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All-Pairs Shortest Paths in Geometric Intersection Graphs

  • Timothy M. Chan
  • Dimitrios Skrepetos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

Keywords

Shortest paths Geometric intersection graphs Intersection searching data structures Disk graphs 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Illinois at Urbana-ChampaignChampaignUSA
  2. 2.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

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