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.ICALP.2018.89
URN: urn:nbn:de:0030-drops-90937
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9093/
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Barequet, Gill ; Eppstein, David ; Goodrich, Michael T. ; Mamano, Nil

Stable-Matching Voronoi Diagrams: Combinatorial Complexity and Algorithms

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Abstract

We study algorithms and combinatorial complexity bounds for stable-matching Voronoi diagrams, where a set, S, of n point sites in the plane determines a stable matching between the points in R^2 and the sites in S such that (i) the points prefer sites closer to them and sites prefer points closer to them, and (ii) each site has a quota indicating the area of the set of points that can be matched to it. Thus, a stable-matching Voronoi diagram is a solution to the classic post office problem with the added (realistic) constraint that each post office has a limit on the size of its jurisdiction. Previous work provided existence and uniqueness proofs, but did not analyze its combinatorial or algorithmic complexity. We show that a stable-matching Voronoi diagram of n sites has O(n^{2+epsilon}) faces and edges, for any epsilon>0, and show that this bound is almost tight by giving a family of diagrams with Theta(n^2) faces and edges. We also provide a discrete algorithm for constructing it in O(n^3+n^2f(n)) time, where f(n) is the runtime of a geometric primitive that can be performed in the real-RAM model or can be approximated numerically. This is necessary, as the diagram cannot be computed exactly in an algebraic model of computation.

BibTeX - Entry

@InProceedings{barequet_et_al:LIPIcs:2018:9093,
  author =	{Gill Barequet and David Eppstein and Michael T. Goodrich and Nil Mamano},
  title =	{{Stable-Matching Voronoi Diagrams: Combinatorial Complexity and Algorithms}},
  booktitle =	{45th International Colloquium on Automata, Languages, and  Programming (ICALP 2018)},
  pages =	{89:1--89:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9093},
  URN =		{urn:nbn:de:0030-drops-90937},
  doi =		{10.4230/LIPIcs.ICALP.2018.89},
  annote =	{Keywords: Voronoi diagram, stable matching, combinatorial complexity, lower bounds}
}

Keywords: Voronoi diagram, stable matching, combinatorial complexity, lower bounds
Collection: 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)
Issue Date: 2018
Date of publication: 04.07.2018

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