Some results on the computation of Voronoi diagrams on a mesh with multiple broadcasting

Abstract

Voronoi diagram is one of the most fundamental and important geometric data structures. Voronoi diagram was historically defined for a set of points on the plane. The diagram partitions the plane into regions, one per site. The region of a site s consists of all points closer to s than to any other sites on the plane. Concepts of Voronoi diagram are often attributed to Voronoi (J. Reine Angew. Math. 133 (1907) 97) and Dirichlet (J. Reine Angew. Math. 40 (1850) 209). As a result of these early works, often the name Voronoi diagram and Dirichlet tessellation is used. Due to the importance of Voronoi diagrams, it is important that algorithms are devised to compute these structures in an efficient manner. Of course, this will create new opportunities for the applicability of these data structures. Towards this end, this paper presents new results for the computation of Voronoi diagrams for a set of n points, or n disjoint circles on the plane, on a mesh with multiple broadcasting (MMB) of size n× n. The algorithm runs in O( log 2 n) time.