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Conditions for Correct Sensor Network Localization Using SDP Relaxation

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Discrete Geometry and Optimization

Part of the book series: Fields Institute Communications ((FIC,volume 69))

Abstract

A Semidefinite Programming (SDP) relaxation is an effective computational method to solve a Sensor Network Localization problem, which attempts to determine the locations of a group of sensors given the distances between some of them. In this paper, we analyze and determine new sufficient conditions and formulations that guarantee that the SDP relaxation is exact, i.e., gives the correct solution. These conditions can be useful for designing sensor networks and managing connectivities in practice. Our main contribution is threefold: First, we present the first non-asymptotic bound on the connectivity (or radio) range requirement of randomly distributed sensors in order to ensure the network is uniquely localizable with high probability. Determining this range is a key component in the design of sensor networks, and we provide a result that leads to a correct localization of each sensor, for any number of sensors. Second, we introduce a new class of graphs that can always be correctly localized by an SDP relaxation. Specifically, we show that adding a simple objective function to the SDP relaxation model will ensure that the solution is correct when applied to a triangulation graph. Since triangulation graphs are very sparse, this is informationally efficient, requiring an almost minimal amount of distance information. Finally, we analyze a number of objective functions for the SDP relaxation to solve the localization problem for a general graph.

Research supported in part by AFOSR Grant FA9550-12-1-0396.

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Author information

Authors and Affiliations

  1. Department of Management Science and Engineering, Huang Engineering Center, Stanford University, 475 Via Ortega, Stanford, CA, 94305, USA

    Davood Shamsi

  2. IBM Research, Smarter Cities Technology Centre, Dublin, Ireland

    Nicole Taheri

  3. IBM Corporation, 1001 East Hillsdale Boulevard, Suite 400, Foster City, CA, 94404-1642, USA

    Zhisu Zhu

  4. Department of Management Science and Engineering, Huang Engineering Center 308, School of Engineering, Stanford University, 475 Via Ortega, Stanford, CA, 94305, USA

    Yinyu Ye

Authors
  1. Davood Shamsi
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  2. Nicole Taheri
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  3. Zhisu Zhu
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  4. Yinyu Ye
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Corresponding author

Correspondence to Nicole Taheri .

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Editors and Affiliations

  1. , Mathematics & Statistics, University of Calgary, University Drive NW, Calgary, T2N 1N4, Alberta, Canada

    Karoly Bezdek

  2. , Department of Computing, McMaster University, Main Street West 1280, Hamilton, L8S 4K1, Ontario, Canada

    Antoine Deza

  3. School of Engineering, Department of Management Science and Eng, Stanford University, Via Ortega 475, Stanford, 94305, California, USA

    Yinyu Ye

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Shamsi, D., Taheri, N., Zhu, Z., Ye, Y. (2013). Conditions for Correct Sensor Network Localization Using SDP Relaxation. In: Bezdek, K., Deza, A., Ye, Y. (eds) Discrete Geometry and Optimization. Fields Institute Communications, vol 69. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00200-2_16

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