success of Monte Carlo integration arises from and relies on the fact that it places the integration grid (samples) in regions of high probability. Note that, if π(θ) has standard form, e.g. Gaussian, it is straightforward to sample form it using easily available routines, otherwise, we need resort to more sophisticated techniques, such as MCMC and SMC, which I’ll present in the following. 3.1

MCMC

MCMC is a strategy for generating random draws θn with aid of a Markov chain, which is constructed to be irreducible, aperiodic and have the target distribution as the invariant distribution. One way to design these samplers is to ensure that detailed balance is satisﬁed, which means π(θn )T (θn−1 |θn ) = π(θn−1 )T (θn |θn−1 ),

(8)

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where T is the transition matrix that is associated with the Markov chain. Note that detailed balance is a suﬃcient, but not necessary, condition. The concepts of irreducibility, aperiodicity and invariance can be referred to [21]. Now I brieﬂy introduce the Metropolis-Hastings (MH) algorithm, which is the most popular and general MCMC method. Given an invariant distribution π(θ) and a proposal distribution q(θ |θ), an MH step consists of sampling a proposal value θ given the current value θ according to q(θ |θ). The Markov chain then moves towards θ with acceptance probability Pr(θ, θ ) = min{1, [π(θ)q(θ |θ)]−1 π(θ )q(θ|θ )}, otherwise it remains at θ. Although simple, the MH algorithm requires careful design of the proposal distribution q(θ |θ). Many MCMC algorithms arise by considering speciﬁc choices of this distribution, among which the Metropolis algorithm and independent sampler are two simple instances. The Metropolis algorithm assumes a symmetric random walk proposal q(θ |θn ) = q(θn |θ ) and, hence the acceptance probability is just Pr(θn , θ ) = min{1, [π(θn )]−1 π(θ )}. For the independent sampler, the proposal is chosen to be independent of the current state, which means q(θ |θn ) = q(θ ). Hence, the acceptance probability is Pr(θn , θ ) = min{1,

π(θ )q(θn ) }. π(θn )q(θ )

(9)

Note that, for the MH algorithm, the normalizing constant of the target distribution is not required. We only need to evaluate the target distribution up to a constant proportionally. However, the success of the algorithm often depends on the choice of proposal distribution. In general, it is possible to use adaptive algorithm to generate data-driven proposal distributions, such as the approaches proposed in [22,23]. 3.2

SMC

SMC methods, a.k.a. particle ﬁltering in engineering community, allow us to perform the simulation strategy to handle scenarios involving non-stationary target distribution, in which the target distribution is denoted πt (·) and changes along with time t (note that t can be an artiﬁcial time variable, such as that used in the simulated annealing optimization method [24]). Such SMC methods are very useful in on-line approximation of probability distributions using samples (particles). These time-variant target distributions often arise in cases involving real-time signal processing, where data arrives sequentially, or the temperature based optimization context, in which each t corresponds to a speciﬁed temperature value. Some properties of such SMC methods are worth highlighting. First, the sequential processing strategy to deal with non-stationarity in signal is appealing, as information from the recent past is given greater weighting than information from the distant past. Secondly, diﬀerently from MCMC, which is actually a batch-mode method, SMC doesn’t require storing all the data, hence it leads to much computation simplicity.

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Suppose that we have an initial distribution π0 = p(θ0 ), a dynamic model p(θt |θt−1 ) for t ≥ 1, and measurement model p(yt |θ0:t , y1:t−1 ) for t ≥ 1. Here θ0:t denotes the states up to time t. The basic idea behind SMC can be simply (n) (n) presented as follows. Starting with a weighted set of samples {θt−1 , ωt−1 }N n=1 , (ω denotes importance weight), which approximately distributed according to πt−1 (θ), it generates a new particle set, each particle component drawn from a n suitably chosen proposal distribution as follows θtn ∼ q(θt |θt−1 , yt ). To maintain consistence, it then updates each importance weight as follows ωtn ∝

n ωt−1 p(yt |xnt )p(xnt |xnt−1 ) , q(xt |xnt−1 , yt )

(10)

N n n n N where n=1 ωt = 1. The updated particle set {θt , ωt }n=1 is then used to approximate πt = p(θt |y1:t ). There are some practical issues to be considered for SMC. The ﬁrst one is the selection of q(θt |xnt−1 , yt ). It’s possible to select the dynamic prior as the proposal, i.e. q(θt |xnt−1 , yt ) = p(xt |xnt−1 ), then the updated importance weight follows to be n p(yt |xnt ). However, this proposal is far from an optimal one. Another ωtn ∝ ωt−1 disturbing issue for SMC is the so-called sample degeneracy problem, which means, after some iterations, all but one particles have zero importance weights, which will destroy the ensemble property. To get around of this, a resampling step is required, in which particles with bigger importance weights will be multiplied, while those with negligible importance weights will be discarded. This resampling procedure results in equal importance weights 1/N to each sample. See details of discussions on the above issues in [25]. And refer to [26] for presentation of a more general SMC framework. 3.3

Applications in Sensor Networks

[27] presents a mechanism of blending the SMC method with a sensor scheduling strategy to track a smart target. People have also adapted SMC methods to deal with distributed state estimation [28] and sensor fusion [29].

4

Discussions

This paper gives a very brief introduction to Beyesian techniques, along with a summarization of applications of these techniques in sensor network related areas. Only ideas and methods which I think are the most important or necessary for sensor networks are covered here, while interested readers are referred to a large amount of literature for other related issues, such as stochastic optimization method based on simulation strategy. The rapid-developing discipline of sensor networks is characterized by large number of sensor nodes, massive data sets, complex models and many and varied applications. The development of Bayesian statistics coincides the requirement of a coherent framework to handle such complex models and the massive data sets.

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As described above, a few solutions based on Bayesian techniques have been proposed recently to handle problems in sensor networks, but there is still great room for innovation in developing Bayesian methods for sensor networks. A promising direction is Bayesian statistical learning/estimation of locations of sensor nodes, for which a bunch of machine learning algorithms should be involved. It’s also possible to handle localization problems by Bayesian experimental design theory [30], which has been developed richly in statistical community.

References 1. Berger, J.O.: Statistical decision theory and Bayesian analysis. Springer, New York (1985) 2. Bishop, C.M.: Pattern recognition and machine learning. Springer, New York (2006) 3. Baldi, P., Brunak, S.: Bioinformatics: the machine learning approach. The MIT Press, Cambridge (2001) 4. Heckerman, D., Geiger, D., Chickering, D.M.: Learning Bayesian networks: The combination of knowledge and statistical data. Machine learning 20(3), 197–243 (1995) 5. Cyert, R.M., DeGroot, M.H.: Bayesian analysis and uncertainty in economic theory. Rowman & Littleﬁeld Pub Inc. (1987) 6. Kim, C.J., Nelson, C.R.: Has the US economy become more stable? A Bayesian approach based on a Markov-switching model of the business cycle. Review of Economics and Statistics 81(4), 608–616 (1999) 7. Borne, K., Accomazzi, A., Bloom, J., Brunner, R., et al.: Astroinformatics: A 21st Century Approach to Astronomy. In: AGB Stars and Related Phenomenastro2010: The Astronomy and Astrophysics Decadal Survey, vol. 2010, 6p. (2009) 8. Ristic, B., Arulampalam, S., Gordon, N.: Beyond the Kalman ﬁlter: Particle ﬁlters for tracking applications. Artech House Publishers, Norwood (2004) 9. Stone, L.D., Corwin, T.L., Barlow, C.A.: Bayesian multiple target tracking. Artech House Publishers, Norwood (1999) 10. Vermaak, J., Godsill, S.J., Perez, P.: Monte Carlo ﬁltering for multi-target tracking and data association. IEEE Transactions on Aerospace and Electronic systems 44(1), 309–332 (2005) 11. Wang, X., Chen, R., Liu, J.S.: Monte Carlo Bayesian signal processing for wireless communications. The Journal of VLSI Signal Processing 30(1), 89–105 (2002) 12. Dellaert, F., Fox, D., Burgard, W., Thrun, S.: Monte carlo localization for mobile robots. In: IEEE International Conference on Robotics and Automation, pp. 1322– 1328. IEEE Press, New York (1999) 13. Fox, D., Burgard, W., Dellaert, F., Thrun, S.: Monte carlo localization: Eﬃcient position estimation for mobile robots. In: Proceedings of the National Conference on Artiﬁcial Intelligence, pp. 343–349. John Wiley & Sons Ltd., Chichester (1999) 14. Liu, J.S.: Monte Carlo strategies in scientiﬁc computing. Springer, Heidelberg (2008) 15. Madigan, D., Raftery, A.E.: Model selection and accounting for model uncertainty in graphical models using Occam’s window. Journal of the American Statistical Association 89(428), 1535–1546 (1994) 16. Hoeting, J.A., Madigan, D., Raftery, A.E., Volinsky, C.T.: Bayesian model averaging: A tutorial. Statistical science 14(4), 382–401 (1999)

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17. Silberstein, A., Puggioni, G., Gelfand, A., Munagala, K., Yang, J.: Suppression and failures in sensor networks: A Bayesian approach. In: Proceedings of the 33rd international conference on Very large data bases, pp. 842–853 (2007) 18. Ihler, A.T., Fisher III, J.W., Moses, R.L., Willsky, A.S.: Nonparametric belief propagation for self-calibration in sensor networks. In: Proceedings of the 3rd international symposium on Information processing in sensor networks, pp. 225–233. ACM, New York (2004) 19. Biswas, R., Thrun, S., Guibas, L.J.: A probabilistic approach to inference with limited information in sensor networks. In: Proceedings of the 3rd international symposium on Information processing in sensor networks, pp. 269–276. ACM, New York (2004) 20. Beichl, I., Sullivan, F.: The metropolis algorithm. Computing in Science & Engineering 2(1), 65–69 (2000) 21. Gilks, W.R., Richardson, S., Spiegelhalter, D.J.: Markov chain Monte Carlo in practice. Chapman & Hall/CRC, Boca Raton (1996) 22. Keith, J.M., Kroese, D.P., Sofronov, G.Y.: Adaptive independence samplers. Statistics and Computing 18(4), 409–420 (2008) 23. Tierney, L., Mira, A.: Some adaptive Monte Carlo methods for Bayesian inference. Statistics in Medicine 18(1718), 2507–2515 (1999) 24. Kirkpatrick, S.: Optimization by simulated annealing: Quantitative studies. Journal of Statistical Physics 34(5), 975–986 (1984) 25. Arulampalam, M.S., Maskell, S., Gordon, N., Clapp, T., Sci, D., Organ, T., Adelaide, S.A.: A tutorial on particle ﬁlters for online nonlinear/non-GaussianBayesian tracking. IEEE Transactions on signal processing 50(2), 174–188 (2002) 26. Del Moral, P., Doucet, A., Jasra, A.: Sequential monte carlo samplers. Journal of the Royal Statistical Society: Series B(Statistical Methodology) 68(3), 411–436 (2006) 27. Liu, B., Ji, C., Zhang, Y., Hao, C.: Blending Sensor Scheduling Strategy with Particle Filter to Track a Smart Target. Wireless Sensor Network 1, 300–305 (2009) 28. Coates, M.: Distributed particle ﬁlters for sensor networks. In: Proceedings of the 3rd international symposium on Information processing in sensor networks. ACM, New York (2004) 29. Rosencrantz, M., Gordon, G., Thrun, S.: Decentralized sensor fusion with distributed particle ﬁlters. In: Proc. of UAI (2003) 30. Chaloner, K., Verdinelli, I.: Bayesian experimental design: A review. Statistical Science 10(3), 273–304 (1995)

Fuzzy C-Means Clustering Based Robust and Blind Noncoherent Receivers for Underwater Sensor Networks Bin Li1, Zheng Zhou1, Weixia Zou1, and Shubin Wang1,2 1

Wireless Network Lab, Beijing University of Posts and Telecommunications Key Lab of Universal Wireless Communications, MOE 2 Inner Mongolia University Inner Box.96, BUPT, Beijing 100876, China [email protected]

Abstract. Attributed to complex and dynamic propagations in underwater acoustic sensors network, the multipath signals has posed great challenges in the receiver designing in past decades of years. In this paper, we adopt the UWB channel model in underwater networks and suggest a blind non-coherent receiver. Some differentiated features have been developed to represent the multipath signals. Then, we formulate the underwater signal detection as a data mining process. Fuzzy c-means clustering (FCM) algorithm is finally adopted to perform blind signal detection. Numerical simulations show that our scheme outperforms the traditional noncoherent techniques. Keywords: Underwater sensors, Saleh-Valenzuela (SV) channel, Noncoherent receiver, Features extraction, Fuzzy C-means Clustering.

1 Introduction Underwater sensor networks are envisioned to enable applications for oceanographic data collection, pollution monitoring, offshore exploration, disaster prevention, assisted navigation and tactical surveillance applications [1]. To make these significant applications viable, there is an urgent need to realize underwater communications at a low complexity and low power architecture. However, the intensive multipath propagations in ocean environments have made the traditional simple receiver structures ineffectual. Usually, in order to improve the receiving performance in the presence of dense multipath, coherent receivers is suggested combined with channel estimations. Alternatively, the channel adaptive equalization should be adopted to combat the undesirable influence from the nonideal channel. In both cases, accurate channel state information (CSI) is required. For RAKE receivers, channel response should be accurately obtained. It is apparent that the estimation algorithms, aiming at extract the position and amplitude of each resolvable multipath components, is computational complex [13]. Even this CSI can be acquired, hardware realization that integrates dozens of correlator is still impractical. To the channel equalizations, a substantial part of power should be allocated to the training sequences to trace the time-variant G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 314–321, 2010. © Springer-Verlag Berlin Heidelberg 2010

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ocean channels [2,3]. So, this scheme considerably scarifies energy efficiency, and hence seriously shortens the longevity of underwater sensors. To overcome these challenges, Liang introduced collaborative conception to the target detection in underwater networks. This method achieves better performance under the intensive multipath environments [4]. However, collaborative schemes may be not applicable for certain highly dynamic applications. Although the noncoherent receivers, such as energy detection (ED), have been attached extensive investigations recently serving as a suboptimal alternative [5], it can be foreseen that ED may experience remarkable performance degradation as a noise-sensitive technique, given that an accurate estimation of noise variance is impractical. In this paper, we develop a novel noncoherent receiver for underwater sensors in the presence of intensive multipath propagations. Our scheme firstly establishes a characteristic representation of multipath ocean channels. Different features that fully reflect the received signals are extracted and form a feature hyperspace. This highdimension space is further mapped to 2-D plane, and the blind underwater signal detection is transformed to a data clustering problem. The fuzzy c-means clustering algorithm is adopted to perform this function. Our scheme is robust to channel noisy uncertainty and superior to other noncoherent method.

