Spanners for geometric intersection graphs

Martin Furer, Shiva Prasad Kasiviswanathan

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

    6 Citations (Scopus)

    Abstract

    A disk graph is an intersection graph of a set of disks with arbitrary radii in the plane. In this paper, we consider the problem of efficient construction of sparse spanners of disk (ball) graphs with support for fast distance queries. These problems are motivated by issues arising from topology control and routing in wireless networks. We present the first algorithm for constructing spanners of ball graphs. For a ball graph in ℝk, we construct a (1 + ε)-spanner with O(nε-k+1) edges in O(n2ℓ+εε-k log S) expected time, using an efficient partitioning of the space into hypercubes and solving intersection problems. Here ℓ = 1-1/(⌊k/2⌋+2), ε is any positive constant, and S is the ratio between the largest and smallest radius. For the special case where all the balls have the same radius, we show that the spanner construction has complexity almost equivalent to the construction of a Euclidean minimum spanning tree. Previously known constructions of spanners of unit ball graphs have time complexity much closer to n2. Additionally, these spanners have a small vertex separator (hereditary), which is then exploited for fast answering of distance queries. The results on geometric graph separators might be of independent interest.

    Original languageEnglish (US)
    Title of host publicationAlgorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings
    Pages312-324
    Number of pages13
    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

    Spanners
    Geometric Graphs
    Intersection Graphs
    Ball
    Spanner
    Graph in graph theory
    Separators
    Separator
    Radius
    Query
    Topology Control
    Minimum Spanning Tree
    Wireless networks
    Hypercube
    Unit ball
    Topology
    Time Complexity
    Wireless Networks
    Partitioning
    Euclidean

    All Science Journal Classification (ASJC) codes

    • Theoretical Computer Science
    • Computer Science(all)

    Cite this

    Furer, M., & Kasiviswanathan, S. P. (2007). Spanners for geometric intersection graphs. In Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings (pp. 312-324). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4619 LNCS).
    Furer, Martin ; Kasiviswanathan, Shiva Prasad. / Spanners for geometric intersection graphs. Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings. 2007. pp. 312-324 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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    title = "Spanners for geometric intersection graphs",
    abstract = "A disk graph is an intersection graph of a set of disks with arbitrary radii in the plane. In this paper, we consider the problem of efficient construction of sparse spanners of disk (ball) graphs with support for fast distance queries. These problems are motivated by issues arising from topology control and routing in wireless networks. We present the first algorithm for constructing spanners of ball graphs. For a ball graph in ℝk, we construct a (1 + ε)-spanner with O(nε-k+1) edges in O(n2ℓ+εε-k logℓ S) expected time, using an efficient partitioning of the space into hypercubes and solving intersection problems. Here ℓ = 1-1/(⌊k/2⌋+2), ε is any positive constant, and S is the ratio between the largest and smallest radius. For the special case where all the balls have the same radius, we show that the spanner construction has complexity almost equivalent to the construction of a Euclidean minimum spanning tree. Previously known constructions of spanners of unit ball graphs have time complexity much closer to n2. Additionally, these spanners have a small vertex separator (hereditary), which is then exploited for fast answering of distance queries. The results on geometric graph separators might be of independent interest.",
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    Furer, M & Kasiviswanathan, SP 2007, Spanners for geometric intersection 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. 312-324, 10th International Workshop on Algorithms and Data Structures, WADS 2007, Halifax, Canada, 8/15/07.

    Spanners for geometric intersection graphs. / Furer, Martin; Kasiviswanathan, Shiva Prasad.

    Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings. 2007. p. 312-324 (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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    Furer M, Kasiviswanathan SP. Spanners for geometric intersection graphs. In Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings. 2007. p. 312-324. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).