License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2016.24
URN: urn:nbn:de:0030-drops-67948
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6794/
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Chan, Timothy M. ; Skrepetos, Dimitrios

All-Pairs Shortest Paths in Unit-Disk Graphs in Slightly Subquadratic Time

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LIPIcs-ISAAC-2016-24.pdf (0.5 MB)

Abstract

In this paper we study the all-pairs shortest paths problem in (unweighted) unit-disk graphs. The previous best solution for this problem required O(n^2 log n) time, by running the O(n log n)-time single-source shortest path algorithm of Cabello and Jejcic [Comput. Geom., 2015] from every source vertex,where n is the number of vertices. We not only manage to eliminate the logarithmic factor, but also obtain the first (slightly) subquadratic algorithm for the problem, running in O(n^2 sqrt{ frac{log log n}{log n} }) time. Our algorithm computes an implicit representation of all the shortest paths, and, in the same amount of time, can also compute the diameter of the graph.

BibTeX - Entry

@InProceedings{chan_et_al:LIPIcs:2016:6794,
  author =	{Timothy M. Chan and Dimitrios Skrepetos},
  title =	{{All-Pairs Shortest Paths in Unit-Disk Graphs in Slightly Subquadratic Time}},
  booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
  pages =	{24:1--24:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-026-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{64},
  editor =	{Seok-Hee Hong},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6794},
  URN =		{urn:nbn:de:0030-drops-67948},
  doi =		{10.4230/LIPIcs.ISAAC.2016.24},
  annote =	{Keywords: unit-disk graphs, all-pairs shortest paths, computational geometry}
}

Keywords: unit-disk graphs, all-pairs shortest paths, computational geometry
Collection: 27th International Symposium on Algorithms and Computation (ISAAC 2016)
Issue Date: 2016
Date of publication: 07.12.2016

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