2 Underwater Channel and System 2.1 Underwater Channel Modeling The underwater acoustic channel exhibits phenomena such as signal fading and phase and amplitude fluctuation due to the interactions with the boundaries and the scattering from inhomogeneities within the ocean medium. Multipath interference is an important phenomenon in UWA networks, causing frequency selective fading in underwater channels. Early research in underwater channel modeling assumes Rayleigh fading in nature [6], whose limitation is lately observed in [7]. Geng and Zielinski [7] proposed that the underwater acoustic channel consists of several distinct paths (eigenpaths) over which a signal can propagate from transmitter to receiver [8].Based on the following considerations, Liang adopts a modified UWB SalehValenzuela (S-V) channel model to depict underwater acoustic channels [4]. Firstly, the frequency range of underwater communication is inferior to 10 kHz. Given the carrier frequency of 550 Hz in shallow water and 2 kHz in deep in short range transmission, the fractional bandwidth (fH-fL)/(( fH+fL)/2) of underwater system is much greater than 0.20-0.25, which can be reasonably referred to as UWB [9]. Beside, S-V multipath channel is based on the observation that usually multipath contributions generated by the same pulse arrive at the receiver grouped into clusters [10]. Additionally, similar to [9], two Poisson models are assumed in the modeling of the path arrivals in SV channel. In this paper we use the channel model defined in [11] by IEEE802.15.3a as the underwater channel during our following analysis. The time domain expression is given by

(

h(t ) = X ∑ l =0 ∑ m = 0 α m ,lδ t − Tl − τ m ,l L −1

M −1

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where h(t) is the channel impulse response, P is the number of clusters, M is the number of rays of each cluster, αm,l is the fading coefficient of the mth path of the lth cluster, X is the channel fading factor, Tl is the arrival time of the lth cluster and τm,l is the delay of the mth path of the lth cluster relative to Tl. Tl and τm,l has poisson distribution and αm,l and Xk are log-normal random variables. 2.2 Underwater Transmitter For elaboration simplicity, we employ the pulse amplitude modulation in this paper. Although the carrier modulation is always adopted in practice to best match channel property, our simplified transmitter can be still viewed as the baseband signal representation. Given the simplicity in implementations, on-off keying (OOK) is employed which enables the noncoherent receivers.

(

x(t ) = Eb N s ∑ i = 0 di / N w t − iT f P −1

s

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where Eb is the bit energy, NS is the number of pulses used to represent one bit d(i) (i=0,1,2 …,P-1) are the transmitted data of length P taking values of {0,1}. The demodulation of received underwater signal is to probe whether there is efficient power available in current time bin. Consequently, the detection problem can be formulated as the following two hypotheses. ⎧⎪ h ( n ) + w ( n ) y (n) = ⎨ ⎪⎩ w ( n )

H1 H0

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3 Non-coherent Receiver As a low-complexity and low-power method, energy detection has drawn great interest in recent literatures. However, ED is far from a suboptimal alterative considering no channel characteristic information is exploited. Moreover, its performance totally relies on the optimal decision threshold, which is closed related with the channel noise variance. Starting from the noisy received waveform y(n), our proposed method deals with the underwater demodulation as a two-groups data clustering in following four steps. 1) Construct the features; 2) Extract the features; 3) Combine the features and 4) detect the feature-based underwater signals. 3.1 Feature Construction

To simplify the analysis, in our demodulator, we assume that: 1) the synchronization has been accurately acquired; 2) Sampling frequency is set to the Nyquist rate, that is, we sample only once in a resolved multipath. In this case, an observation interval consists of N samples, denoted by a vector y(n), n=0,1,2,…,N-1. We may construct an autorelation matrix A according to:

A = yT y

(4)

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In order to further enhance the difference between received multipath signal and channel noise, we perform unitary transformation on A:

B = AT A

(5)

We denote the principal diagonal elements of B with a, while the elements immediately below this diagonal with b. Also, b can be viewed as the diagonal elements of a dimension-decreased matrix, which is the cofactor of B(N,N).

diag(B), b1u˄n1˅ diag(Bp )

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Now, the correlation between a and b can be obtained which comprehensively reflects the characteristic of underwater channels.

c = a ⊗ b2

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where ⊗ denotes the correlation operation. Before further proceeding, we also smooth c(k) through a moving average filter which can be realized by a matrix H.

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From Fig.1, it can be observed some distinguishing features exist in the two channel states “1” and “0”. So, we define d as the characteristic spectrum. Certain quantifiable features may be extracted according to the following process. Firstly, we find the equivalent energy which corresponds to the area below d, is quite concentrated at the centre of the waveform when the underwater signal has been received. Otherwise, the equivalent energy is relatively dispersed. So, we define the first feature as energy concentration

F1 = ∑ k =n −1− K c(k ) n −1+ K

∑

2 n −1 k =1

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where the interested range is limited by K. In this paper, K can be determined by the imbalance property of d(k). That means K can be obtained once the left value d(n1+K) surpassed the right value d(n-1-K) by δ.

K = minarg{c(n -1+ k) − c(n -1− k ) > δ }

(10)

k

Secondly, it should also be noted that the slop of d(k) in Fig.1 (a) is much more obvious than in Fig.1 (b) in the middle range, and consequently, variance in this range is supposed to be much distinctive. So, we define the second feature as the middle section variance:

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F2 = ∑k =n−1−K c(k ) −1 2K −1× ∑k =n−1−K c(k ) n−1+ K

n−1+ K

)

2

(11)

Thirdly, the energy presents remarkable imbalance during the outside range. Therefore, we may reasonably employ the energy imbalance as the third feature: F3 = ∑ k = n −1+ K c ( k ) − ∑ k =1 c ( k ) 2 n −1

n −1− K

(12)

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At last, the total received energy can also differentiate the UWB multipath signals from the channel noise: F4 = ∑ k =1 y 2 ( k ) 2 n −1

(13)

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3.3 Feature Combination

We have constituted a full feature set that is dedicated to separate two channel states. The demodulation of a received signal can be formulated to determine a separating hyper-plane dividing the two group points in this multidimensional features space. However, high-dimensional problem is either computational complex or has an intolerable convergence. So, we further reduce the feature dimension to a 2-D plane to simplify complexity. v1 = F2 , v2 = η1 F1 + η 2 F4 + η3 F3

(14)

where the combining coefficients ηi is chosen according to equal combing (EC) rule. 3.4 Feature-Based Blind Signal Detection

Based on 3.3, we note that detection process is equivalent to a data clustering problem given a 2-D feature plane. So, we may blindly divide the received data into 2 groups (clusters) such that similar data objects belong to the same cluster and dissimilar data objects to different clusters using fuzzy c-means clustering algorithm. Differentiated from traditional clustering algorithms, membership degrees between zero and one are used in fuzzy clustering instead of crisp assignments of the data during cluster [12]. A feature vector [v1 v2] belongs to one cluster to some degree that is specified by a membership matrix U. The FCM analyzes the feature data and produces outputs: (1) the membership matrix U, (2) the cluster centers {center_i, i=1,2}. Four steps in the FCM function are depicted below. Step 1: Initialize the membership matrix U with random values between 0 and 1. The size of U is K×P, K and P are both equal to 2 in our analysis. The element of membership matrix U, uij, is the probability that the jth feature vector belongs to the

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ith cluster (1≤i≤K and 0≤j≤P−1). Notice that elements of U must satisfy the constraint below: K ∑i=1uij = 1, 0 ≤ j ≤ P -1 (15) Step 2: Calculate fuzzy cluster centers for all K clusters according to Center _ i = ∑ j =1 v j uij P−1

∑

P−1 j =1

uij , 0 ≤ i ≤ K

(16)

where vj is the feature vector derived from spectrum harmonics coefficients [12]. Step 3: Compute the cost function defined below:ï J (U , center _1, center _ 2 ) = ∑ i =1 ∑ j =1 uij m Dij 2

(17)

Dij = center _ i − v j

(18)

K

P −1

ïï ï ï

where (18) is the Euclidean distance between the ith cluster center and the jth feature vector. The iteration terminates if improvement of J over previous iteration is below a pre-specified threshold. Otherwise, the algorithm proceeds with next step.ï Step 4: Compute a new membership matrix U whose element ï ïï is adjusted by

(

uij = ∑ k =1 Dij Dkj K

)

2

1− m

1 ≤ i ≤ K , 0 ≤ j ≤ P -1

(19)

The above iteration process from Step 2 to Step 4 continues on the received data set, until either the predefined maximum iterations are reached or the objective function converges.ï

4 Numerical Simulations and Evaluations In this part, we evaluate the receiving performance of this presented algorithm through numerical simulations. In our simulations, 802.15.3a channel modeling is employed as underwater channels. The underwater sensors adopt OOK modulations. The sampling frequency is sent to Nyquist rate, and the interesting observation window focus on the front multipath components that contains 95% energy. In Fig.2 (a), we show the convergence of the objective function J. In this simulation, we set the channel SNR to 12.5dB, the maximum iterations is 150 and the stop criterion on the relative minimum improvement of J is 10-5. It can be observed that the clustering algorithm reach its convergence after 12 iterations. Therefore, our algorithm is computationally efficient. The 2-D feature plane is also obtained. We find that our scheme can essentially cluster the received signal into the right groups in a fully blind fashion. The error clustering number is only 2 and the corresponding bit error rate is about 1.3×10-5. By contrast, even the noise variance has been accurately estimated, the BER of ED is still 4.7×10-3. The BER curves derived from numerical simulations on 150000 symbols have been plotted in Fig.4. When the channel noise is accurately estimated, our algorithm obtains a processing gain about 1.4dB compared to ED. However, if noise uncertainty is considered, additional gain about 4-5dB can be achieved when employing our

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clustering-based noncoherent demodulator. So, except for the virtue of lowcomplexity, our method also is robust to the noise uncertainty, which is significant to underwater communications.

5 Conclusions We address the low-complexity receiver designing for underwater sensor networks in this paper. Our presented algorithm contains two phases that is the features extraction

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and data clustering. Multiple features fully exploring the intrinsic character of underwater channel are firstly obtained. Then, fuzzy c-means cluster algorithm is adopted to perform blind signal detection. Simulations further validate the advantages of our algorithm to other traditional noncoherent methods. Our future investigations include the further improvements on clustering algorithms as well as other realistic scenarios in the presence of Doppler shifting and narrow-band interferences. Acknowledgments. This work was supported by NSFC (60772021, 60972079, 60902046), the Research Fund for the Doctoral Program of Higher Education (20060013008, 20070013029) and the National High-tech Research and Development Program (863 Program) (2009AA01Z262) and Important National Science & Technology Specific Projects(2009ZX03006-006/-009). This research was also partly supported by the Ministry of Knowledge Economy, Korea, under the ITRC support program supervised by the Institute for Information Technology Advancement (IITA2009-C1090-0902-0019).

References 1. Akyildiz, I.F., Pompili, D., Melodia, T.: Underwater acoustic sensor networks: research challenges. Elsevier’s Journal of Ad Hoc Networks 3(3), 257–279 (2005) 2. Stojanovic, M., Catipovic, J., Proakis, J.: Adaptive multichannel combining and equalization for underwater acoustic communications. J. Acoust. Society Amer. 94(3), 1621–1631 (1993) 3. Stojanovic, M.: Efficient processing of acoustic signals for high-rate information transmission over sparse underwater channels. Elsevier’s Journal of Physical Communication 1, 146–161 (2008) 4. Liang, Q., Cheng, X.: Underwater acoustic sensor networks: Target size detection and performance analysis. Elsevier’s Journal of Ad Hoc Networks 7, 803–808 (2009) 5. Witrisal, K., Leus, G., Janssen, G.J.M., et al.: Noncoherent Ultra-wideband systems. IEEE Signal Processing Magazine, 48–66 (July 2009) 6. Sandsmark, G.H., Solstad, A.: Adaptive beam forming and adaptive equalization for high-speed underwater acoustic data transmission. In: Proceedings of Underwater Defence Tech. Conference, pp. 707–712 (1991) 7. Geng, X., Zielinski, A.: An eigenpath underwater acoustic communication channel model. In: Proceedings OCEANS 1995. MTS / IEEE. Challenges of Our Changing Global Environment. Conference, vol. 2, pp. 1189–1196 (1995) 8. Jobst, W., Zabalgogeazcoa, X.: Coherence estimates for signals propagating through acoustic channels with multiple paths. Journal of the Acoustic Society of America 65(3), 622–630 9. Win, M.Z., Scholtz, R.A.: Impulse radio: How it works. IEEE Commun. Lett. 2(2), 36–38 (1998) 10. Saleh, A.A., Valenzuela, R.A.: A statistical model for indoor multipath propagation. IEEE Journal on Selected Areas in Communications 5(2), 128–137 (1987) 11. IEEE 802.15.SG3a, Channel modeling sub-committee report final, IEEE P802.15-02/490r1SG3a (2003) 12. Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification, 2nd edn. John Wiley & Sons, Inc., Chichester (2001) 13. Proakis, J.G.: Digital Communications, 4th edn. McGrawHill, New York (2001)

Research on Enhanced Spectrum Efficiency for BWA Networks Xu-hui Wang1 and Cheng-lin Zhao2 1 School of Economics and Management, BUPT, Beijing, China Key Laboratory of Universal Wireless Communication, Ministry of Education, China Wireless Network Lab, Beijing University of Posts and Telecommunications, 100876, China [email protected] 2

Abstract. Relay is considered as a method to ensure capacity and coverage in the broadband wireless access (BWA) system. This paper proposes a type of MAC layer simulator architecture for the relay enhanced BWA networks, and then addresses the issue on the network topology when relay is introduced. Furthermore, downlink co-operative communication in such relay scenario is discussed on. The results show that the proposed relay enhanced BWA system outperforms the single-hop one on spectrum efficiency. Keywords: spectrum efficiency, BWA networks, relay.

1 Introduction Broadband wireless access (BWA) technologies promise to provide wireless data transportation of high quality for various services [1][2][3]. For the radio frequency scarcity, the radio frequency band allocated to BWA applications is usually above 2GHz [4], on which the radio signals experience serious path loss and channel fading. To compensate for these negative effects, engineers have to increase the transmit power level or set more base stations (BSs) in the service region, which leads to higher infrastructure deployment costs and larger inter-cell interferences. To solve these problems, the relay station (RS) has been introduced in BWA networks [5][6][7]. The norm relay traditionally appears in ad hoc networks, where the communications between two nodes via other nodes which are called relay nodes. Relay systems can be classified into two systems, the amplify-and-forward (A&F) and the decode-and-forward (D&F). Relay node based on A&F acting as an analog repeater, has limited the effect on the improvement of system throughput, yet it complicates the interference mitigation scheme and network planning for its increasing system noise level. The relay system based on D&F has been enhanced in radio resource management functions, and can probably utilize the spatial diversity scheme for improvement of network capacity and coverage. More and more research effects have focused on the D&F relay. The concept relay is transferred into BWA networks, in which RS is an independent communication entity besides BS and mobile subscriber station (MS). In BWA systems, the relay node RS is under BS’s supervision, requiring changes in both the G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 322–329, 2010. © Springer-Verlag Berlin Heidelberg 2010

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protocols and network platforms, not just the simple accession to conventional singlehop wireless access system structure. IEEE 802.16 family of standards is a typical series of BWA protocols, which are considered as an alternative to 3G and migration towards 4G [8]. Currently one of their work groups is taking effort to draft an updating version based on existent IEEE 802.16d/e standard. They are aiming at introducing relay into IEEE 802.16 networks to practice more economic deployment and more flexible radio resource management. Some related works are as follows. [9] is concerted mainly on the architecture of IEEE 802.16e simulator, focusing on MAC and PHY layer functions, but certain network infrastructure is little mentioned. Yet the network architecture is an important element to the performance evaluation of BWA system. [4] outlines the concepts of relay, addressing several issues on deployment for BWA system, such as the cooperative use of relays and the system scenario. But it fails to illustrate a specified example based on certain BWA standard, for example, IEEE 802.16e. [6] presents concepts to integrate multi-hop relay into the IEEE 802.16 system with proposed scheduling scheme. Certain resource allocation methods are discussed based on OFDM MAC frame structure. In [10][11], when RS is introduced, some efforts are taken to combine different nodes to transport data for the same MS, which is called co-operative communication. In this paper we consider a relay enhanced architecture based on IEEE 802.16e system by revising the OFDMA TDD frame structure to meet the two-hop relay requirements. Based on it, we propose a new BS-RS cell model as the basic unit to organize co-operative communication in the relay enhanced BWA networks. Then the comparative results with non-relay BWA systems are put forward and analyzed.

