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Building (1 − ε) Dominating Sets Partition as Backbones in Wireless Sensor Networks Using Distributed Graph Coloring

  • Dhia Mahjoub
  • David W. Matula
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6131)

Abstract

We recently proposed in [19,20] to use sequential graph coloring as a systematic algorithmic method to build (1 − ε) dominating sets partition in Wireless Sensor Networks (WSN) modeled as Random Geometric Graphs (RGG). The resulting partition of the network into dominating and almost dominating sets can be used as a series of rotating backbones in a WSN to prolong the network lifetime for the benefit of various applications. Graph coloring algorithms in RGGs offer proven constant approximation guarantees on the chromatic number. In this paper, we demonstrate that by combining a local vertex ordering with the greedy color selection strategy, we can in practice, minimize the number of colors used to color an RGG within a very narrow window of the chromatic number and concurrently also obtain a domatic partition size within a competitive factor of the domatic number. We also show that the minimal number of colors results in the first (δ + 1) color classes being provably dense enough to form independent sets that are (1 − ε) dominating. The resulting first (δ + 1) independent sets, where δ is the minimum degree of the graph, are shown to cover typically over 99% of the nodes (e.g. ε< 0.01), with at least 20% being fully dominating. These independent sets are subsequently made connected through virtual links using localized proximity rules to constitute planar connected backbones. The novelty of this paper is that we extend our recent work in [20] into the distributed setting and present an extensive experimental evaluation of known distributed coloring algorithms to answer the (1 − ε) dominating sets partition problem. These algorithms are both topology and geometry-based and yield O(1) times the chromatic number. They are also shown to be inherently localized with running times in O(Δ) where Δ is the maximum degree of the graph.

Keywords

Domatic partition problem (1 − ε) dominating sets partition Wireless Sensor Network Graph coloring Distributed Algorithm 

