Mollified Zone Diagrams and Their Computation

Abstract

The notion of the zone diagram of a finite set of points in the Euclidean plane is an interesting and rich variation of the classical Voronoi diagram, introduced by Asano, Matoušek, and Tokuyama [1]. In this paper, we define mollified versions of zone diagram named territory diagram and maximal territory diagram. A zone diagram is a particular maximal territory diagram satisfying a sharp dominance property. The proof of existence of maximal territory diagrams depends on less restrictive initial conditions and is established via Zorn’s lemma in contrast to the use of fixed-point theory in proving the existence of the zone diagram. Our proof of existence relies on a characterization which allows embedding any territory diagram into a maximal one. Our analysis of the structure of maximal territory diagrams is based on the introduction of a pair of dual concepts we call safe zone and forbidden zone. These in turn give rise to computational algorithms for the approximation of maximal territory diagrams. Maximal territory diagrams offer their own interesting theoretical and computational challenges, as well as insights into the structure of zone diagrams. This paper extends and updates previous work presented in [4].