2 Relay Terminology Unlike the former relay concepts in ad hoc or mesh networks, relay has several special meanings for BWA system. RS helps BS and MS to transmit their messages and data if they have no good direct radio link. As the radio propagation characteristics at high radio frequencies, the propagation loss is serious in the environment of NLOS, such as the metropolitan environment. In such scenario, it has to significantly increase the density of BSs, while the solution leads to extensive cost for network planning. There are several special issues addressed for IEEE 802.16 relay application. First, as MS in BWA system connects to the fixed BS associating with backbone networks, the introduction of RS requires the new proposed RS to act not only as a simple bidirectional repeater, but has to involve more management functions, such as relay path selection and radio resource management. Also, RS is defined in IEEE 802.16 networks just on the wireless access aspects, not a whole solution to the entire network infrastructure. And then, the aim for relay enhanced BWA system is to provide subscribers with ubiquitous wireless access services which is application-oriented development, compared with ad hoc and mesh technology which are mainly for organizing networks without fixed infrastructure support. The last one is that relay in IEEE 802.16 supports the point-to- multipoint (PMP) mode, and it can reuse the deployed BWA network on the best advantage.

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Fig. 1. RS Protocol Stack

Figure 1 shows a proposed architecture for relay protocol stack [12]. If RS plays its major role in the BWA networks just by forwarding the data from BS or MS, it is called transparent RS. In this case, BS performs as the centralized scheduler for any links in its cell, such as BS-MS, BS-RS and RS-MS. But there is another type of RS which acts as a distributed scheduler when no direct link is available for BS and MS. Under the management of BS, RS will extensively re-allocate the radio resource in the relay access link between RS and MS. Such RS is called non-transparent RS. In the design for relay protocol stack, there are two principles: one is to ensure the scalability of system functions, and the other is to minimize the system control overhead.

3 Enhanced BWA Architecture 3.1 MAC Layer Simulator Architecture According to [9][13], we propose a system-level platform to evaluate BWA system performance with certain MAC layer and OFDMA PHY layer technologies. As it is required, QoS guarantee scheme is involved to distinguish requirements of different services. Types of transportation merits are classified with different QoS requirements. Scheduler allocates resources to satisfy different services with the consideration of current radio link quality. Both BS and MS adopt schedulers for the resource allocation. With service-classified scheduler, call admission control (CAC) and Service Flow (SF) management, QoS guarantee can be provided on MAC layer. Meanwhile, we introduce both HHO and FBSS to manage the MS mobility. MAC layer simulator is embedded in the BWA system-level evaluation platform. Besides incorporating radio resource management schemes in MAC layer, the platform is also an open test bed. It is able to include many new radio link technologies to meet the transportation requirements. Adaptive Modulation and Coding (AMC) with Hybrid ARQ as the link adaptation adjustment can provide the BWA system with more reliable and adaptive response to time-varying channels. Also to calculate how the channel influences upon the BWA system-level performance, link-to-system mapping methodology has been introduced, such as EESM. EESM aims to incorporate multi-carrier transportation effects into empirical formula. Using several iterations to get more precise results, the system-level platform gets more accurate channel information for packet-based simulation that matches the practical situation [13].

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3.2 Relay Based Network Planning In this part we present a relay enhanced cell unit on a two-hop OFDMA frame structure. First, as the scalability of IEEE 802.16e OFDMA PHY layer [1], the frame can be reused to meet the relay link requirements by inserting relay zones into both downlink (DL) and uplink (UL) subframes. We first start with the introduction of two-hop OFDMA frame structure by defining some basic terminologies in relay links. When a MS is under the management of BS, it will communicate directly with the BS or with the RS affiliated to BS. The radio link between MS and its up streaming station, BS or RS, is called the access link, while the link between BS and RS is called the relay link correspondingly. Thus, we associate MS that has access link to BS with the symbol, and for MS accessing to RS.

Fig. 2. MAC layer simulator architecture

Fig. 3. Two-hop relay frame structure

Fig. 4. The 19 cell evaluation infrastructure with the proposed RCU

It is assumed that the RS-MS link and BS-RS link share the same frequency resource, which is called In-band Relay. And there is a new type of transporting zone defined in the DL/UL subframe for relay. As shown in Figure 3, access zone is for a BS to communicate with its affiliated RS(s) and its own access MS(s), while relay zone

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is for the communication between RS and its MS(s). In the case of downlink subframe on TDD mode, BS and RS utilize the same temporal and frequency resource of OFDMA PHY layer to transport data. For relay zone is appended in the IEEE 802.16e frame structure, it is easier to be expanded to adapt to the multi-hop relay scenarios. A relay enhanced cell unit, shown in Figure 4, is introduced to assist BWA networks. Before the description of the relay network architecture, several assumptions are listed as follows to simplify the modeling complexity. • • •

RS’s coverage is of the same size that of BS. BS cell is divided into 3 sectors, using directional antenna to cover 120° range within each. Omni-directional antenna is used in RS cell.

BS involves the directional antenna to cover 120 degree range in the cell. Three directional antennas make the cell consisting of three sectors, each with one antenna. MS which is near the cell border experiences a significant path loss in the direction between antenna main lobes on received signal quality, for it is out of the scale of the main lobes. Thus, in the proposed relay enhanced cell unit, 3 RSs are introduced to enhance the BS cell. Each one is placed in the location near the vicinity of the cell border facing directional antennas’ main lobes to establish BS-RS relay link. Such BS-RS node topology is defined as relay cell unit (RCU) in this paper. RSs in RCU enhance the cell link quality for MS transportation. The area between main lobes of directional antennas can be filled up by the coverage of RSs from the abutting RCUs. Figure 5 illustrates an example.

Fig. 5. Co-operative Coverage between RCUs

In the BSi centered RCU, there are three assisting RSs which are RSi1 , RSi 2 and RSi 3 . The abutting RSs from surrounding RCUs help to pad the fading zones between

links of BSi to its affiliated RS. These RSs, RSa1 , RSb 2 and RSc 3 , are shown here. Each RS uses one third the frequency resources of the cell, while BS has the frequency reuse pattern 1×3×1. Based on RCU, the two-hop relay connection scenario is involved in the simulation platform that MS connects to BS via at most one-hop relay or to BS directly if it gets better signal quality in the BS service area.

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4 Performance Evaluation and Analysis Hereafter, we present simulation results for the proposed RCU and briefly evaluate its performance in comparison with two non-relay network planning approaches: one is with the radio frequency reuse pattern 1×3×1, and the other is 1×3×3. We apply these network topologies in the 19-cell system evaluation infrastructure, and the parameters are described in detail in [13]. As Figure 6 shows, in the radio spectrum efficiency results, systems with frequency reuse have much better performance compared with systems involving no frequency reuse. There are significant improvements in spectrum efficiency for topologies with different frequency resource allocated in each sector. As observed, RCU doesn’t outperform a lot the non-relay pattern 1×3×3. It is largely due to the assumption that RS covers as large area as BS does, in which RCU coverage is approximately 4 times the size of the non-relay cell on the same transmit power level, that is, RCU expands the service area significantly without extensive power requirements. Moving away from comparison in spectrum efficiency, Figure 7 shows that the proposed RCU can improve the system throughput significantly. MS can receive the forwarding signals by RS from BS in the RS cell where it can not connect to BS with certain link quality requirements without relay. Thus, RCU can enlarge the good reception coverage in turn to improve the system throughput. Another element to the throughput improvement is that RS provides better link for MS to choose. So the corresponding more efficient modulation and coding scheme can be applied which results in better system performance.

Fig. 6. Spectrum efficiency in relay and nonrelay mode

Fig. 7. Throughput in relay and non-relay mode

Also, comparing the performances of the two frequency reuse patterns for nonrelay mode, we will find specified pattern is suitable for either high spectrum efficiency or high system throughput, but it can not achieve the goals together. And this can be observed that RCU is taking a balance between the two key systematic merits with good performance. Besides, there are several other advantages for the application of RCU in our BWA networks. For example, RS in its service area utilizes part of the radio frequency resources of BS. If the moving MS switches its access station

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between RS and BS in the RCU, the handover procedure will be much simpler than the normally defined one. In such so-called intra-cell handover, no further MS registration and networking entry procedures are needed, for MS is under the management of the same BS. Usually RS will only be expected to cover a region of the diameter which is much less than that of BS. It is in turn to significantly reduce the transmit power requirements. Practically, we have to calculate certain coverage between budgeting the BS and RS link. Plenty of research optimization remains to be done.

5 Co-operation in Relay Cell The co-operative communication can be deployed when RS has been introduced into BWA system. There are two radio propagation features we can exploit: broadcasting and achieving diversity through independent channels. MS might receive signals from both BS and RS in the RCU at the same time, in this case, MS combines the signals received to gain certain downlink co-operative communication diversity gain. Here, we deploy a scenario of downlink co-operative data transportation.

Fig. 8. Proposed DL co-operative communication

In the proposed two-hop relay frame structure, BS shares the same frequency resource with its affiliated RSs. In the DL relay zone, while RS is broadcasting its downlink data to its affiliated MSs, BS is inactive in such temporal and frequency resource. However, MS active in the RS coverage can also receive BS’s signals after its synchronization. In order to associate BS with RS’s transporting to the MS, it is necessary to allow BS to transmit in DL relay zone. As shown in Figure 8, when a MS in RS service area is suitable for receiving BS’s signal to achieve DL co-operation gain, upon the centralized scheduling on BS, BS will send the signals on DL Relay Zone with the same resource allocation as RS send

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to the MS, where central scheduling is needed. BS and RS will transmit the same signals to the same MS using the same temporal and frequency resource. Certain issues should be addressed further, such as synchronization, availability of channel state information, and appropriate cluster formation should be considered.

6 Conclusion We propose a MAC layer simulator and system-level evaluation platform with a proposed RCU for the relay enhanced BWA network topology. Certain adaptations to such architecture are illustrated in the OFDMA TDD frame structure. Discussions above are concerned mainly on how to guarantee the capacity and coverage while keeping the complexity of former system with relay introduced in BWA systems. For the economic consideration, RS usually covers smaller area than BS does, but in this paper we assume they cover the same. Also, since RS has the same transmit power level as BS, there is more interference in the networks.

References 1. IEEE Std 802.16e: IEEE Standard for Local and metropolitan area networks, Part 16: Air Interface for Fixed and Mobile Broadband Wireless Access Systems (February 2006) 2. Tafazolli, R.: Technologies for the Wireless Future. John Wiley & Son, NY (2004) 3. Ghosh, A., Wolter, D., Andrews, J., Chen, R.: Broadband wireless access with WiMAX/802.16: Current performance benchmarks and future potential. IEEE Communication Magazine 42(2), 129–136 (2005) 4. Pabst, R., Walke, B.H., Schultz, D.C., et al.: Relay-based deployment concepts for wireless and mobile broadband radio. IEEE Communication Magazine 42(9), 80–89 (2004) 5. Berlemann, L., Pabst, R., Walke, B.: Efficient multimode protocol architecture for complementary radio interfaces in relay-based 4G networks. IEEE Wireless Communication 12(3), 15–23 (2006) 6. Hoymann, C., Klagges, K., Schinnenburg, M.: Multihop communication in relay enhanced IEEE 802.16 networks. In: PIMRC 2006 Conference Proceedings, the 17th Annual IEEE International Symposium on Personal Indoor and Mobile Radio communications, Helsinki, Finland (September 2006) 7. IEEE C802.16j-06/026r2, Baseline Document for Draft Standard for Local and Metropolitan Area Networks, Part 16: Air Interface for Mobile Broadband Wireless Systems Multihop Relay Specification (February 2007) 8. Santhi, K.R., Kumran, G.S.: Migration to 4G: mobile IP based solution. In: Proceedings of AICT-ICIW Conference, p. 76 (2006) 9. Kwon, T., Lee, H., Choi, S., et al.: Design and implementation of a simulator based on a cross-layer protocol between MAC and PHY layers in a WiBro Compatible IEEE 802.16e OFDMA system. IEEE Communication Magazine 43(12), 136–146 (2005) 10. IEEE C802.16j-06/230r1, Efficient resource utilization scheme on the basis of precoding and cooperative transmission in downlink (November 2006) 11. IEEE C802.16j-07/124r1, Cooperative relaying in downlink for IEEE 802.16j (January 2007) 12. IEEE C802.16j-07/096, MMR Protocol Stack and Definition of RS Types (March 2007) 13. WiMAX Forum, WiMAX System Evaluation Methodology (January 2007)

Improved Channel Estimation Based on Compressed Sensing for Pulse Ultrawideband Communication System Dejian Li, Zheng Zhou, Feng Zhao, Weixia Zou, and Bin Li Key Lab of Universal Wireless Communications, MOE Wireless Network Lab, Beijing University of Posts and Telecommunications Beijing 100876, China [email protected], [email protected]

Abstract. Exploiting the rich multipath of ultrawideband (UWB) system with channel estimating for Rake receivers is a key point to achieve high performance. This paper presents an improved approach of channel estimation for pulse UWB communication system based on the theory compressed sensing (CS). The Matching Pursuit algorithm is used to identify the channel taps under the compressed sensing framework with prior knowledge of the power delay profile (PDP) of the channel. The MP algorithm itself is also modified to achieve higher performance of estimation based on the principle of probabilistic MP (PMP). Simulation results show that the proposed approach outperforms the already existing CS-MP channel estimation scheme. Keywords: channel estimation, compressed sensing(CS), Matching Pursuit (MP), Power Delay Profile (PDP), UWB.

1 Introduction The very fine time resolution of the pulse UWB signal results in a very large number of resolvable multipath components [1]. The multipath components can be captured by a number of correlators which are sampled at the delays related to specific multipath components, each of those correlators is called “Rake finger”. This implemention requires accurate multipath delays and amplitudes estimation. The compressed sensing(CS) theory [2] outlines a novel strategy to jointly compress and detect a sparse signal with fewer sampling resources than traditional method. The signal reconstruction problem can be solved by linear programming techniques like basis pursuit (BP) or greedy algorithms such as matching pursuit (MP). CS has been used to solve the channel estimation problem[3][4][5], signal detection[6], positioning of UWB system. Channel estimation for pulse UWB based on CS was first proposed in [3]. In [4], two approaches denoted by CS-MP for pulse UWB channel estimation were developed under a data-aided framework. In the approach for Rake receiver, from a set of random projections of the received pilot signal, Matching Pursuit(MP) algorithm is used to identify the strongest atoms in the projected signal that, in turn, are related to the strongest propagation paths that composite the multipath UWB channel. Similarly, basis pursuit denoising (BPDN) was used to solve the channel estimation with noise [5] for UWB system. Comparing to G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 330–337, 2010. © Springer-Verlag Berlin Heidelberg 2010

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the CS maximum likelihood (ML) channel estimator [6], CS-MP estimator requires much lower computational complexity. The computational complexity of the ML channel estimator increases as the number of significant multipath components increase and become prohibitively expensive for NLOS propagation scenario [4]. In this paper, we modified the MP algorithm under the CS framework and propose a novel strategy to improve the performance of channel estimation using Power Delay Profile (PDP) information. The partial channel state information PDP can enhance the reliability of channel estimation with a modified MP algorithm. Assuming the knowledge of the average power delay profile (APDP), the optimum receiver for 2PPM, resulting in an energy detector where the receive signal is weighted with the APDP, is derived in [7]. For the transmitted reference receiver, similar work was developed in [8][9]. Inspired by this, we introduce an approach using APDP to weight the atoms of the dictionary in MP. In addition, to achieve a higher performance, we modified the atom selection scheme of the MP algorithm according APDP based on the principle of Probabilistic MP(PMP) [10]. The rest of paper is organized as follows: Section II describes our system model and gives a review of CS-MP channel estimator presented by [4]. Section III depicts using a modified MP to improve the performance of CS-MP estimation. Simulation results are presented and discussed in Section IV. Section V contains the conclusions.

