On Linear-Sized Farthest-Color Voronoi Diagrams

Abstract

Given a collection of k sets consisting of a total of n points in the plane, the distance from any point in the plane to each of the sets is defined to be the minimum among distances to each point in the set. The farthest-color Voronoi diagram is defined as a generalized Voronoi diagram of the k sets with respect to the distance functions for each of the k sets. The combinatorial complexity of the diagram is known to be Θ(kn) in the worst case. This paper initiates a study on farthest-color Voronoi diagrams having O(n) complexity. We introduce a realistic model, which defines a certain class of the diagrams with desirable geometric properties observed. We finally show that the farthest-color Voronoi diagrams under the model have linear complexity.