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.SoCG.2019.60
URN: urn:nbn:de:0030-drops-104649
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10464/
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Wang, Haitao ; Xue, Jie

Near-Optimal Algorithms for Shortest Paths in Weighted Unit-Disk Graphs

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LIPIcs-SoCG-2019-60.pdf (0.6 MB)

Abstract

We revisit a classical graph-theoretic problem, the single-source shortest-path (SSSP) problem, in weighted unit-disk graphs. We first propose an exact (and deterministic) algorithm which solves the problem in O(n log^2 n) time using linear space, where n is the number of the vertices of the graph. This significantly improves the previous deterministic algorithm by Cabello and Jejcic [CGTA'15] which uses O(n^{1+delta}) time and O(n^{1+delta}) space (for any small constant delta>0) and the previous randomized algorithm by Kaplan et al. [SODA'17] which uses O(n log^{12+o(1)} n) expected time and O(n log^3 n) space. More specifically, we show that if the 2D offline insertion-only (additively-)weighted nearest-neighbor problem with k operations (i.e., insertions and queries) can be solved in f(k) time, then the SSSP problem in weighted unit-disk graphs can be solved in O(n log n+f(n)) time. Using the same framework with some new ideas, we also obtain a (1+epsilon)-approximate algorithm for the problem, using O(n log n + n log^2(1/epsilon)) time and linear space. This improves the previous (1+epsilon)-approximate algorithm by Chan and Skrepetos [SoCG'18] which uses O((1/epsilon)^2 n log n) time and O((1/epsilon)^2 n) space. Because of the Omega(n log n)-time lower bound of the problem (even when approximation is allowed), both of our algorithms are almost optimal.

BibTeX - Entry

@InProceedings{wang_et_al:LIPIcs:2019:10464,
  author =	{Haitao Wang and Jie Xue},
  title =	{{Near-Optimal Algorithms for Shortest Paths in Weighted Unit-Disk Graphs}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{60:1--60:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Gill Barequet and Yusu Wang},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10464},
  URN =		{urn:nbn:de:0030-drops-104649},
  doi =		{10.4230/LIPIcs.SoCG.2019.60},
  annote =	{Keywords: Single-source shortest paths, Weighted unit-disk graphs, Geometric graph algorithms}
}

Keywords: Single-source shortest paths, Weighted unit-disk graphs, Geometric graph algorithms
Collection: 35th International Symposium on Computational Geometry (SoCG 2019)
Issue Date: 2019
Date of publication: 11.06.2019

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