2 Problem Formulation Consider a single-user system that sends a series of pilot symbols for channel estimation prior to data transmission. The received signal can be modeled as N f −1 L

M −1

y (t ) =

∑ b ∑ ∑ α s(t − τ

m=0

m

j = 0 l =1

l

l

− jT f − mTs ) + n(t )

(1)

where { bm } are the pilot symbols taking values ±1 , α l and τ l are the gain and delay of the lth path respectively, s(t) is the received Gaussian pulse with duration Tp and T f is the frame time. If restrict the analysis to real-valued UWB channel models

where there is no pulse distortion, s(t) equals the transmitted Gaussian pulse. There are N f frames per symbol for a symbol period of Ts = N f T f and n(t) is the zero-mean additive white Gaussian noise (AWGN) with double-sided power spectral density N 0 / 2 . Furthermore, the ultra-short pulse of duration Tp is spread over frame interval T f as a result of multipath. The pilot symbols are transmitted over an observation

interval of T0 = MTs in which the dense multipath channel is assumed to be static. Let h(t) be the impulse response of the UWB channel which has been modeled as L −1

h(t ) = ∑ α l δ (t − τ l )

(2)

l =0

where δ (⋅) is the dirac delta function. The goal of channel estimation is to estimate the channel parameters {α l , τ l }lL=1 related to the various propagation paths. For rake receivers, only Lc most significant paths of all L paths have to be estimated.

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A CS-MP UWB channel parameter estimation followed by a Rake receiver was developed in [4]. Designing a dictionary of parameterized waveforms can increase the sparsity of the underlying signal. The atoms of the dictionary should be generated by shifting versions (with minimum step Δ ) of the transmitted Gaussian pulse, leading to a set of parameterized waveform given by di (t ) = s (t − iΔ), i = 0,1, 2,...

(3)

D = {d 0 (t ), d1 (t ), d 2 (t ),...,}

(4)

The dictionary is defined as

Both parameters t and Δ should be discretized in practice, setting Δ to a multiple of the sampling period. In a case when pulse overlapping is allowed, i.e., τ i − τ j < Tp ,

Δ is set to a value shorter than the time support of the basic pulse. Without loss of generality, we let the polarity of all pilot symbols to be 1, i.e. bm = 1, ∀m . Let y be a N-point discrete time representation of the y(t) in (1).Using the discrete time dictionary Ψ which is defined by uniformly sampling the atoms of the dictionary D , Ψ = {d 0 ,d1 ,d 2 , ...,d L } .

(5)

Then we have y = ΨΘ + n w

(6)

where n w is the discrete AWGN. (6) shows that y can be more compactly presented. Define a measurement matrix Φ as a K × N random matrix with entries i.i.d. taken from a normal distribution with zero mean and unit variance. Let g = Φy be the random projected signal, we get the measurements g = ΦΨΘ + Φn w

(7)

where Θ is a sparse vector Θ = [θ1 , θ 2 ,..., θ Z ] , Z is the number of atoms in the dictionary Ψ . V = ΦΨ = {v1 , v 2 ,..., v Z } is the holographic dictionary. The MP l1-norm minimization algorithm is then applied on the random projected signal g and the holographic dictionary V for signal reconstruction. The MP algorithm is showed in details in Table 1 [4]. The MP algorithm outputs a sparse vector ˆ = [θˆ , θˆ ,..., θˆ ] that contains the signal contribution of the various propagation Θ 1 2 Z paths. The reconstruction step in the MP algorithm can be thought of as a weighted sum of the elements in the dictionary,

ˆ. yˆ = ΨΘ

(8)

Since each atom in the dictionary is a shifted version of the transmitted pulse, θˆi is an estimate of the path gain related to the ith propagation path. The number of iteration N iter is set to a value much smaller than necessary for signal reconstruction.

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Table 1. Matching Pursuit algorithm Step 1 Step 2

ˆ = 0, Θ ˆ ∈ RZ ; Initialize: The residual error e0 = y ; The approximation Θ Iteration counter t = 1. Select the atom in the holographic dictionary that best match the residual error. et −1, vi lt = arg max vi i =1,2,..., Z

Step 3

Update the residual error and the estimate of the coefficient for the selected vector et = et −1 −

Step 4 Step 5

et −1 , vlt vlt

vlt , θˆlt = θˆlt +

2

If t < Niter and et

2

et −1 , vlt vlt

> ε y 2 then set t=t+1 and go to step 2 ; otherwise, go to step 5

ˆ Reconstruct the signal estimate as: yˆ = ΨΘ

3 Sequential Weighted Matching Pursuit 3.1 WMP: MP with Weighted Dictionary Using APDP

The noise component of the received pilot waveforms may drive the MP algorithm to misplacing the strongest atoms in the received pilot signal and to a wrong estimate of the atoms’ contribution to the signal. Consequently, this leads to errors in the estimation of the UWB channel since each atom found in the random projected signal represents the signal contribution of a path. Improving the performance of combating noise of MP is required. From (4) we get a dictionary of delayed versions of transmitted pulse. For different delayed atoms of the dictionary, they have identical pulse energy. Intuitively, for the propagation paths with large delay, they have weak energy with high probability. If a weak path contaminated by noise is wrongly selected as a spurious atom by MP according Step 2 in Table 1, we wish the detrimental effect will be small for channel estimation. If an atom corresponding to a weak path but wrongly selected due to the influence of AWGN by step2 also has low energy, the adverse effect can be reduced in the signal reconstruction process. To remedy the mistake of wrongly identifying an atom, APDP can be used to weight the atoms to obtain a new dictionary. APDP is the mean power of the different paths. Most atoms of the dictionary except the path that has the largest mean power are down-weighted with APDP, then “less of them” will be used in the signal reconstruction. Let weighting function ω (t ) be a function of APDP as PAPDP (t ) ,

ω (t ) = f ( PAPDP (t ))

(9)

f is a proper function of APDP and we let f be a fourth root operator in our simulations. Thus, we can get a weighted dictionary Dw ,

Dw = {ω (t )d 0 (t ), ω (t )d1 (t ), ω (t )d 2 (t ),...,}

(10)

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Assumes that ω is the discrete counterpart of ω (t ) and is an L points vector corresponding to the L paths of the channel. Then, we get the discrete dictionary, Ψ w = {ω0 d 0 , ω1d1 , ω2 d 2 , ..., ωL d L }.

(11)

If a weak path is identified as a strong path, the corresponding ωs will reduce the contribution of this atom in the signal reconstruction, like getting a ω θˆ as the estis s

mation output of the step for MP that using non-weighted dictionary. The MP algorithm using a weighted dictionary is called WMP hence. Note that in the new scheme, except the dictionary Ψ of MP is changed to a weighted dictionary Ψ w and leading to a change to the holographic dictionary, all steps of the new method are identical with MP. However, we stress that the output ˆ of WMP can not be viewed as the direct estimated channel parameter like that in Θ ˆ does not have the MP, since the WMP is applied to weighted dictionary Ψ w and Θ physical meaning as in MP. After the WMP ends its iterations, reconstruct the signal ˆ . To get the estimated channel parameters, a conversion is reestimate as: yˆ = Ψ w Θ ˆ ′ is the channel estimation result. ˆ = ΨΘ ˆ ′. Θ quired, that is yˆ = Ψ Θ w

3.2 SWMP: WMP with Sequential Atom Selection Scheme

We stress that our objective is not the reconstruction of the received noisy pilot signal. However, we aim at extracting the parameters of paths that have the largest power from the pilot signal contaminated by noise. We state that “best atom” with maximum inner product with the pilot signal selected by MP for each step has poor effect for denoising, which is a similar statement as PMP. PMP is a randomization of the MP algorithm for denoising [10]. In PMP, multiple random expansions are obtained as estimates for a given signal. Although most of the random expansions generated by PMP are poorer estimates for the signal than those obtained by MP, the final estimate, obtained as an expected value computed by means of an ergodic average, can improve the result obtained by MP in denoising situations. Random selection of expansions offers an advantage over deterministic selection: the probability distributions employed in random selection may be adjusted to take into account empirical observations or additional knowledge in special-case situations. The key procedure of PMP is to modify the atom selection of Step 2 of Table 1 to select the atoms at random. It selects the Gabor window function parameters (atoms) based on a conditional probability. More details about PMP can be found in [10]. In this paper, we introduce the basic principle of PMP to obtain a modified MP. According to APDP, despite of LOS or NLOS scenario, the most significant paths arrive early with high probability. From the perspective of denoising and making estimation of channel parameters more correctly, a greedy atom selection scheme is not necessary anymore. In CS-WMP channel estimation, since the atom selection sequence of WMP represents the significance of these paths, we wish WMP can select the atoms that corresponding to the paths that has true high gain firstly. In other word, select atoms according to the sequence of selecting atoms of using CS-MP to recover a noiseless received pilot signal is ideal. However, using step 2 of Table 1 can

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not select the atoms sequentially according to their gain due to the influence of noise. Ideally, like in PMP, we anticipate to select the atoms at random based on probability of the paths gains. However, constructing a pdf that depicts the probability of paths gain with delay is difficult as APDP is only partial channel statement. We propose a simple but effective atom selection scheme, that is, select the atoms from the start of the dictionary sequentially. The sequential atom selection scheme with WMP is named SWMP. The main parameter of SWMP N iter is set to on the order of the most significant number of paths Lc. The target residual energy is no longer used to end the WMP algorithm since we are not interested in signal reconstruction. To summarize, SWMP is presented in Table 2. Table 2. Sequential weighted matching pursuit algorithm

Step 1

ˆ = 0, Θ ˆ ∈ RZ Initialize: The residual error e0 = y ; The approximation Θ Iteration counter t = 1, Niter = Z

Step 2

Select the atom in the holographic dictionary sequentially A t = A t −1 + 1, with A 0 = 0

Step 3

Update the residual error, the estimate of the coefficient for the selected vector et = et −1 −

et −1 , vlt

et −1 , vlt vlt , θˆlt = θˆlt + 2 vlt vlt then set t = t + 1 and go to step 2 ; otherwise, go to step 5

Step 4

If t < Niter

Step 5

ˆ Reconstruct the signal estimate as: yˆ = Ψ wΘ

4 Simulation Results We examine the performance of the CS-SWMP and CS-WMP channel estimation approaches compared to CS-MP presented in [4] by simulations. The system is formed as stated in Section II, N f = 10 , T f = 100 ns, s(t) is modeled as the first derivative of Gaussian pulse that has been normalized to have unit energy and a pulse duration of 1 ns. We use the IEEE 802.15.3a in [11] as the channel model for testing the UWB Rake receiver. Note that the IEEE 802.15.3a channel model has a important parameter which represents the time resolution, i.e., the bin duration ts. We set the time resolution of the channel to ts = 0.5ns which is shorter than the pulse duration. Thus, pulse overlapping occurred. The negligible taps at the tail of the multipath impulse response are cut off to make the maximum delay spread of the multipath equal to 35 ns for CM1(LOS) and 50 ns for CM2(NLOS) scenario. The remaining channel taps are normalized such that the channel energy is set to one. We obtain the APDP from 200 random channel realizations for CM1 and CM2. Also, we evaluate the performance of the proposed WMP and SWMP over 200 channel realizations. The sampling frequency before the projection stage in all the simulation is 16 GHz. Further, a 2-PAM modulation scheme is adopted in our simulation. For Rake receiver, the number of fingers is set to 30 for LOS and 40 for NLOS, one pilot symbol is used for channel estimation. All BER curves shown next are depicted as a function of signalto-noise ratio defined as E p N0 , where E p is the transmitted pulse energy.

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As stated in [4], the maximum iteration number Niter of WMP and MP are both set to 1.5Lc, where Lc is the number of Rake’s fingers. For SWMP, Niter is set to 2.5Lc , which is a little higher than that used in MP and WMP.

(a)

(b)

Fig. 1. (a) RMSE of CS-SWMP, CS-WMP, CS-MP as SNR(dB) varies for Lc = 30, LOS and Lc = 40, NLOS with K/N = 0.5. (b) BER performance for different number of projections.

(a)

(b)

Fig. 2. BER performance of CM1 (a) and CM2 (b) for CS-SWMP, CS-WMP, CS-MP with number of pilot symbols M = 1, K/N = 0.5.

Fig.1 (a) shows a normalized measure of the root mean squared estimation error (RMSE) of the Lc most significant paths. As seen, the performance of channel estimator using CS-WMP and CS-SWMP are both improved for LOS and NLOS scenarios. The improvement in NLOS scenario for RMSE is not significant as in LOS scenario, since the CS-MP channel estimation is unable to resolve the strongest paths among the multiple closely spaced propagation paths. This phenomenon is also seen Rake receiver operation curve in Fig. 2 (a) and Fig. 2 (b). Fig. 1 (b). shows the BER performance of CS-Rake receiver using CS-MP and CSSWMP channel estimation for different number of projections K/N = 0.3 and K/N = 0.6. The data burst is set to 20000 bits. As expected, the CS-SWMP’s performance improves as the number of projections increases.

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Fig. 2 (a) and Fig. 2 (b) depict the BER performance of CS-Rake receiver for CM1 and CM2 respectively. We can see that CS-SWMP channel estimation based Rake receiver outperforms CS-MP based Rake receiver.

5 Conclusion In this paper, we modified MP to improve the performance of CS-MP channel estimator based on APDP and PMP. In the proposed approach, the Rake receiver has no other prior information except APDP. We showed that CS-WMP with a dictionary weighted with APDP and CS-SWMP with sequential atom selection scheme based on WMP both outperform CS-MP through simulation. The computational complexity of CS-SWMP is low since its number of iterations is still on the order of number of Rake’s fingers. Although the proposed approach was designed for Rake receiver, they are easy to applied for transmit reference (TR) receiver.

Acknowledgments

＆

This work was supported by NSFC (60772021), National 863 Program (2009AA01Z262), Important National Science Technology Specific Projects (2009ZX03006-0 06/009) and Korean Ministry of Knowledge Economy Project (IITA-2009-C1090-090 2-0019).

