Faster approximation of distances in graphs

Piotr Berman, Shiva Prasad Kasiviswanathan

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    22 Scopus citations

    Abstract

    Let G = (V, E) be a weighted undirected graph on n vertices and m edges, and let dG be its shortest path metric. We present two simple deterministic algorithms for approximating all-pairs shortest paths in G. Our first algorithm runs in Õ(n2) time, and for any u, v ε V reports distance no greater than 2dG(u,v) + h(u,v). Here, h(u, v) is the largest edge weight on a shortest path between u and v. The previous algorithm, due to Baswana and Kavitha that achieved the same result was randomized. Our second algorithm for the all-pairs shortest path problem uses Boolean matrix multiplications and for any u, v ε V reports distance no greater than (1 + ε)dG(u, v) + 2h(u, v). The currently best known algorithm for Boolean matrix multiplication yields an O(n 2.24+o(1)ε-3log(nε-1)) time bound for this algorithm. The previously best known result of Elkin with a similar multiplicative factor had a much bigger additive error term. We also consider approximating the diameter and the radius of a graph. For the problem of estimating the radius, we present an almost 3/2-approximation algorithm which runs in Õ(m√n + n2) time. Aingworth, Chekuri, Indyk, and Motwani used a similar approach and obtained analogous results for the diameter approximation problem, Additionally, we show that if the graph has a small separator decomposition a 3/2-approximation of both the diameter and the radius can be obtained more efficiently.

    Original languageEnglish (US)
    Title of host publicationAlgorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings
    PublisherSpringer Verlag
    Pages541-552
    Number of pages12
    ISBN (Print)3540739483, 9783540739487
    DOIs
    StatePublished - 2007
    Event10th International Workshop on Algorithms and Data Structures, WADS 2007 - Halifax, Canada
    Duration: Aug 15 2007Aug 17 2007

    Publication series

    NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    Volume4619 LNCS
    ISSN (Print)0302-9743
    ISSN (Electronic)1611-3349

    Other

    Other10th International Workshop on Algorithms and Data Structures, WADS 2007
    CountryCanada
    CityHalifax
    Period8/15/078/17/07

    All Science Journal Classification (ASJC) codes

    • Theoretical Computer Science
    • Computer Science(all)

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