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# Stable-Matching Voronoi Diagrams: Combinatorial Complexity and Algorithms

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LIPIcs.ICALP.2018.89.pdf
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## Cite As

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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