A Simple Linear Time Algorithm for Computing a (2k — 1)-Spanner of O(n1+1/k) Size in Weighted Graphs

  • Surender Baswana
  • Sandeep Sen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2719)

Abstract

Let G(V,E) be an undirected weighted graph with |V| = n, and |E| = m. A t-spanner of the graph G(V,E) is a sub-graph G(V,ES) such that the distance between any pair of vertices in the spanner is at most t times the distance between the two in the given graph. A 1963 girth conjecture of Erdős implies that Ω(n1+1/k) edges are required in the worst case for any (2k − 1)-spanner, which has been proved for k = 1, 2, 3, 5. There exist polynomial time algorithms that can construct spanners with the size that matches this conjectured lower bound, and the best known algorithm takes O(mn1/k) expected running time. In this paper, we present an extremely simple linear time randomized algorithm that constructs (2k − 1)-spanner of size matching the conjectured lower bound.

Our algorithm requires local information for computing a spanner, and thus can be adapted suitably to obtain efficient distributed and parallel algorithms.

Keywords

Graph algorithms Randomized algorithms Shortest path 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    I. Althofer, G. Das, D. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete and Computational Geometry, 9:81–100, 1993.CrossRefMathSciNetGoogle Scholar
  2. 2.
    B. Awerbuch. Complexity of network synchronization. Journal of Ass. Compt. Mach., pages 804–823, 1985.Google Scholar
  3. 3.
    H. J. Bandelt and A. W. M. Dress. Reconstructing the shape of a tree from observed dissimilarity data. Journal of Advances of Applied Mathematics, 7:309–343, 1986.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    B. Bollobas. In Extremal Graph Theory, page 164. Academic Press, 1978.Google Scholar
  5. 5.
    J.A. Bondy and M. Simonovits. Cycle of even length in graphs. Journal of Combinatorial Theory, Series B, 16:97–105, 1974.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Edith Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM J. Comput., 28:210–236, 1998.MATHCrossRefGoogle Scholar
  7. 7.
    P. Erdos. Extremal problems in graph theory. In In Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963), pages 29–36. Publ. House Czechoslovak Acad. Sci., Prague, 1964, 1963.Google Scholar
  8. 8.
    Shay Halperin and Uri Zwick. Unpublished result. 1996.Google Scholar
  9. 9.
    David Peleg and A. A. Schaffer. Graph spanners. Journal of Graph Theory, 13:99–116, 1989.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    David Peleg and Eli Upfal. A trade-off between space amd efficiency for routing tables. Journal of Assoc. Comp. Mach., 36(3):510–530, 1989.MATHMathSciNetGoogle Scholar
  11. 11.
    Mikkel Thorup and Uri Zwick. Approximate distance oracle. In Proceedings of 33rd ACM Symposium on Theory of Computing (STOC), pages 183–192, 2001.Google Scholar
  12. 12.
    Uri Zwick. Personal communication.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Surender Baswana
    • 1
  • Sandeep Sen
    • 1
  1. 1.Department of Computer Science and EngineeringI.I.T. DelhiNew DelhiIndia

Personalised recommendations