Advertisement

Abstract

In the network localization problem the locations of some nodes (called anchors) as well as the distances between some pairs of nodes are known, and the goal is to determine the location of all nodes. The localization problem is said to be solvable (or uniquely localizable) if there is a unique set of locations consistent with the given data. Recent results from graph rigidity theory made it possible to characterize the solvability of the localization problem in two dimensions.

In this paper we address the following related optimization problem: given the set of known distances in the network, make the localization problem solvable by designating a smallest set of anchor nodes. We develop a polynomial-time 3-approximation algorithm for this problem by proving new structural results in graph rigidity and by using tools from matroid theory.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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. Trans. Mobile Computing (to appear)Google Scholar
  2. 2.
    Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 78–89. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Eren, T., Goldenberg, D., Whiteley, W., Yang, Y.R., Morse, A.S., Anderson, B.D.O., Belhumeur, P.N.: Rigidity, Computation, and Randomization in Network Localization. In: Proc. of the IEEE INFOCOM Conference, Hong-Kong, March 2004, pp. 2673–2684 (2004)Google Scholar
  4. 4.
    Fekete, Z., Jordán, T., Uniquely localizable networks with few anchors, EGRES TR-2006-07, http://www.egres.hu/egres/
  5. 5.
    Hendrickson, B.: Conditions for unique graph realizations. SIAM J. Comput. 21(1), 65–84 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hopcroft, J.E., Tarjan, R.E.: Dividing a graph into triconnected components. SIAM J. Comput. 2, 135–158 (1973)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Jackson, B., Jordán, T.: Connected rigidity matroids and unique realizations of graphs. J. Combinatorial Theory, Ser. B. 94, 1–29 (2005)zbMATHCrossRefGoogle Scholar
  8. 8.
    Jackson, B., Jordán, T., Szabadka, Z.: Globally linked pairs of vertices in equivalent realizations of graphs. Discrete and Computational Geometry 35, 493–512 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Laman, G.: On graphs and rigidity of plane skeletal structures. J. Engineering Math. 4, 331–340 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lorea, M.: Hypergraphes et matroides. Cahiers Centre Etud. Rech. Oper. 17, 289–291 (1975)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Lovász, L.: The matroid matching problem. In: Algebraic methods in graph theory, Proc. Conf. Szeged (1978)Google Scholar
  12. 12.
    Lovász, L., Yemini, Y.: On generic rigidity in the plane. SIAM J. Algebraic Discrete Methods 3(1), 91–98 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Makai, M.: Matroid matching with Dilworth truncation. In: Proc. EuroComb 2005, Berlin (2005)Google Scholar
  14. 14.
    Recski, A.: Matroid theory and its applications in electric network theory and in statics, Akadémiai Kiadó, Budapest (1989)Google Scholar
  15. 15.
    Schrijver, A.: Combinatorial Optimization. Springer, Berlin (2003)zbMATHGoogle Scholar
  16. 16.
    Man-Cho So, A., Ye, Y.: Theory of semidefinite programming for sensor network localization. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (2005)Google Scholar
  17. 17.
    Whiteley, W.: Some matroids from discrete applied geometry. In: Bonin, J.E., Oxley, J.G., Servatius, B. (eds.) Matroid theory, Seattle, WA. Contemp. Math., vol. 197. Amer. Math. Soc., Providence (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Zsolt Fekete
    • 1
  • Tibor Jordán
    • 2
  1. 1.Computer and Automation Institute, Hungarian Academy of Sciences 
  2. 2.Department of Operations ResearchEötvös UniversityBudapestHungary

Personalised recommendations