References 1. Molisch, A.F.: Ultrawideband propagation channels-theory, measurement, and modeling. IEEE Transactions on Vehicular Technology 54(5) (2005) 2. Candes, E.J., Wakin, M.B.: An Introduction To Compressive Sampling. IEEE Signal Processing Magazine 25(2) (2008) 3. Paredes, J.L., Arce, G.R., Wang, Z.: Compressed Sensing for Ultrawideband Impulse Radio. In: IEEE Conference on Acoustics, Speech and Signal Processing, vol. 3, pp. 553–556 (2007) 4. Paredes, J.L., Arce, G.R.: Ultra-Wideband Compressed Sensing: Channel Estimation. IEEE Journal of Selected Topics in Signal processing 1(3), 383–395 (2007) 5. Zhang, P., Hu, Z., Qiu, R.C.: A Compressed Sensing Based Ultra-Wideband Communication System. In: IEEE International Conference on Communications, pp. 1–5 (2009) 6. Liu, T.C.-K., Dong, X., Lu, W.-S.: Compressed Sensing Maximum Likelihood Channel Estimation for Ultra-Wideband Impulse Radio. In: IEEE International Conference on Communications, pp. 1–5 (2009) 7. Weisenhorn, M., Hirt, W.: ML receiver for pulsed UWB signals and partial channel state information. In: IEEE International Conference on Ultra-Wideband, pp. 180–185 (2005) 8. Chao, Y.-L., Scholtz, R.A.: Weighted correlation receivers for ultrawideband transmitted reference systems. In: Proc. IEEE GLOBECOM 2004, pp. 66–70 (December 2004) 9. Franz, S., Mitra, U.: Generalized UWB transmitted reference systems. IEEE J. Select. Areas Commun. 24(4), 780–786 (2006) 10. Ferrando1, S.E., Doolittle, E.J., Bernal, A.J., Bernal, L.J.: Probabilistic matching pursuit with Gabor dictionaries. Journal of Signal Processing, Elsevier 80(10), 2099–2120 (2000) 11. Molisch, A., Foerster, J., Pendergrass, M.: Channel models for ultrawideband personal area networks. IEEE wireless Commun. Mag. 10(6), 14–21 (2003)

Compressive Sensing Using Singular Value Decomposition Lei Xu and Qilian Liang Department of Electrical Engineering, University of Texas at Arlington, Arlington TX 76013, USA [email protected], [email protected]

Abstract. Using Singular Value Decomposition (SVD), we develop an algorithm for signal recovery in compressive sensing. If the signal or sparse basis is properly chosen, an accurate estimate of the signal could be obtained by a simple and eﬃcient signal recovery method even in the presence of additive noise. The theoretical and simulation results show that our approach is scalable both in terms of number of measurements required for stable recovery and computational complexity. Keywords: compressive sensing, Singular Value Decomposition, sparse basis.

1

Introduction

Compressive sensing (CS) [1][2] is an emerging framework that a signal vector which is K-sparse in a speciﬁc domain can be completely characterized by M measurements (M > K) with M << N , where N is the traditional Nyquist based number of samples required. The major algorithmic challenge in compressive sensing is to approximate a signal given a vector of noisy samples. There are three rough categories of signal recovery algorithms: convex relaxation, combinatorial algorithms and greedy pursuits. The convex relaxation algorithms succeed with a very small number of measurements, however, it tends to be computationally burdensome. Many of the combinatorial algorithms are extremely fast, but they require a large number of somewhat unusual samples that may not be easy to acquire. Greedy pursuits are intermediate in their running time and sampling eﬃciency but has its own disadvantages. In this paper, we provide a new algorithm-the CS-SVD algorithm for signal recovery in compressive sensing by introducing the concept of SVD (Singular Value Decomposition). We use SVD to study the compressive sensing framework and develop a simple and straitforward recovery algorithm in the presence of additive noise. This paper is organized as follows. In Section 2, we provide the necessary background on CS and SVD. In Section 3, we describe the proposed CS-SVD algorithm. In Section 4, we provide the simulation result illustrating the performance of our algorithm. Section 5 concludes with discussion and extensions. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 338–342, 2010. c Springer-Verlag Berlin Heidelberg 2010

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Preliminaries Compressive Sensing

The recent results of compressive sensing have shown that the information from a signal may be captured with a small set of nonadaptive, linear measurements as long as the signal is sparse in some basis or frame [3][4]. We acquire a signal vector x ∈ N via linear measurements y = Φx + e

(1)

where Φ is an M × N measurement matrix modeling the sampling system, y ∈ M is the vector of samples observed, and e is an M × 1 vector that represents measurement errors. If x is termed as K−sparse in the sparsity basis Ψ , i.e, θ = Ψ T x containing no more than K nonzero elements, we need to acquire only M = O(Klog(N/K)) random measurements to recover the signal x. The convex relaxation algorithm is a powerful method for CS signal recovery. Here, one can use l1 minimization with relaxed constrains for reconstruction [5]: min|| x||l1

subject to

||Φ x − y||l2 ≤

(2)

where bounds the amount of noise in the data. The convex relaxation algorithm could obtain a small number of measurement, but results in large computational complexity. In Section 4, the signal recovery performance of the convex relaxation algorithm will be provided to be compared with the performance of the proposed algorithm. 2.2

Singular Value Decomposition

We introduce the concept of the extremely useful singular value decomposition. Theorem 1 (Singular Value Decomposition (SVD))[6]: If A is a real mby-n matrix, then there exist orthogonal matrices U = [u1 , ..., um ] ∈ m×m and V = [v1 , ..., vn ] ∈ n×n

(3)

such that U T AV = diag(σ1 , ..., σp) ∈ m×n , p = min(m, n)

(4)

where σ1 ≥ σ2 ≥ ... ≥ σp ≥ 0. According to the above theorem, assuming that Σ = U T AV = diag(σ1 , ..., σp ), then A = U ΣV T . We could also obtain the important properties of orthogonal matrices that U T U = diag(1, ..., 1) ∈ I m×m and V T V = diag(1, ..., 1) ∈ I n×n . These direct but signiﬁcant properties would be used in the following section.

3

The CS-SVD Algorithm

Applying the concept of SVD to CS, we propose a new algorithm-the CS-SVD algorithm for signal recovery in this section. If the signal or the sparse representation

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scheme of signal is properly chosen, x is K-sparse in the basis Ψ V (V would be mentioned in the latter part), i.e., x = Ψ V θ with ||θ ||0 = K, the measurements M required to recover the original signal is only K, and the recover algorithm is easy and very straitforward. 3.1

Signal Model

We study the signal vector x which is obtained by linear measurements y = Φx + e = ΦΨ θ + e

(5)

where Φ is an M × N measurement matrix modeling the sampling system, x is K−sparse when expanded in the sparsity basis Ψ , i.e,. x = Ψ θ and y ∈ M is the vector of samples obtained. We assume that A = ΦΨ . Based on the deﬁnition and properties of SVD that Σ = U T AV = diag(σ1 , σ2 , ..., σp ), we use SVD to decompose the matrix A so that A = ΦΨ = U ΣV T . Without loss of generality, we let θ = V H θ, then θ = V θ . The equation (5) could be expressed as y = ΦΨ θ + e = U ΣV T V θ + e 3.2

(6)

Recovery Algorithm

Motivated by how SVD works in MIMO system, we multiply y by U T and have equation (6) turn out to be U T y = U T (ΦΨ θ + e) = U T U ΣV T V θ + U T e = Σθ + e

(7)

where e = U T e. As it is known that Σ = diag(σ1 , σ2 , ..., σp ) ∈ M×N , p = min(M, N ), where σ1 ≥ σ2 ≥ ... ≥ σp ≥ 0. We could express the equation (7) as the following vector U T y = [y1 , y2 , ..., yp ]T + e = [σ1 θ1 , σ2 θ2 , ..., σp θp ]T + e

(8)

Once Φ and Ψ are known, Σ is known and ﬁxed. We use l2 minimization for reconstruction as: min||θ ||l2

subject to

|θi −

yi 2 | ≤ , i = 1, 2, ..., p σi

(9)

Since the value of θi , i = 1, 2, ..., p is ﬁxed as shown in (9), min ||θ ||l2 is just to satisfy that θi = 0, i > p and P ≥ K. It is also easy to see that ||θ ||l2 = ||V H θ||2 = θ H (V H )H V H θ = θH θ = ||θ||l2 As a result, min ||θ ||l2 is equivalent to min ||θ||l2 .

(10)

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The original signal x could be recovered by x = Ψ θ = Ψ V θ

(11)

We observe that p = min(M, N ) and M is usually less than N in CS, hence, we could obtain that p = M here. Stating diﬀerently, the number of measurements M depends on the number and positions of nonzero values in θ where x is K-sparse when represented in the basis Ψ V , i.e., x = Ψ V θ . For the sake of simplicity, but without loss of generality, we assume that θi = 0, i > p, if the sparsity basis and measurements matrix are properly chosen, i.e., x = Ψ V θ . Using the proposed scheme, we only have to capture M (M ≥ p) measurements to recover the signal vector x.

4

Simulation Results

In this section, we simulate the signal recovery from noisy data. We compare the mean-squared error (MSE) for the proposed algorithm with the MSE for the convex relaxation algorithm as we increase the number of measurements M for ﬁxed SN R = 20dB. Here, SN R is the power ratio of signal acquired to measurements error. We set N = 500 and K = 40, where N is the number of samples of the signal vector x and x has sparsity K in the properly chosen basis Ψ V , i.e., x = Ψ V θ and θi = 0, i > p. According to the Fig. 1, it is easy to see that the lower values of M results in higher error. Increasing the value of M , we see the expected decay in MSE. Comparing the two lines in the Fig. 1, the CS-SVD algorithm could provide stable and much better recovery performance than that using the convex relaxation algorithm.

Convex relaxation SVD 0.4

0.35

MSE

0.3

0.25

0.2

0.15

0.1

50

100

150

200

250

300

350

400

M

Fig. 1. MSE versus M for ﬁxed N = 500, K = 40 and SN R = 20dB

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Conclusion

Motivated by the concept of SVD, we propose a new CS technique-the CS SVD algorithm in this paper. The theoretical result shows that our method requires less measurements than standard state-of-the art compressive sensing techniques and provide a simpler and more eﬃcient recovery scheme. The simulation results demonstrate that our method provides considerable gains over convex relaxation algorithm in terms of number of measurements required for stable recovery. In the future, we will further study how to choose the sparse basis and measurements matrix in our scheme, since the sparse representation of the signal has a great eﬀect on the performance of the proposed scheme.

References 1. Donoho, D.: Compressed sensing. IEEE Trans. on Inform. Theory 6(4), 1289–1306 (2006) 2. Candes, E.: Compressive sampling. In: Int. Congress of Mathematics, vol. 3, pp. 489–509 (2006) 3. Candes, E., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. on Inform. Theory, 489–509 (February 2006) 4. Candes, E., Tao, T.: Near-optimal signal recovery from random projections: universal encoding strategies. IEEE Trans. on Inform. Theory, 5406–5425 (December 2006) 5. Santosa, F., Symes, W.W.: Linear inversion of band-limited reﬂection seismograms. SIAM J. Sci. Statist. Comput. 7(4), 1307–1330 (1986) 6. Golub, G.H., Van Loan, C.F.: Matrix Computations. John Hopkins University Press, Maryland (1996)

The Wideband Spectrum Sensing Based on Compressed Sensing and Interference Temperature Estimation Ting Jiang and Shijun Zhai Key Laboratory of Universal Wireless Communication, Ministry of Education, Wireless Network Laboratory, BUPT, Beijing, China [email protected], [email protected]

Abstract. Future cognitive radios will be capable of scanning a wide band of frequencies, in the order of a few GHz, and the employment of adaptive waveforms according to the estimated spectrum of the licensed systems challenges traditional spectral estimation methods, which typically operate at or above Nyquist rates. We research the problem of estimating the spectrum of the wideband signal received at the cognitive radio sensing receiver using compressed sensing coupled with interference temperature estimation to determine the spectrum occupancy of the licensed system. First, we achieve coarse identifica-tion of frequency boundaries of each sub-band via compressed sensing tech-niques at a Sub-Nyquist rate, which is much less than Nyquist rate. Then we use MTM-SVD method to compute the interference temperature value of each subband, and compare these values with threshold. The multitaper spectrum estimation has high accuracy and near-optimality. And the simulation result shows that we can achieve the interference temperature estimation of each sub-band accurately. Keywords: Wideband Spectrum Sensing, Compressed Sensing, Interference Temperature Estimation, Cognitive Radio, MTM-SVD.

1 Introduction It has been widely recognized that utilization of radio spectrum by licensed wireless systems, e.g., TV broadcasting, aeronautical telemetry, is quite low [1]. This means that at a given spatial region and time, there are the frequency bands with no signal occupancy. Such empty spectrum can be available for secondary access by means of cognitive radios. Cognitive radios employ spectrum sensing to determine frequency bands that are vacant of licensed transmissions and restrict their secondary transmissions to such empty portions to meet regulatory requirements of limiting harmful interference to licensed systems. Compressed Sensing (CS) is a method for the acquisition of the sparse signals at the rates significantly lower than the Nyquist sampling rate; the signal reconstruction is a solution to an l1-norm optimization problem [2], [3]. In [4] ~ [6] the spectrum G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 343–351, 2010. © Springer-Verlag Berlin Heidelberg 2010

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sensing schemes based on the principle of CS were presented for wide-band CR sensing receiver. Following the approach in [4], an estimate of the original signal spectrum is then made based on CS reconstruction using a wavelet edge detector, it has however been observed that such a sensing scheme does not result in a sufficiently high detection probability at low signal-to-noise ratios (SNRs) commonly encountered in the wireless channel fading conditions[5]. In this paper, we address the problem of estimating the spectrum of the wide-band signal received at the cognitive radio sensing receiver using compressed Sensing coupled with interference temperature estimation. First, we achieve coarse identification of frequency boundaries of each sub-band via compressed sensing techniques at a Sub-Nyquist rate, which is much less than Nyquist rate. Then we use MTM-SVD method to compute the interference temperature value of each sub-band. The multitaper spectrum estimation has high accuracy and near-optimality. The remainder of the paper is organized as follows. In Section 2, we reviewed the basic principles of CS. In Section 3 we analyses the singularity of the power spectral densities over a wide frequency base on the compressive spectrum sensing scheme of [4]. In section 4, we compute the interference temperature value of each sub-band integrating multitaper method with singular value decomposition. Simulation results are shown in Section 5, with conclusions drawn in Section 6.

2 Preliminaries We refer to [2] ~ [6], and describe a scheme for wide-band spectrum sensing based on the CS principles. Let the analog signal x ( t ) ,

0 ≤ t ≤ T be represented as a finite

weighted sum of the basis functions (e.g. Fourier, Wavelet) ψ i ( t ) as follows N

x(t ) = ∑ siψ i (t ) i =1

where only a few basis coefficients

(1)

si are much larger than zero due to the sparsity of

x ( t ) . In particular, with a discrete-time CS framework, we consider the acquisition

N ×1 vector x = Ψs , where Ψ is the N × N sparsity basis matrix and s an N ×1 vector with K << N non-zero (and large enough) entries si . It has been of an

shown that

x can be recovered using M = KO ( log N ) non-adaptive linear pro-

jection measurements on to an M × N basis matrix Φ that is incoherent with Ψ [4]. An example construction of Φ is by choosing elements that are drawn independently from a random distribution, e.g., Gaussian, Bernoulli. The measurement vector y can be written as

y = Φx = ΦΨs

(2)

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Reconstruction is achieved by solving the following l1-norm optimization problem

sˆ = arg min s 1

s.t. y = ΦΨs

(3)

The linear programming technique, e.g., basis pursuit [3], or iterative greedy algorithms [2] can be used to solve (3).

