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Distributed Spanner Construction in Doubling Metric Spaces

  • Mirela Damian
  • Saurav Pandit
  • Sriram Pemmaraju
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4305)

Abstract

This paper presents a distributed algorithm that runs on an n-node unit ball graph (UBG) G residing in a metric space of constant doubling dimension, and constructs, for any ε> 0, a (1 + ε)-spanner H of G with maximum degree bounded above by a constant. In addition, we show that H is “lightweight”, in the following sense. Let Δ denote the aspect ratio of G, that is, the ratio of the length of a longest edge in G to the length of a shortest edge in G. The total weight of H is bounded above by O(logΔ) · wt(MST), where MST denotes a minimum spanning tree of the metric space. Finally, we show that H satisfies the so called leapfrog property, an immediate implication being that, for the special case of Euclidean metric spaces with fixed dimension, the weight of H is bounded above by O(wt(MST)). Thus, the current result subsumes the results of the authors in PODC 2006 that apply to Euclidean metric spaces, and extends these results to metric spaces with constant doubling dimension.

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References

  1. 1.
    Awerbuch, B., Goldberg, A., Luby, M., Plotkin, S.: Network decomposition and locality in distributed computation. In: IEEE Symposium on Foundations of Computer Science, pp. 364–369 (1989)Google Scholar
  2. 2.
    Chan, H.T.-H., Gupta, A., Maggs, B.M., Zhou, S.: On hierarchical routing in doubling metrics. In: SODA 2005: Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, pp. 762–771 (2005)Google Scholar
  3. 3.
    Hubert Chan, T.-H.: Personal Communication (2006)Google Scholar
  4. 4.
    Chan, T.-H.H., Gupta, A.: Small hop-diameter sparse spanners for doubling metrics. In: SODA 2006: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, pp. 70–78 (2006)Google Scholar
  5. 5.
    Czumaj, A., Zhao, H.: Fault-tolerant geometric spanners. Discrete & Computational Geometry 32(2), 207–230 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Damian, M., Pandit, S., Pemmaraju, S.: Local approximation schemes for topology control. In: PODC 2006: Proceedings of the twenty-fifth annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing (2006)Google Scholar
  7. 7.
    Das, G., Heffernan, P., Narasimhan, G.: Optimally sparse spanners in 3-dimensional euclidean space. In: ACM Symposium on Computational Geometry, pp. 53–62 (1993)Google Scholar
  8. 8.
    Das, G., Narasimhan, G.: A fast algorithm for constructing sparse euclidean spanners. Int. J. Comput. Geometry Appl. 7(4), 297–315 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gudmundsson, J., Levcopoulos, C., Narasimhan, G.: Fast greedy algorithms for constructing sparse geometric spanners. SIAM J. Comput. 31(5), 1479–1500 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Har-Peled, S., Mendel, M.: Fast construction of nets in low dimensional metrics, and their applications. In: SCG 2005: Proceedings of the 21st annual symposium on Computational geometry, pp. 150–158 (2005)Google Scholar
  11. 11.
    Krauthgamer, R., Gupta, A., Lee, J.R.: Bounded geometries, fractals, and low-distortion embeddings. In: FOCS 2003: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 534–543 (2003)Google Scholar
  12. 12.
    Krauthgamer, R., Lee, J.R.: Navigating nets: simple algorithms for proximity search. In: SODA 2004: Proceedings of the 15th annual ACM-SIAM symposium on Discrete algorithms, pp. 798–807 (2004)Google Scholar
  13. 13.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: On the locality of bounded growth. In: PODC 2005: Proceedings of the twenty-fourth annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing, pp. 60–68 (2005)Google Scholar
  14. 14.
    Li, X.-Y., Wang, Y.: Efficient construction of low weighted bounded degree planar spanner. International Journal of Computational Geometry and Applications 14(1–2), 69–84 (2004)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Linial, N.: Locality in distributed graph algorithms. SIAM J. Comput. 21(1), 193–201 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Rajaraman, R.: Topology control and routing in ad hoc networks: A survey. SIGACT News 33, 60–73 (2002)CrossRefGoogle Scholar
  17. 17.
    Talwar, K.: Bypassing the embedding: algorithms for low dimensional metrics. In: STOC 2004: Proceedings of the 36th annual ACM symposium on Theory of computing, pp. 281–290 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Mirela Damian
    • 1
  • Saurav Pandit
    • 2
  • Sriram Pemmaraju
    • 2
  1. 1.Department of Computer ScienceVillanova UniversityVillanovaUSA
  2. 2.Department of Computer ScienceThe University of IowaIowa City

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