Stable-matching voronoi diagrams: Combinatorial complexity and algorithms

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 R² 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 o ce problem with the added (realistic) constraint that each post o ce 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+ε) faces and edges, for any ε > 0, and show that this bound is almost tight by giving a family of diagrams with (n²) faces and edges. We also provide a discrete algorithm for constructing it in O(n³ + n²f(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.