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Zone Diagrams: Existence, Uniqueness, and Algorithmic Challenge by Asano, Tetsuo; Matoušek, JiřÍ; Tokuyama, Takeshi is a Computer Science article available to read on EtoBox.

What is Zone Diagrams: Existence, Uniqueness, and Algorithmic Challenge about?

A zone diagram is a new variation of the classical notion of the Voronoi diagram. Given points (sites) p 1 , . . . , pn in the plane, each 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 x ∈ R 2 that lie closer (nonstrictly) to p i than to the union of all the other R j , j = 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.

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Author

Asano, Tetsuo; Matoušek, JiřÍ; Tokuyama, Takeshi

Publisher

Society for Industrial and Applied Mathematics; Society for Industrial & Applied Mathematics (SIAM) (ISSN 0097-5397)

Published

2007

Language

EN

Field

Computer Science (Physical Sciences)