Zone Diagrams: Existence, Uniqueness, and Algorithmic Challenge
Tetsuo Asano, Jiřı́ Matoušek, Takeshi Tokuyama · SIAM Journal on Computing · 2007
A zone diagram is a new variation of the classical notion of the Voronoi diagram. Given points (sites) ${\mathbf p}_1,\ldots,{\mathbf p}_n$ in the plane, each ${\mathbf p}_i$ is assigned a region $R_i$, but in contrast to the ordinary Voronoi diagrams, the union of the $R_i$ has a nonempty complement, the neutral zone. The defining property is that each $R_i$ consists of all ${\mathbf x}\in{\mathbb{R}}^2$ that lie closer (nonstrictly) to ${\mathbf p}_i$ than to the union of all the other $R_j$, $j e i$. Thus, the zone diagram is defined implicitly, by a “fixed-point property,” and neither its existence nor its uniqueness seem obvious. We establish existence using a general fixed-point result (a consequence of Schauder's theorem or Kakutani's theorem); this proof should generalize easily to related settings, say higher dimensions. Then we prove uniqueness of the zone diagram, as well as convergence of a natural iterative algorithm for computing it, by a geometric argument, which also relies on a result for the case of two sites in an earlier paper. Many challenging questions remain open.