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References

  1. 1.
    Alzoubi, K.M., Wan, P., Frieder, O.: Message-optimal connected dominating sets in mobile ad hoc networks. In: Proc. of MOBIHOC 2002, pp. 157–164 (2002)Google Scholar
  2. 2.
    Avin, C.: Fast and efficient restricted delaunay triangulation in random geometric graphs. In: Proc. of Workshop on Combinatorial and Algorithmic Aspects of Networking, CAAN 2005 (2005)Google Scholar
  3. 3.
    Caragiannis, I., Fishkin, A.V., Kaklamanis, C., Papaioannou, E.: A tight bound for online colouring of disk graphs. Theoretical Computer Science 384, 152–160 (2007)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chatterjea, S., Nieberg, T., Zhang, Y., Havinga, P.: Energy-efficient data acquisition using a distributed and self-organizing scheduling algorithm for wireless sensor networks. In: Aspnes, J., Scheideler, C., Arora, A., Madden, S. (eds.) DCOSS 2007. LNCS, vol. 4549, pp. 368–385. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Couture, M., Barbeau, M., Bose, P., Carmi, P., Kranakis, E.: Location oblivious distributed unit disk graph coloring. In: Prencipe, G., Zaks, S. (eds.) SIROCCO 2007. LNCS, vol. 4474, pp. 222–233. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Diaz, J., Penrose, M.D., Petit, J., Serna, M.J.: Linear orderings of random geometric graphs. In: Widmayer, P., Neyer, G., Eidenbenz, S. (eds.) WG 1999. LNCS, vol. 1665, pp. 291–302. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  7. 7.
    Feige, U., Halldorsson, M.M., Kortsarz, G., Srinivasan, A.: Approximating the domatic number. J. of Computing 32(1), 172–195 (2003)MathSciNetGoogle Scholar
  8. 8.
    Finocchi, I., Panconesi, A., Silvestri, R.: Experimental analysis of simple, distributed vertex coloring algorithms. In: Proc. of SODA 2002, pp. 606–615 (2002)Google Scholar
  9. 9.
    Fotakis, D., Nikoletseas, S., Papadopoulou, V., Spirakis, P.: Hardness results and efficient approximations for frequency assignment problems: Radio labelling and radio coloring. In: Proc. of Workshop on Algorithmic Issues in Communication Networks, vol. 20(2), pp. 121–180 (2001)Google Scholar
  10. 10.
    Funke, S., Milosavljevic, N.: Infrastructure-establishment from scratch in wireless sensor networks. In: Prasanna, V.K., Iyengar, S.S., Spirakis, P.G., Welsh, M. (eds.) DCOSS 2005. LNCS, vol. 3560, pp. 354–367. Springer, Heidelberg (2005)Google Scholar
  11. 11.
    Gavoille, C., Klasing, R., Kosowski, A., Kuszner, L., Navarra, A.: On the complexity of distributed graph coloring with local minimality constraints. Networks 54(1), 12–19 (2009)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Gräf, A., Stumpf, M., Weinβenfels, G.: On coloring unit disk graphs. Algorithmica 20, 277–293 (1998)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. CRC Press, Boca Raton (1998)MATHGoogle Scholar
  14. 14.
    Islam, K., Akl, S.G., Meijer, H.: Distributed generation of a family of connected dominating sets in wireless sensor networks. In: Krishnamachari, B., Suri, S., Heinzelman, W., Mitra, U. (eds.) DCOSS 2009. LNCS, vol. 5516, pp. 343–355. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    Islam, K., Akl, S.G., Meijer, H.: Maximizing the lifetime of a sensor network through domatic partition. In: Proc. of the 34th IEEE Conference on Local Computer Networks (LCN) (2009)Google Scholar
  16. 16.
    Kothapalli, K., Scheideler, C., Onus, M., Richa, A.: Constant density spanners for wireless ad-hoc networks. In: Proc. of the SPAA 2005, pp. 116–125 (2005)Google Scholar
  17. 17.
    Lenzen, C., Suomela, J., Wattenhofer, R.: Local algorithms: Self-stabilization on speed. In: Proc. of 11th International Symposium on Stabilization, Safety and Security of Distributed Systems (SSS), pp. 17–34 (2009)Google Scholar
  18. 18.
    Lin, Z., Wang, D., Xu, L., Gao, J.: A coloring based backbone construction algorithm in wireless ad hoc network. In: Chung, Y.-C., Moreira, J.E. (eds.) GPC 2006. LNCS, vol. 3947, pp. 509–516. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Mahjoub, D., Matula, D.W.: Experimental study of independent and dominating sets in wireless sensor networks. In: Liu, B., Bestavros, A., Du, D.-Z., Wang, J. (eds.) WASA 2009. LNCS, vol. 5682, pp. 32–42. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Mahjoub, D., Matula, D.W.: Employing (1 − ε) dominating set partitions as backbones in wireless sensor networks. In: Proc. of the 11th Workshop on Algorithm Engineering and Experiments (ALENEX), pp. 98–111 (2010)Google Scholar
  21. 21.
    Marathe, M., Breu, H., Ravi, S., Rosenkrantz, D.: Simple heuristics for unit disk graphs. Networks 25, 59–68 (1995)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Matula, D.W., Beck, L.: Smallest-last ordering and clustering and graph coloring algorithms. J. of the ACM 30(3), 417–427 (1983)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Matula, D.W., Marble, G., Isaacson, J.: Graph Coloring Algorithms. In: Graph Theory and Computing, pp. 109–122. Academic Press, London (1972)Google Scholar
  24. 24.
    Matula, D.W., Sokal, R.: Properties of gabriel graphs relevant to geographic variation research and the clustering of points in the plane. Geographical Analysis 12, 205–222 (1980)Google Scholar
  25. 25.
    Misra, R., Mandal, C.A.: Efficient clusterhead rotation via domatic partition in self-organizing sensor networks. Wireless Communications and Mobile Computing 9(8), 1040–1058 (2008)CrossRefGoogle Scholar
  26. 26.
    Misra, R., Mandal, C.A.: Rotation of cds via connected domatic partition in ad hoc sensor networks. IEEE Trans. Mob. Comput. 8(4), 488–499 (2009)CrossRefGoogle Scholar
  27. 27.
    Moscibroda, T., Wattenhofer, R.: Maximizing the lifetime of dominating sets. In: Proc. of 5th IEEE WMAN 2005 (2005)Google Scholar
  28. 28.
    Pandit, S., Pemmaraju, S.V., Varadarajan, K.: Approximation algorithms for domatic partitions of unit disk graphs. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX 2009. LNCS, vol. 5687, pp. 312–325. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  29. 29.
    Parthasarathy, S., Gandhi, R.: Distributed algorithms for coloring and domination in wireless ad hoc networks. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 447–459. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  30. 30.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM, Philadelphia (2000)MATHGoogle Scholar
  31. 31.
    Pemmaraju, S.V., Pirwani, I.A.: Energy conservation via domatic partitions. In: Proc. of MobiHoc 2006, pp. 143–154 (2006)Google Scholar
  32. 32.
    Schneider, J., Wattenhofer, R.: A log-star distributed maximal independent set algorithm for growth-bounded graphs. In: Proc. of PODC 2008, pp. 35–44 (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Dhia Mahjoub
    • 1
  • David W. Matula
    • 1
  1. 1.Bobby B. Lyle School of EngineeringSouthern Methodist UniversityDallasUSA

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