3 The Wideband Spectrum Coarse Sensing Based on Compressed Sensing Figure 1 depicts the wideband spectrum sensing based on compressed sensing. The analog baseband signal

x ( t ) is sampled using an AIC. An AIC may be conceptually

viewed as an ADC operating at the Nyquist rate, followed by compressed sensing. Denote the N × 1 stacked vector at the output of the ADC by

xk = [ xkN

xkN +1 L xkN + N −1 ]

T

k = 0,1, 2 L

(4)

M × N compressed sensing matrix by Φ A . The output of the AIC denoted by the M × 1 vector

and the

yk = [ ykM

ykM +1 L ykM + M −1 ]

T

k = 0,1, 2 L

(5)

is given by

yk = Φ A xk x t

AIC

yk

Autocorrelation

(6)

rk

Fig. 1. The wideband spectrum sensing based on compressed sensing

The respective N × N and M × M autocorrelation matrices of the compressed signal and the input signal vectors in (5) and (4) are related as follows

Ry = E ⎡⎣ yk ykH ⎤⎦ = Φ A Rx Φ HA Where H denotes the Hermitian. The elements of the matrices in (7) are given by:

⎡⎣ Ry ⎤⎦ = ry ( i − j ) = ry* ( j − i ) , [ Rx ]ij = rx ( i − j ) = rx* ( j − i ) . ij

(7)

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Denote the respective to (4) and (5) as follows

2 N × 1 and 2 M × 1 autocorrelation vectors corres-ponding

rx = ⎡⎣0 rx ( − N + 1) L rx ( 0 ) L rx ( N − 1) ⎤⎦

T

ry = ⎡⎣0 ry ( − M + 1) L ry ( 0 ) L ry ( M − 1) ⎤⎦

(8) T

(9)

where the zeros are artificially inserted. To pose the CS reconstruction in the form of (3), we need to first relate the autocorrelation vectors in (8) and (9). After some matrix algebraic operations, we obtain the following result.

ry = Φr x Where

(10)

Φ is given as

⎡Φ Φ Φ=⎢ A 1 ⎣Φ AΦ 3 Denoting the

Φ AΦ 2 ⎤ ⎥ Φ AΦ 4 ⎦

(11)

( i, j ) -th element of Φ A by φi*, j , the

M × N matrix Φ A has its

( i, j ) -th element given by

i = 1, j = 1L , N ⎧0 ⎡Φ ⎣ A ⎤⎦ i , j = ⎨φ ⎩ M + 2−i i ≠ 1, j = 1,L, N And the

，Φ ，Φ ，Φ = hankel ([ 0 ] , ⎡⎣0 φ

N × N matrices Φ1

Φ1

2

N ×1

(

3

4

are

* 1,1

L φ1,* N −1 ⎤⎦

* Φ 2 = hankel ⎡⎣φ1,1 L φ1,* N ⎤⎦ , ⎡⎣φ1,* N

(

Φ 3 = toeplitz [ 0 N ×1 ] , ⎡⎣0 φ1, N

(

(12)

)

01×( N −1) ⎤⎦

L φ1,2 ⎤⎦

)

Φ 4 = toeplitz ⎡⎣φ1,1 L φ1, N ⎤⎦ , ⎡⎣φ1,1 01×( N −1) ⎤⎦

) )

hankel (c, r ) is a hankel matrix (i.e., symmetric and constant across the antidiagonals) whose first column is c and whose last row is r , toeplitz (c, r ) is a where

toeplitz matrix (i.e., symmetric and constant across the diagonals) whose first column is c and whose first row is r , ON ×1 is a column of N zeros, and O1×( N −1) is a row of

N − 1 zeros.

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rx has a sparse representation in the edge spectrum domain. With the 2 N × 1 discrete component vector zs corresponding to the It has been shown in [4], [5] that

edge spectrum, we have

rx = Gz s where

(13)

G = ( ΓFW ) . The 2 N × 2 N matrices W and F respectively demote the −1

discrete counterparts of a wavelet-based smoothing and a Fourier transform. The 2 N × 2 N matrix Γ represents a derivative operation approximated by a first-order difference and is given as

⎡ 1 0 L 0⎤ ⎢ −1 1 L 0 ⎥ ⎥ Γ=⎢ ⎢0 O O ⎥ ⎢ ⎥ ⎣ 0 L −1 1 ⎦ 2 N × 2 N

(14)

Now using (10) and (12), we can formulate the CS reconstruction of the edge spectr-um as an l1-norm optimization problem

zˆs = arg min zs zs

1

s.t. ry = ( ΦG ) z s

(15)

The optimization problem above is solved using the BP algorithm described in the following section for a single CR case.

4 The Wideband Spectrum Fine Sensing Based on Interference Temperature Estimation The interference temperature mechanism is to quantify and manage the inter-ference source of the wireless communications environment through the receiver’s interference temperature. With the cognitive radio being receiver-centric, it is necessary that the receiver be provided with a reliable spectral estimate of the interference temperature. In [7], Simon Haykin proposed to consider from two aspects: 1) Use a large number of sensors (such as antenna) to properly sensing the RF environment, wherever it is feasible. The large number of sensors is needed to account for the wireless environment is very complex and the spatial difference is large. 2) Use the multitaper method to estimate the power spectrum of the interference temperature due to the cumulative distribution of both the internal sources of noise and the external sources of RF energy. In light of the findings reported in [21], this estimate is near-optimal.

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{ X n }n =1 N

Suppose

are time series received at sub-band

{ }

f n −1 ≤ f ≤ f n } , wn( k )

N

n =1

( k = 1, 2,L , K ) is

Bn : { f ∈ Bn : K

an orthonormal sequence of

Slepian sequences, the associated eigenspectrum defined by the Fourier transforms N

Yk ( f ) = ∑ wn( k ) X ( n ) e− j 2π fn , k = 0,1,L, K − 1

(16)

n =1

M denote the total number of sensors deployed in the wireless environment. Let Yk ( f ) denote the k -th eigenspectrum computed by the m -th sensor. We Let

( m)

may then construct the

M -by- K spatio-temporal complex-valued matrix A ( f )

⎡ w1Y1(1) ( f ) w1Y2(1) ( f ) ⎢ ( 2) w2Y2( 2) ( f ) ⎢ w2Y1 ( f ) A( f ) = ⎢ M M ⎢ ⎢w Y (M ) ( f ) w Y (M ) ( f ) M 2 ⎣ M 1 Each row of

w1YK(1) ( f ) ⎤ ⎥ L w2YK( 2) ( f ) ⎥ ⎥ O M ⎥ L wM YK( M ) ( f ) ⎥⎦ L

(17)

A ( f ) is produced by using different Slepian sequences, each col-

umn is computed using different sensor. And the

{wm }m=1 represent variable weights M

accounting for the relative areas of sensors. Each entry in the matrix

A ( f ) is produced by two contributions, one due to the

additive internal noise in the sensor and the other due to the input RF stimuli, and the primary contribution of interest is that due to RF stimuli. An effective tool for denoising is the singular value decomposition (SVD). r

A ( f ) = ∑ σ k ( f ) uk ( f ) vkH ( f ) = U ∑ V H

(18)

k =0

⎡S ⎣0

where ∑ = ⎢

0⎤ , S = ⎡⎣σ 1 ( f ) , σ 2 ( f ) , L , σ r ( f ) ⎤⎦ , σ k ( f ) is the k th 0 ⎥⎦

singular value of matrix

A ( f ) , the column vector uk ( f ) of orthonormal matrices

U is the associated left singular vector, and vk ( f ) is the associated right singular vector; the superscript H denotes Hermitian transposition. Let the singular values of matrix

A ( f ) be ordered. The largest eigenvalue σ max ( f ) or a linear combina2

tion of the largest two or three eigenvalues provides an estimation of the interference temperature.

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The multitaper spectrum estimation almost reaches to the nonparametric spectrum estimation’s Cramer-Rao bound (CRB). So this method is recognized as the best by many people [7].

5 The Simulation Results We consider, at base band, a wide frequency band of interest ranging from -50 to 50 MHz, containing 10 non-overlapping channels of equal bandwidth of 10 MHz, each channel is possibly occupied by a licensed system transmission signal that use OFDM modulation. Each 10MHz OFDM has 1024 frequency tones and a cyclic prefix length of 256. The number of OFDM symbols used for spectrum is 1. The sampling rate is 8 × 10 MHz. The received signal is corrupted by the additive white Gaussian noise 2

(AWGN), the noise level is nw

= 10dB . Fig.2 (a) illustrates the PSD

estimated

based on the Nyquist sampling. Fig.2 (b) illustrates the PSD estimated based on compressive sampling, the compression ratio M / N is set to 30% with reference to the Nyquist rate. Fig.2 (c) shows the estimated edge vector

zˆs in (15) using a Basic

Pursuit recovery from the CS measurements with M/N=0.3. MSE performance: We analyze the relationship between the normalized MSE of the estimated PSD and compression rate M / N . The normalized MSE is defined as

{

MSE = E Sˆx − S x

2 2

Sx

2 2

}

(19)

S x denotes the PSD estimate vector based on the periodogram using the signals sampled at Nyquist rate, Sˆ x denotes the PSD estimate based on compressed where

sensing. We can see from figure 3 that the signal recovery quality (via Basic Pursuit) improves as the compression rate M / N increases. The MSE performance is good enough when the compression rate M / N is greater than 50%. Figure 4 shows the Interference Temperature estimation for each sub-band with MTM joint SVD. It has good smoothness and stability and estimates the existence of “frequency spectrum hole” easily by the limit.

(a)

(b)

(c)

Fig. 2. (a) Spectrum estimation based on Nyquist sampling; (b) spectrum estimation based on the compressive sampling ;( c) The edge detection based on the wavelet transform

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Fig. 3. MSE performance

Fig. 4. The interference temperature estimation for each sub-band with MTM – SVD

6 Conclusion In this paper, we presented a wideband spectrum sensing based on the compressed sensing and the interference temperature estimation scheme. We use compressed sensing for the wideband spectrum coarse sensing. The frequency boundaries estimation is done based on CS reconstruction, we can represent compressible signals at a rate significantly below the Nyquist rate. Then we use interference temperature estimation for wideband spectrum fine sensing. Multitaper spectrum estimation almost reaches the Cramer-Rao bound; its accuracy could satisfy the requirements completely. So how to reduce the computation and enhance the efficiency will be a major research topic in the future.

Acknowledgment This work was supported by Important National Science & Technology Specific Projects (2009ZX03006-006/009).

References 1. Federal Communications Commission - First Report, and Order and Further Notice of Proposed Rulemaking, Unlicensed operation in the TV broadcast bands, FCC 06-156 (October 2006) 2. Candes, E., Romberg, J., Tao, T.: Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. on Information Theory 52, 489–509 (2006)

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3. Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM Journal on Scientific Computing 43(1), 129–159 (2001) 4. Polo, Y.L., Wang, Y., Pandharipande, A., Leus, G.: Compressive Wide-Band Spectrum Sensing. In: International Conference on Acoustics, Speech, and Signal Processing (to appear, 2009) 5. Wang, Y., Pandharipande, A., Polo, Y.L., Leus, G.: Distributed compressive wide-band spectrum sensing. In: Information Theory and Applications Workshop, pp. 178–183 (February 2009) 6. Tian, Z., Giannakis, G.B.: Compressed sensing for wideband cognitive radios. In: Proc. of the International Conference on Acoustics, Speech, and Signal Processing, p.IV/1357– IV/1360 (April 2007) 7. Haykin, S.: Cognitive radio: brain-empowered wireless communications. IEEE JSAC 23(2), 201–220 (2005)

The Applications of Compressive Sensing to Radio Astronomy Feng Li, Tim J. Cornwell, and Frank De hoog Australian Commonwealth Scientiﬁc and Research Organization (CSIRO)

Abstract. Compressive sensing/sampling (CS) has been one of the most active research in signal and image processing since it was proposed. The importance of CS is that it provides a high performance sampling theory for sparse signals or signals with sparse representation. CS has shown outstanding performances in many applications. In this paper we discuss two potential applications of CS in radio astronomy: image deconvolution and Faraday rotation measure synthesis. Both theoretical analysis and experimental results show that CS will bring radio astronomy to a brand new stage.

1

Introduction

Since compressive sensing/sampling (CS) was proposed, it has attracted very substantial interests. This new theory also has been applied in many research areas including: a single pixel camera [1], high performance Magnetic Resonance Imaging [2], a new spectral imaging prototype [3], Modulated Wideband Converter [4], image codec on Herschel satellite [5] and so on. There are some pioneers who have applied CS to the deconvolution problem in radio astronomy. For example, Wiaux et.al. in [6] compare the CS based deconvolution methods with the Hogbom CLEAN method [7] by assuming the target signal is sparse. In a subsequent paper, Wiaux and other authors propose a new spread spectrum technique for radio interferometry [8] by using the non-negligible and constant component of the antenna separation in the pointing direction. Since CS can measure a spare signal or a signal with a sparse representation with far fewer measurements than Nyquist-Shannon sampling theory, this promotes us to apply CS on the image deconvolution problem and the Faraday rotation measure synthesis problem which are cased by incomplete sampling. The contributions of this paper are: (1) a new CS based image deconvolution method, in which the Isotropic Undecimated Wavelet Transform (IUWT) [9] is used as a dictionary of basis functions, is introduced for extended sources and compared with traditional deconvolution methods; (2) a new application of CS for Faraday rotation measure synthesis is proposed.

2

Image Deconvolution in Radio Astronomy and CS

The fundamental principle is the van Cittert-Zernike theorem which says that the spatial coherence function is the Fourier transform of the sky brightness. A G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 352–359, 2010. c Springer-Verlag Berlin Heidelberg 2010

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radio interferometer (i.e. a pair of antennas) is used to measure the coherence of the sky within the same region. As described in the van Cittert-Zernike theorem, the radio telescope array measures the Fourier coeﬃcients of the sky brightness. This can be described as: M F I = V, (1) where M is the UV coverage mask matrix comprises of 0 and 1, denotes the Hadamard (element by element) product operator, F is the Fourier transform matrix, I is the sky brightness image in a vector format and V is the measured visibility data i.e. the Fourier coeﬃcients in a vector format. Since we can not capture all the visibility data using the telescope array, there are many 0s in the mask matrix M ; there are many unknowns in V . This can also be understood as an interpolation problem for V in the Fourier domain. Due to the global and highly oscillatory nature of the Fourier Transform, it is diﬃcult to estimate the Fourier coeﬃcient by using nearby Fourier coeﬃcients. If we transform (1) into the image domain, then it becomes a deconvolution problem. A sparse signal in the image domain, it will be spread out in the Fourier domain in which it is acquired. Based on the incoherent deﬁnition in [10], therefore, incoherent is 1 in this case. CS can be applied to radio astronomy in a straightforward manner, because the partial Fourier sensing matrix is perfect incoherent to the image domain [10]. This allows reconstruction of point sources and compact sources, these being sparse in the image domain. However, it will not work eﬀectively for some extended sources such as galaxies, gas emissions and nebulae. Fortunately, a sparse representation can also be achieved by using some dictionary functions such as the Total Variation, the Wavelet transform and the Undecimated Wavelet transform [9]. Therefore, as long as we can ﬁnd a suitable dictionary of basis functions ψ which can decompose the expanded sources into a sparse representation, the deconvolution procedure can be carried out as follows: (2) min αl1 s.t. M F ψα − V l2 ≤ , where I = ψα, α is a sparse representation with respect to the basis function dictionary ψ and describes the uncertainty about the observation V as in the situation where the measurements are contaminated with noise. In this paper, the isotropic undecimated wavelet transform (IUWT) [9] is used as the dictionary. IUWT algorithm is well known in the astronomical data analysis, because UDWT or RDWT (Undecimated/ Redundant Wavelet Transform) can well preserve the translation-invariance property, and most of sources in the universe are isotropic [11]. These L1 norm based problems with a sparse representation on IUWT are solved by using the inpainting C++ toolbox from Arnaud Woiselle and Jean-Luc Starck (private communication). A comparison is carried out between the IUWT-based-CS based deconvolution method introduced above and the traditional deconvolution methods. This experiment is based on simulating the Australian Square Kilometer Array Pathﬁnder (ASKAP) radio telescope. The UV coverage for this test can be seen in the leftmost image in ﬁgure 1 and its relevant dirty beam (point spread function whose maximum is 1) is shown in the second image from the left in

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ﬁgure 1. In this UV coverage, the white dots illustrate that the visibility data can be measured by the array. The true image is shown in the third image from the left in ﬁgure 1. Gaussian noise of the standard deviation of 0.01Jy is added to the true image, the relevant dirty map can be seen in the rightmost image in ﬁgure 1.

