Advertisement

Conditions for Correct Sensor Network Localization Using SDP Relaxation

Chapter
Part of the Fields Institute Communications book series (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.

Key words

Sensor network localization Graph realization Semidefinite programming 

Subject Classifications

90C22 90C46 90C90 90B18 

References

  1. 1.
    Alfakih, A.Y.: On the universal rigidity of generic bar frameworks. Contrib. Discrete Math. 5(3), 7–17 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alfakih, A.Y., Khandani, A., Wolkowicz, H.: Solving euclidean distance matrix completion problems via semidefinite programming. Comput. Optim. Appl. 12, 13–30 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Angluin, D., Aspnes, J., Chan, M., Fischer, M.J., Jiang, H., Peralta, R.: Stably computable properties of network graphs. In: Prasanna, V.K., Iyengar, S., Spirakis, P., Welsh, M. (eds.) Proceedings of the First IEEE International Conference Distributed Computing in Sensor Systems, DCOSS 2005, Marina del Rey, CA, USE, June/July, 2005. Lecture Notes in Computer Science, vol. 3560, pp. 63–74. Springer (2005)Google Scholar
  4. 4.
    Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M.J., Peralta R.: Computation in networks of passively mobile finite-state sensors. Distrib. Comput. 18(4), 235–253 (2006)zbMATHCrossRefGoogle Scholar
  5. 5.
    Araújo, F., Rodrigues, L.: Fast localized delaunay triangulation. In: Higashino, T. (ed.) Principles of Distributed Systems, vol. 3544, pp. 81–93. Springer, Berlin/Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Aspnes, J., Eren, T., Goldenberg, D.K., Morse, A.S., Whiteley, W., Yang, Y.R., Anderson, B.D.O., Belhumeur, P.N.: A theory of network localization. IEEE Trans. Mob. Comput. 5(12), 1663–1678, (2006)CrossRefGoogle Scholar
  7. 7.
    Aspnes, J., Goldenberg, D., Yang, Y.R.: On the computational complexity of sensor network localization. In: First International Workshop on Algorithmic Aspects of Wireless Sensor Networks. Lecture Notes in Computer Science, vol. 3121 pp. 32–44. Springer (2004)Google Scholar
  8. 8.
    Badoiu, M., Demaine, E.D., Hajiaghayi, M., Indyk, P.: Low-dimensional embedding with extra information. Discrete Comput. Geom. 36(4), 609–632 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Belk M., Connelly, R.: Realizability of graphs. In: Discrete and Computational Geometry, Springer-Verlag, vol. 37, pp. 7125–7137 (2007)MathSciNetGoogle Scholar
  10. 10.
    Biswas, P., Lian, T., Wang, T., Ye, Y.: Semidefinite programming based algorithms for sensor network localization. In: IPSN, ACM Transactions on Sensor Networks (TOSN), Berkeley, pp 46–54 (2004).Google Scholar
  11. 11.
    Biswas P., Ye, Y.: Semidefinite programming for ad hoc wireless network localization. In: IPSN, Berkeley, pp. 46–54 (2004)Google Scholar
  12. 12.
    Bruck, J., Gao, J., Jiang, A.: Localization and routing in sensor networks by local angle information. In: MobiHoc, the 6th ACM international symposium on Mobile ad hoc networking and computing, pp. 181–192. (2005)Google Scholar
  13. 13.
    Bulusu, N., Heidemann, J., Estrin, D.: Gps-less low-cost outdoor localization for very small devices. IEEE Pers. Commun. 7(5), 28–34 (2000)CrossRefGoogle Scholar
  14. 14.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86(1–3), 165–177 (1991)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Connelly, R.: Generic global rigidity. Discrete Comput. Geom. 33(4), 549–563 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Crippen G.M., Havel, T.F.: Distance geometry and molecular conformation. In: Chemometrics Series, Research Studies Press Ltd., Taunton, Somerset, England, volume 15, 1988.Google Scholar
  17. 17.
    Doherty, L., Pister, K.S.J., El Ghaoui, L.: Convex position estimation in wireless sensor networks. In: IEEE INFOCOM, Anchorge, vol. 3, pp. 1655–1663 (2001)Google Scholar
