Faster approximation of distances in graphs

Piotr Berman, Shiva Prasad Kasiviswanathan

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

    17 Citations (Scopus)

    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
    Pages541-552
    Number of pages12
    StatePublished - Dec 1 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

    Fingerprint

    Distance in Graphs
    Shortest path
    Boolean Matrix
    Approximation
    Matrix multiplication
    Radius
    Separator
    Shortest Path Problem
    Approximation Problem
    Deterministic Algorithm
    Weighted Graph
    Graph in graph theory
    Error term
    Undirected Graph
    Approximation algorithms
    Approximation Algorithms
    Separators
    Multiplicative
    Decompose
    Metric

    All Science Journal Classification (ASJC) codes

    • Theoretical Computer Science
    • Computer Science(all)

    Cite this

    Berman, P., & Kasiviswanathan, S. P. (2007). Faster approximation of distances in graphs. In Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings (pp. 541-552). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4619 LNCS).
    Berman, Piotr ; Kasiviswanathan, Shiva Prasad. / Faster approximation of distances in graphs. Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings. 2007. pp. 541-552 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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    title = "Faster approximation of distances in graphs",
    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 {\~O}(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 {\~O}(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.",
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    Berman, P & Kasiviswanathan, SP 2007, Faster approximation of distances in graphs. in Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4619 LNCS, pp. 541-552, 10th International Workshop on Algorithms and Data Structures, WADS 2007, Halifax, Canada, 8/15/07.

    Faster approximation of distances in graphs. / Berman, Piotr; Kasiviswanathan, Shiva Prasad.

    Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings. 2007. p. 541-552 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4619 LNCS).

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

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    N2 - 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.

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    Berman P, Kasiviswanathan SP. Faster approximation of distances in graphs. In Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings. 2007. p. 541-552. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).