(a) UV coverage

(b) Dirty beam

(c) True

(d) Dirty map

Fig. 1. Data preparations

The deconvolution results of diﬀerent methods are displayed in ﬁgure 2. We can see that the results of the IUWT based CS deconvolution method show the sharpest edges in both the mode image and the restored image. Interestingly, readers might notice that in the last row in ﬁgure 2, the model image is much sharper than the restored image. Astronomers adopt the restored image rather than the model image, because the traditional deconvolution methods, for example, the Hogbom CLEAN can not provide reliable deconvolution results, and they try to make the restored images approximate the dirty maps. However, the combination of ASKAP and this new deconvolution method might change this, because ASKAP can provide a good UV coverage and CS based deconvolution methods are more robust than the traditional deconvolution methods.

3

Faraday Rotation Measure Synthesis and CS

Faraday rotation is a physical phenomenon. The angle of linear polarization radiation which propagates through magnetic ﬁelds is rotated as a function of frequency. As introduced in [14,15], Faraday rotation measure synthesis is an important tool to study magnetic ﬁelds of galaxies. In [16], Burn introduced the Faraday dispersion function F (φ) , which describes the intrinsic polarized ﬂux per unit Faraday depth φ, and its relationship with the complex polarized emission P (λ2 ) as follows: ∞ 2 F (φ)e2iφλ dφ, (3) P (λ2 ) = −∞

where λ is the wavelength. Note that, P can also be written as P = Q + iU , where Q and U represent the emission of Stokes Q and U , respectively. The Faraday depth φ of a source is deﬁned as:

The Applications of Compressive Sensing to Radio Astronomy

355

(a) Hogbom CLEAN [7]

(b) Multiscale CLEAN [12]

(c) MEM [13]

(d) IUWT based CS Fig. 2. Deconvolution results with noise of the standard deviation 0.01. They are ordered as follows: the columns from left to right are the model images, the residual images and the restored images, respectively. The model images and the restored images are displayed with range 0 Jy/pixel to 1 Jy/pixel; the residual images are truncated and displayed with range -0.01 Jy/pixel to +0.01 Jy/pixel.

356

F. Li, T.J. Cornwell, and F. De hoog

φ(r) = 0.81

observor

source

ne Bdr,

(4)

where B is the magnetic ﬁeld strength in micro-Gauss, ne is the electron density in cm−3 , and dr is an inﬁnitesimal path length in parsecs. The Faraday depth φ can be expressed in rad m−2 . The Faraday depth can either be positive (which implies a magnetic ﬁeld pointing towards the observer) or negative (which implies a magnetic ﬁeld pointing away from the observer). In order to study the magnetic ﬁelds of galaxies, we need to reconstruct the Faraday dispersion function, which is, in general, a complex-valued function of φ. From (3) , we can easily inverse the expression and have: ∞ 2 F (φ) = P (λ2 )e−2iφλ dλ2 . (5) −∞

However, the problem is that we can not observe the polarized emission at wavelengths where λ2 < 0. Even though for the wavelength range λ2 > 0, it is impossible to observe at all wavelengths. In [15], Brentjens and Bruyn proposed a new synthesis method by bringing an observing window function, W (λ2 ) in which there are none zero values at the observable wavelength and zeros otherwise. Therefore, the observed complex polarized emission can be described as: P (λ2 ) = W (λ2 )P (λ2 ).

(6)

Based on Brentjens & Bruyn’ method, the reconstructed Faraday rotation measure synthesis can be written in discrete format as follows: F (φ) ≈ K

N

2 2 P (λ2n )e−2iφ(λn −λ0 ),

(7)

n=1

where N is the number of channels of the observation, K is the sum of the window function W (λ2 ), and the parameter λ20 is the mean of the sampled values between λ21 and λ2N within the observation window W (λ2 ). By using this method, the Faraday dispersion function can be easily reconstructed provided that there is a good window function (spectral coverage) along the line of sight. However, the reconstructed results generally include some sidelobes. The relationship between the Faraday dispersion function and the observed polarized emission is a Fourier pair if λ2 = πu, where u is a wavelength related parameter. The task of the Faraday rotation measure synthesis is to reconstruct the Faraday dispersion function based on the incomplete Fourier coeﬃcients (the observed polarized emission). We can apply CS to this problem in a straightforward manner, if we assume that the Faraday dispersion function is a sparse signal. Since the Faraday dispersion function is a complex-valued function, we have the following solution for this problem: min {R(F (φ)), I(F (φ))l1 } s.t. P (λ2 ) = W (λ2 )P (λ2 ),

(8)

where R(F (φ)) and I(F (φ)) are the real and the imagery parts of the Faraday dispersion function, respectively.

The Applications of Compressive Sensing to Radio Astronomy

(a) True emission; horizontal axis is φ (rad m−2 )

(b) Complete polarized emission; horizontal axis is φ

(c) Observed (126 channels); horizontal axis is λ2

(d) Brentjens & Bruyn’ method; horizontal axis is φ

(e) Our CS method; horizontal axis is φ

(f) Our CS method based on random measurements (126 channels); x axle is φ

357

Fig. 3. We test our method on the standard data used in [15]. (a) is the true F (φ); (b) is the relevant polarized emission of the true F (φ); (c) is the measured emission through the 126 observing channels between 3.6cm to 50cm; (d) is the reconstructed by Brentjens & Bruyn’ method; (e) shows the reconstructed Faraday dispersion function by our CS based rotation measure method. As an extension test, if we use the same number of observing channels (126), but these channels are randomly selected from where λ2 are between (0,1) rather than (0.0362 , 0.52 ), the reconstructed F (φ) by our CS method is shown in ﬁgure 3(f). Black solid line indicates the real value; the red solid line indicates the imagery part and the blue dashed line indicates the amplitude.

358

F. Li, T.J. Cornwell, and F. De hoog

In order to test this CS based Faraday rotation measure method, we use the standard test introduced in [15]. There are three sources in a line of sight and the true F (φ) is shown in ﬁgure 3(a). We can see that the Faraday dispersion function in this test is a real value plot. Its relevant polarized emission can be seen in ﬁgure 3(b). Within this test, we assume that there are 126 observing channels within 3.6cm to 50cm. Through these 126 observing channels, what we measured in reality is shown in ﬁgure 3(c). Based on these incomplete observed polarization emission, the reconstructed Faraday dispersion function by Brentjens & Bruyn’ method is shown in ﬁgure 3(d). Figure 3(e) shows the reconstructed Faraday dispersion function by our CS based rotation measure method. Note that, black solid line shows the real value; the red solid line shows the imagery part and the blue dashed line shows the amplitude. As mentioned in [17], the magnitude of F (φ) indicates the polarized emission of the region with Faraday depth φ and its phase deﬁnes the intrinsic position angle. For the study of polarized emission of galaxies, the magnitude of F (φ) is very important, thus, Heald in [14] proposed a similar CLEAN method to get ride of the sidelobes. They just applied the matching pursuit idea to this one dimensional signal, to deconvolve the reconstructed Faraday dispersion function. However, for the study of orientation of the magnetic ﬁeld perpendicular to the line of sight, the phase information of F (φ) is crucial. Unfortunately, Brentjens & Bruyn’ method can not reconstruct a reliable phase information of F (φ). We can easily see in this test that our CS based rotation measure synthesis method not only can reconstruct better magnitude of F (φ) but also a more close approximation of the true phase than Brentjens & Bruyn’ method, because our CS reconstructed F (φ) shows no imagery part. We can see that there are still some distortion with the second and the third sources from the left in our reconstructed F (φ). The distortion depends on the number of observing channels, the randomness of these observing channels and the sparsity of the Faraday dispersion function. From the CS point of view, randomness is one of the key aspects. Suppose we use the same number of observing channels (126) in the previous test, but these channels are randomly selected from where λ2 are between (0,1) rather than (0.0362, 0.52 ), the reconstructed F (φ) by our CS method is shown in ﬁgure 3(f). Apparently, this reconstructed F (φ) is almost the same as the true Faraday dispersion function.

4

Conclusion

Compressive sensing can sample a signal with higher eﬃciency than the NyquistShannon sampling if the signal is sparse or there is sparse representation of the signal with respect to a given basis function dictionary. For radio astronomy, two potential applications of compressive sensing are discussed in this paper. A new CS based deconvolution method is introduced for deconvolving extended sources. Results show that this new deconvolution method provides outstanding reconstructions than those traditional deconvolution methods. Then, a new application of CS for Faraday rotation measure is discussed and tested. Results illustrate that this CS based Faraday rotation measure synthesis method

The Applications of Compressive Sensing to Radio Astronomy

359

can reconstruct both magnitude and phase information of a Faraday dispersion function accurately.

References 1. Wakin, M., Laska, J., Duarte, M., Baron, D.: An architecture for compressive imaging. In: International Conference on Image Processing (2006) 2. Lustig, M., Donoho, D., Pauly, J.: Sparse mri: The application of compressed sensing for rapid mr imaging. Magnetic Resonance in Medicine (2007) 3. Wagadarikar, A., John, R., Willett, R., Brady, D.: Single disperser design for coded aperture snapshot spectral imaging. Applied optics (2008) 4. Mishali, M., Eldar, Y.C., Dounaevsky, O., Shoshan, E.: Xampling: Analog to digital at sub-nyquist rates. eprint arXiv (2009) 5. Bobin, J., Starck, J.: Compressed sensing in astronomy and remote sensing: a data fusion perspective. In: Proceedings of SPIE (2009) 6. Wiaux, Y., Jacques, L., Puy, G., Scaife, A., Vandergheynst, P.: Compressed sensing imaging techniques for radio interferometry. arXiv astro-ph (2009) 7. H¨ ogbom, J.: Aperture synthesis with a non-regular distribution of interferometer baselines. Astronomy and Astrophysics Supplement 15, 417 (1974) 8. Wiaux, Y., Puy, G., Boursier, Y., Vandergheynst, P.: Spread spectrum for imaging techniques in radio interferometry. arXiv astro-ph (2009) 9. Starck, J., Fadili, J., Murtagh, F.: The undecimated wavelet decomposition and its reconstruction. IEEE Transactions on Image Processing 16, 297 (2007) 10. Candes, E., Romberg, J., Tao, T.: Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. eprint arXiv, 9186 (2004) 11. Starck, J., Murtagh, F.: Astronomical image and data analysis. Springer, Heidelberg (2006) 12. Cornwell, T.: Multi-scale clean deconvolution of radio synthesis images. arXiv astro-ph (2008) 13. Cornwell, T., Evans, K.: A simple maximum entropy deconvolution algorithm. Astronomy and Astrophysics 143, 77–83 (1985) 14. Heald, G.: The faraday rotation measure synthesis technique. Cosmic Magnetic Fields: From Planets 259, 591 (2009) 15. Brentjens, M., de Bruyn, A.: Faraday rotation measure synthesis. Astronomy and Astrophysics 441, 1217 (2005) 16. Burn, B.: On the depolarization of discrete radio sources by faraday dispersion. Monthly Notices of the Royal Astronomical Society 133, 67 (1966) 17. Frick, P., Sokoloﬀ, D., Stepanov, R., Beck, R.: Wavelet-based faraday rotation measure synthesis. Monthly Notices of the Royal Astronomical Society: Letters 401, L24 (2010)

Compressive Sensing for Autoregressive Hidden Markov Model Signal Ji Wu1 , Qilian Liang1 , and Zheng Zhou2 1

Department of Electrical Engineering, University of Texas at Arlington, USA [email protected], [email protected] 2 School of Information and Communication Engineering Beijing University of Posts and Telecommunications Beijing, 100876, China [email protected]

Abstract. Compressive sensing(CS) is an emerging ﬁled based on the revelation that a small collection of linear projections of a sparse signal contains enough information for stable, sub-Nyquist signal acquisition. One challenging problem in compressive sensing is that it is diﬃcult to represent signal in sparse basis, which makes this algorithm sometimes impractical. In this paper, we can setup a new standard compressive sensing problem for autoregressive hidden markov signal by utilizing the original observation vector and the autoregressive coeﬃcients. Keywords: compressive sensing, coeﬃcient estimation, hidden markov model.

1

Introduction

Compressive sensing (CS) is a new method to capture and represent compressible signals at a rate signiﬁcantly below the Nyquist rate. It employs nonadaptive linear projections that preserve the structure of the signal; the signal is then reconstructed from these projections using an optimization process. This leads immediately to new signal reconstruction methods that are successful with surprisingly few measurements, which in turn leads to signal acquisition methods that eﬀect compression as part of the measurement process (hence “compressive sensing”). One challenging problem in compressive sensing is that it is diﬃcult to represent certain signals in sparse basis, which makes this novel algorithm sometimes impractical. We try to apply compressive sensing to signal which follows autoregressive hidden markov model. Typically, it is neither sparse nor approximately sparse. This problem appears impossible to solve at ﬁrst glance. However, by exploring the property of autoregressive hidden markov model and rearranging the sensing matrix Φ, we can setup a new standard compressive sensing problem, i.e., Φ s = y , where Φ , y can be constructed by entries of Φ and y. We can also guarantee s is always sparse. Autoregressive coeﬃcients is important to implement this novel algorithm. G. Pandurangan et al. (Eds.): WASA 2010, LNCS 6221, pp. 360–363, 2010. c Springer-Verlag Berlin Heidelberg 2010

Compressive Sensing for Autoregressive Hidden Markov Model Signal

2

361

Problem Formulation

Our objective is to reconstruct a speech signal xt which can be represented by using autoregressive model from a number of linear and non-adaptive measurements. An typical p-order autoregressive model can be represented as follows xt +

p

αi xt−i = st

(1)

i=1

where αi is the ith corresponding coeﬃcient and st is a sparse signal. We assume the vector s = [s0 , s1 , · · · , sn−1 ]T is k-sparse, that is, there are only k entries in s are non-zero. In order to recovery xt , we need to ﬁnd out the AR coeﬃcients α = [α0 , α1 , · · · , αn−1 ]T and sparse vector s = [s0 , s1 , · · · , sn−1 ]T from the original signal xt and the new measurement y. Note that in standard compressive sensing(CS) scenario, the signal xt is assumed to be sparse in some known orthogonal basis. Sometimes it is diﬃcult to ﬁnd the appropriate sparsity basis. However, in our problem, both the autoregressive model and the basis are unknown and the main contribution of this paper is to solve this new problem eﬃciently. In the standard compressive sensing equation, vector x is the sparse signal. While in our case, s is the sparse signal that needs further investigation. Based on (1), s can be represented by linear combinations of vector x multiply by some coeﬃcients αi . This relation motivates us to design a new sensing matrix Φ which is composed of entries of standard sensing matrix Φ. When we apply Φ to the sparse signal s, the new observation vector y could also be represented by the linear combination of observation vector y. It implies that sensing matrix Φ should follow two conditions: 1) The (m − 1)th row in the sensing matrix Φ is a shift version of the mth row. 2)The (m − 1)th row contains one more non-zero entries than the mth row. In order to meet these two conditions, we can specify the sensing matrix Φ and Φ as follows ⎤ ⎡ gn−m gn−m−1 · · · g0 · · · 00 ⎢ gn−m+1 gn−m · · · g1 g0 · · · 0 ⎥ ⎥ ⎢ (2) Φ = ⎢. ⎥ .. . . .. . . . . ⎣ .. . . . . 0⎦ . gn−1 gn−2 · · · · · · · · · g1 g0 ⎡

gn−m+p−1 ⎢ gn−m+p ⎢ Φ = ⎢ . ⎣ .. gn−1

⎤ g0 · · · 00 g1 g0 · · · 0 ⎥ ⎥ ⎥ .. . . . . . . . 0⎦

gn−m+p−2 gn−m+p−1 .. .