  18. 18.
    Eren, T., Goldenber, E.K., Whiteley, W., Yang, Y.R.: Rigidity, computation, and randomization in network localization. In: IEEE INFOCOM, Hong Kong, vol. 4, pp. 2673–2684 (2004)Google Scholar
  19. 19.
    Gortler, S.J., Healy, A.D., Thurston, D.P: Characterizing generic global rigidity. Am. J. Math. vol. 4 pp. 897–939 (2010)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hendrickson, B.: Conditions for unique graph realizations. SIAM J. Comput. 21(1), 65–84 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Gao, J., Bruck, J., Jiang, A.A.: Localization and routing in sensor networks by local angle information. ACM Trans. Sens. Networks 5(1), 181–192 (2009)Google Scholar
  22. 22.
    Jackson B., Jordan, T.: Connected rigidity matroids and unique realizations of graphs. J. Comb. Theory Ser. B 94(1), 1–29 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Javanmard, A., Montanari, A.:Localization from incomplete noisy distance measurements. (2011). http://arxiv.org/abs/1103.1417v3
  24. 24.
    Krislock, N., Wolkowicz, H.: Explicit sensor network localization using semidefinite representations and clique reductions. SIAM J. Optim. 20(5), 2679–2708 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Li, X.: Wireless Ad Hoc and Sensor Networks: Theory and Applications. Cambridge University Press, New York (2008)CrossRefGoogle Scholar
  26. 26.
    Li, X.Y., Calinescu, G., Wan, P.J., Wang, Y.: Localized delaunay triangulation with application in ad hoc wireless networks. IEEE Transaction on Parallel and Distributed Systems, 14(10), 1035–1047 (2003)CrossRefGoogle Scholar
  27. 27.
    Priyantah, N.B., Balakrishnana, H., Demaine, E.D., Teller, S.: Mobile-assisted localization in wireless sensor networks. In: IEEE INFOCOM, Miami, vol. 1, pp. 172–183 (2005)Google Scholar
  28. 28.
    Recht, B., Fazel, M, Parrilo, P.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52(3), 471–501 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Savvides, A., Han, C.C., Strivastava, M.B., Dynamic fine-grained localization in ad-hoc networks of sensors. In: MobiCom, Rome, pp. 166–179 (2001)Google Scholar
  30. 30.
    So, A.M.C., Ye, Y.: Theory of semidefinite programming for sensor network localization. In: Symposium on Discrete Algorithms, Mathematical Programming, Vancouver, Springer-verlag, 109(2–3), 367–384 (2007)MathSciNetzbMATHGoogle Scholar
  31. 31.
    So, A.M.C., Ye, Y.: A semidefinite programming approach to tensegrity theory and realizability of graphs. In: Symposium on Discrete Algorithms, Miami, pp. 766–775 (2006)Google Scholar
  32. 32.
    Tseng, P.: Second-order cone programming relaxation of sensor network localization. SIAM J. Optim. 18(1), 156–185 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Wang, Z., Zheng, S., Ye, Y., Boyd, S.: Further relaxations of the semidefinite programming approach to sensor network localization. SIAM J. Optim. 19, 655–673 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Yang, Z., Liu, Y., Li, X.Y.: Beyond trilateration: on the localizability of wireless ad-hoc networks. In: IEEE INFOCOM, Rio de Janeiro, pp. 2392–2400 (2009)Google Scholar
  35. 35.
    Zhu, Z., So, A.M.C., Ye,Y.: Universal rigidity: towards accurate and efficient localization of wireless networks. In: IEEE INFOCOM, San Diego (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Department of Management Science and Engineering, Huang Engineering CenterStanford UniversityStanfordUSA
  2. 2.IBM Research, Smarter Cities Technology CentreDublinIreland
  3. 3.IBM CorporationFoster CityUSA
  4. 4.Department of Management Science and Engineering, Huang Engineering Center 308, School of EngineeringStanford UniversityStanfordUSA

Personalised recommendations