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

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Book cover Automata, Languages and Programming (ICALP 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2719))

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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,E S) 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(n 1+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(mn 1/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.

Work was supported in part by a fellowship from Infosys Technologies, Bangalore.

Work was supported in part by an IBM Faculty Partnership award

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Authors and Affiliations

  1. Department of Computer Science and Engineering, I.I.T. Delhi, Hauz Khas, New Delhi, 110016, India

    Surender Baswana & Sandeep Sen

Authors
  1. Surender Baswana
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  2. Sandeep Sen
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Editor information

Editors and Affiliations

  1. Dept. of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Jos C. M. Baeten

  2. School of Industrial and Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive, Atlanta, GA, 30332-0205, USA

    Jan Karel Lenstra

  3. Department of Information Technology, Uppsala University, P.O. Box 337, 75105, Uppsala, Sweden

    Joachim Parrow

  4. Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands

    Gerhard J. Woeginger

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© 2003 Springer-Verlag Berlin Heidelberg

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Baswana, S., Sen, S. (2003). A Simple Linear Time Algorithm for Computing a (2k — 1)-Spanner of O(n 1+1/k) Size in Weighted Graphs. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds) Automata, Languages and Programming. ICALP 2003. Lecture Notes in Computer Science, vol 2719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45061-0_32

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  • DOI: https://doi.org/10.1007/3-540-45061-0_32

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40493-4

  • Online ISBN: 978-3-540-45061-0

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