··· ··· .. .

gn−2

· · · · · · · · · g1 g0

(3)

Compare with those two sensing matrix, it is easy to ﬁnd out Φ is a submatrix of Φ which is composed of last (m − p) rows of Φ. By applying the sensing matrix Φ to sparse signal z(t), we can get a new observation vector y which contains (m − p) entries.

362

J. Wu, Q. Liang, and Z. Zhou

⎡

yp ⎢ yp+1 ⎢ y = ⎢ . ⎣ .. ym−1

yp−1 · · · y1 yp · · · y2 .. . . .. .. . ym−2 · · · ym−p

y0 y1 .. . ym−p−1

⎤⎡

⎤ 1 ⎥ ⎢ α1 ⎥ ⎥⎢ ⎥ ⎥ ⎢ .. ⎥ = Y α ⎦⎣. ⎦

(4)

αp

where α = [1, α1 , · · · , αp ]T . αp is the coeﬃcients of autoregressive model which will be discussed in the next section. Finally, we can formulate a new compressive sensing problem with regard to s = [s0 , s1 , · · · , sn−1 ]T as follows Φ s = y

(5)

where s is the k-sparse signal we interest in. It can be solved eﬃciently by linear programming algorithm.

3

Autoregressive Coeﬃcients Estimation

First, we deﬁne jth autocovariance of signal x: cj =< xt+j xt >, where <> denotes expectance. And rj = cj /c0 . Then we get ⎡ ⎤⎡ ⎤ ⎤ ⎡ α1 r1 1 r1 r2 · · · rp−2 rp−1 ⎢ r2 ⎥ ⎢ r1 ⎢ ⎥ ⎥ 1 r · · · r r 1 p−3 p−2 ⎥ ⎢ α2 ⎢ ⎥ ⎢ ⎥ ⎢ .. ⎢ ⎥ ⎥ ⎢ ⎥ . . . .. .. (6) ⎢. ⎥ ⎢ .. ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎥ ⎢ ⎣ rp−1 ⎦ ⎣ rp−2 rp−3 rp−4 · · · 1 r1 ⎦ ⎣ αp−1 ⎦ rp rp−1 rp−2 rp−3 · · · r1 1 αp or succinctly r = RΛ

(7)

Note that this is a well-posed system(with a square coeﬃcients matrix R), i.e., with the same number of constraints (equations, R’s rows) as unknowns (the elements αj of the unknown vector Λ). Further, R is full-rank and symmetric, so that invertability is guaranteed, Λˆ = R−1 r

4

(8)

Numerical Results

Fig.1 shows the recovered sparse signal st from noisy case, i.e., y = Φ s + n and SN R = 15dB. As you can see, a large number of the reconstructed samples have small value, hence we can setup a threshold to ﬁlter st and it will become sparse again (Most of the samples will force to 0 after ﬁltering). The only distortion is occurred when the amplitude of sample is large. Since few samples have large amplitude and SN R is high, its eﬀect on our signal recovery procedure is truly negligible. We can also say that O(klog(n)) measurements are suﬃcient to reliably recover xt from standard compressive sensing algorithm.

Compressive Sensing for Autoregressive Hidden Markov Model Signal

363

Sparse signal st 4

Estimated with noise Original sparse signal

3 2

Amplitude

1 0

−1 −2 −3 −4 0

100

200

300

400

500 600 Sample Index

700

800

900

1000

Fig. 1. Comparison of estimated sparse singal st from noisy measurements with original signal

Acknowledgment This work was supported in part by National Science Foundation (NSF) under Grants CNS-0721515, CNS-0831902, CCF-0956438, CNS-0964713, and Oﬃce of Naval Research (ONR) under Grants N00014-07- 1-0395 and N00014-07-1-1024.

References 1. Cand´es, E., Romberg, J., Tao, T.: Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. on Information Theory 52(2), 485–509 (2006) 2. Tropp, J., Gilbert, A.C.: Signal recovery from partial information via orthogonal matching pursuit. IEEE Trans. on Information Theory 53(12), 4655–4666 (2007) 3. Donoho, D.: Compressed Sensing. IEEE Trans. on Information Theory 52(4), 1289–1306 (2006) 4. Baranuik, R.: Compressive Sensing. IEEE Signal Processing Magazine 24(4), 118–121 (2007) 5. Cand´es, E., Wakin, M.: An introduction to compressive sensing. IEEE Signal Processing Magazine, 21–30 (March 2008) 6. Baranuik, R., Davenport, M., Devore, R., Wakin, M.: A simple proof of the restricted isometry property for random matrices. Constructive Approximation 28(3), 253–263 (2008) 7. Cand´es, E., Romberg, J.: Sparse and incoherence in compressive sampling. Inverse problem 23(3), 969–985 (2007) 8. Bertsimas, D., Tsitsiklis, J.N.: Introduction to Linear Optimization. Athena Scientiﬁc (February 1997) 9. Donoho, D.L., Huo, X.: Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inf. Theory 47, 2845–2862 10. Cand´es, E., Tao, T.: Near optimal signal recovery form random projections: Universal encoding strategies? (2004) (preprint) 11. Cand´es, E., Romberg, J., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics 59(8), 1207–1223 (2006)

Author Index

Anjali, Tricha Bolla, Jason

203 57

Cao, Jiannong 192 Cheng, Xiuzhen 36 Cheng, Yu 203 Chen, Honghui 110 Chen, Tao 110 Chen, Wenping 21, 80 Chen, Ying 120 Cheung, S.C. 115 Chiang, Kun-Lin 253 Cornwell, Tim J. 352 Cui, Yong 36 Cui, Yonggang 261, 282 De hoog, Frank 352 Deng, Yu 90 Dong, Qunfeng 225 Dong, Wei 136 Du, Junzhao 172 Fan, Suqin 90 Fu, Hong 105

Lam, Nhat X. 57 Lau, Francis C.M. 1 Lee, Junghoon 287, 295 Lee, Wonjun 167 Liang, Qilian 182, 338, 360 Li, Bin 299, 314, 330 Li, Dejian 330 Li, Deying 21, 80 Li, Fan 95 Li, Feng 352 Li, Jiang 157 Li, Shufang 115 Li, Tongtong 235 Liu, Bin 307 Liu, Hui 21, 172 Liu, Limin 261 Liu, Yujun 261, 274, 282 Li, Wei 36 Li, Xue 209 Li, Yingshu 105 Li, Yinjuan 274 Li, Zi 125 Lou, Wei 192 Luo, Xueshan 110 Ma, Huan 21 Ma, Junchao 192 Mao, Lei 172

Gao, Shunxi 90 Guo, Deke 110, 172 Hao, Xiaohong 1 He, Yuan 26 Hong, Weijun 115 Hong, Yi 80 Huang, Liusheng 26, 225 Huang, Minsu 95 Huang, Wei 90 Hua, Qiang-Sheng 1 Huynh, D.T. 57

Nguyen, Trac N.

57

Park, Gyung-Leen Park, Namje 245 Peng, Yang 125

287

Qiang, Sheng 266 Qiao, Daji 125 Qi, Yong 120

Jiang, Ting 343 Jiao, Shuhong 220 Jiao, Xianlong 192

Ren, Jian 235 Roh, Heejun 167

Kang, Mikyung 295 Kim, Kyunghwi 167 Kirachaiwanich, Davis

Shen, Zhen 266 Shih, Hung-Cheng 75 Shila, Devu Manikantan

182

203

366

Author Index

Song, Youjin 245 Su, Ming-Yang 253 Sun, Guodong 146 Tang, Shaojie 68 Tang, Xiaoming 274 Tang, Xing 172 Tan, Haisheng 1 Tian, Ye 157 Wang, Guanghui 11 Wang, Kuochen 75 Wang, Lixin 47 Wang, Nannan 11 Wang, Pei 26, 225 Wang, Shengling 36 Wang, Shubin 299, 314 Wang, Xiaodong 192 Wang, Xiaoming 105 Wang, Xu-hui 322 Wang, Yu 95 Wang, Yuexuan 1 Wan, Peng-Jun 47, 68 Wei, Feng 291 Wei, Ning 172 Wu, Ji 360 Wu, Qiwu 90 Wu, Zhiqiang 209 Xiang, Yingchang 291 Xiao, Mingjun 225

Xie, Bing 209 Xi, Min 136 Xu, Bin 146 Xu, Lei 338 Xu, Xiaohua 68 Xu, Xiaoli 220 Yang, Huiqiang 80 Yao, Frances 47 Yao, Qingsong 120 Yi, Dong-yun 266 Yu, Jiguo 11 Zhai, Shijun 343 Zhang, Bowu 291 Zhang, Gesen 220 Zhang, Lei 235 Zhang, Long 90 Zhang, Wensheng 125 Zhao, Cheng-lin 322 Zhao, Feng 330 Zhao, Xi-bin 136 Zhao, Yiyang 115 Zhao, Yuting 274 Zhong, Xiao 120 Zhou, Chi 209 Zhou, Xiangqian 209 Zhou, Xingming 192 Zhou, Zheng 299, 314, 330, 360 Zhu, Qinghua 80 Zou, Weixia 299, 314, 330

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Advances in Swarm Intelligence: First International Conference, ICSI 2010, Beijing, China, June 12-15, 2010, Proceedings, Part II (Lecture Notes in ... Computer Science and General Issues)

Advances in Swarm Intelligence: First International Conference, ICSI 2010, Beijing, China, June 12-15, 2010, Proceedings, Part I (Lecture Notes in ... Computer Science and General Issues)

Frontiers in Algorithmics: 4th International Workshop, FAW 2010, Wuhan, China, August 11-13, 2010, Proceedings (Lecture Notes in Computer Science Theoretical Computer Science and General Issues)

CONCUR 2010 - Concurrency Theory: 21th International Conference, CONCUR 2010, Paris, France, August 31-September 3, 2010, Proceedings (Lecture Notes ... Computer Science and General Issues)

Artificial Immune Systems: 9th International Conference, ICARIS 2010, Edinburgh, UK, July 26-29, 2010, Proceedings (Lecture Notes in Computer Science Theoretical Computer Science and General Issues)

Algorithms and Complexity: 7th International Conference, CIAC 2010, Rome, Italy, May 26-28, 2010, Proceedings (Lecture Notes in Computer Science)

Advances in Grid and Pervasive Computing: 5th International Conference, CPC 2010, Hualien, Taiwan, May 10-13, 2010, Proceedings (Lecture Notes in ... Computer Science and General Issues)

Language and Automata Theory and Applications: 4th International Conference, LATA 2010, Trier, Germany, May 24-28, 2010, Proceedings (Lecture Notes in ... Computer Science and General Issues)

Computer Science -- Theory and Applications: 5th International Computer Science Symposium in Russia, CSR 2010, Kazan, Russia, June 16-20, 2010, ... Computer Science and General Issues)

Reachability Problems: 4th International Workshop, RP 2010, Brno, Czech Republic, August 28-29, 2010. Proceedings (Lecture Notes in Computer Science Theoretical Computer Science and General Issues)

Network and Parallel Computing: IFIP International Conference, NPC 2010, Zhengzhou, China, September 13-15, 2010, Proceedings (Lecture Notes in ... Computer Science and General Issues)

Mathematical Foundations of Computer Science 2010: 35th International Symposium, MFCS 2010, Brno, Czech Republic, August 23-27, 2010, Proceedings ... Computer Science and General Issues)

Biomedical Simulation: 5th International Symposium, ISBMS 2010, Phoenix, AZ, USA, January 23-24, 2010. Proceedings (Lecture Notes in Computer Science Theoretical Computer Science and General Issues)

Biomedical Simulation: 5th International Symposium, ISBMS 2010, Phoenix, AZ, USA, January 23-24, 2010. Proceedings (Lecture Notes in Computer Science Theoretical Computer Science and General Issues)

Theory and Applications of Satisfiability Testing - SAT 2010: 13th International Conference, SAT 2010, Edinburgh, UK, July 11-14, 2010, Proceedings ... Computer Science and General Issues)

Sequences and Their Applications - SETA 2010: 6th International Conference, Paris, France, September 13-17, 2010. Proceedings (Lecture Notes in ... Computer Science and General Issues)

Unconventional Computation: 9th International Conference, UC 2010, Tokyo, Japan, June 21-25, 2010, Proceedings (Lecture Notes in Computer Science Theoretical Computer Science and General Issues)

Interactive Theorem Proving: First International Conference, ITP 2010 Edinburgh, UK, July 11-14, 2010, Proceedings (Lecture Notes in Computer Science Theoretical Computer Science and General Issues)

WALCOM: Algorithms and Computation: 4th International Workshop, WALCOM 2010, Dhaka, Bangladesh, February 10-12, 2010, Proceedings (Lecture Notes in Computer ... Computer Science and General Issues)

Applications and Theory of Petri Nets: 31st International Conference, PETRI NETS 2010, Braga, Portugal, June 21-25, 2010, Proceedings (Lecture Notes ... Computer Science and General Issues)

Parameterized and Exact Computation: 5th International Symposium, IPEC 2010, Chennai, India, December 13-15, 2010. Proceedings (Lecture Notes in Computer ... Computer Science and General Issues)

Developments in Language Theory: 14th International Conference, DLT 2010, London, ON, Canada, August 17-20, 2010, Proceedings (Lecture Notes in ... Computer Science and General Issues)

Graph Transformations: 5th International Conference, ICGT 2010, Twente, The Netherlands, September 27--October 2, 2010, Proceedings (Lecture Notes in ... Computer Science and General Issues)

Architecture of Computing Systems - ARCS 2010: 23rd International Conference, Hannover, Germany, February 22-25, 2010, Proceedings (Lecture Notes in ... Computer Science and General Issues)

Applied Cryptography and Network Security: 8th International Conference, ACNS 2010, Beijing, China, June 22-25, 2010, Proceedings (Lecture Notes in Computer Science Security and Cryptology)

Evolvable Systems: From Biology to Hardware: 9th International Conference, ICES 2010, York, UK, September 6-8, 2010, Proceedings (Lecture Notes in ... Computer Science and General Issues)

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