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Workshop on Algorithms and Data Structures

WADS 2017: Algorithms and Data Structures pp 253–264Cite as

All-Pairs Shortest Paths in Geometric Intersection Graphs

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10389))

Abstract

We address the All-Pairs Shortest Paths (APSP) problem for a number of unweighted, undirected geometric intersection graphs. We present a general reduction of the problem to static, offline intersection searching (specifically detection). As a consequence, we can solve APSP for intersection graphs of n arbitrary disks in \(O\left( n^2\log n\right) \) time, axis-aligned line segments in \(O\left( n^2\log {\log n}\right) \) time, arbitrary line segments in \(O\left( n^{7/3}\log ^{1/3} n\right) \) time, d-dimensional axis-aligned boxes in \(O\left( n^2\log ^{d-1.5} n\right) \) time for \(d\ge 2\), and d-dimensional axis-aligned unit hypercubes in \(O\left( n^2\log {\log n}\right) \) time for \(d=3\) and \(O\left( n^2\log ^{d-3} n\right) \) time for \(d\ge 4\).

In addition, we show how to solve the Single-Source Shortest Paths (SSSP) problem in unweighted intersection graphs of axis-aligned line segments in \(O\left( n\log n\right) \) time, by a reduction to dynamic orthogonal point location.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Illinois at Urbana-Champaign, Champaign, USA

    Timothy M. Chan

  2. Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada

    Dimitrios Skrepetos

Authors
  1. Timothy M. Chan
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  2. Dimitrios Skrepetos
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Corresponding author

Correspondence to Dimitrios Skrepetos .

Editor information

Editors and Affiliations

  1. Computer Science, University of Toronto , Toronto, Ontario, Canada

    Faith Ellen

  2. Computer Science, Memorial University of Newfoundland , St. John’s, Newfoundland and Labrador, Canada

    Antonina Kolokolova

  3. Carleton University, Ottawa, Ontario, Canada

    Jörg-Rüdiger Sack

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Chan, T.M., Skrepetos, D. (2017). All-Pairs Shortest Paths in Geometric Intersection Graphs. In: Ellen, F., Kolokolova, A., Sack, JR. (eds) Algorithms and Data Structures. WADS 2017. Lecture Notes in Computer Science(), vol 10389. Springer, Cham. https://doi.org/10.1007/978-3-319-62127-2_22

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  • DOI: https://doi.org/10.1007/978-3-319-62127-2_22

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-62126-5

  • Online ISBN: 978-3-319-62127-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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