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.2018.58
URN: urn:nbn:de:0030-drops-87717
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8771/
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Liu, Chih-Hung

A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon

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LIPIcs-SoCG-2018-58.pdf (0.5 MB)

Abstract

The geodesic Voronoi diagram of m point sites inside a simple polygon of n vertices is a subdivision of the polygon into m cells, one to each site, such that all points in a cell share the same nearest site under the geodesic distance. The best known lower bound for the construction time is Omega(n+m log m), and a matching upper bound is a long-standing open question. The state-of-the-art construction algorithms achieve O((n+m)log (n+m)) and O(n+m log m log^2n) time, which are optimal for m=Omega(n) and m=O(n/(log^3n)), respectively. In this paper, we give a construction algorithm with O(n+m(log m+log^2 n)) time, and it is nearly optimal in the sense that if a single Voronoi vertex can be computed in O(log n) time, then the construction time will become the optimal O(n+m log m). In other words, we reduce the problem of constructing the diagram in the optimal time to the problem of computing a single Voronoi vertex in O(log n) time.

BibTeX - Entry

@InProceedings{liu:LIPIcs:2018:8771,
  author =	{Chih-Hung Liu},
  title =	{{A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{58:1--58:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Bettina Speckmann and Csaba D. T{\'o}th},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8771},
  URN =		{urn:nbn:de:0030-drops-87717},
  doi =		{10.4230/LIPIcs.SoCG.2018.58},
  annote =	{Keywords: Simple polygons, Voronoi diagrams, Geodesic distance}
}

Keywords: Simple polygons, Voronoi diagrams, Geodesic distance
Collection: 34th International Symposium on Computational Geometry (SoCG 2018)
Issue Date: 2018
Date of publication: 08.06.2018

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