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Stable-Matching Voronoi Diagrams: Combinatorial Complexity and Algorithms

Authors Gill Barequet, David Eppstein, Michael T. Goodrich, Nil Mamano

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Author Details

Gill Barequet
  • Technion - Israel Inst. of Technology, Haifa, Israel
David Eppstein
  • University of California, Irvine, U.S.
Michael T. Goodrich
  • University of California, Irvine, U.S.
Nil Mamano
  • University of California, Irvine, U.S.

Funding

This work was partially supported by DARPA under agreement no. AFRL FA8750-15-2-0092 and NSF grants 1228639, 1526631,2 1217322, 1618301, and 1616248. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.

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Gill Barequet, David Eppstein, Michael T. Goodrich, and Nil Mamano. Stable-Matching Voronoi Diagrams: Combinatorial Complexity and Algorithms. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 89:1-89:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ICALP.2018.89

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Voronoi diagram
  • stable matching
  • combinatorial complexity
  • lower bounds

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