Divide-and-Conquer Algorithms for Computing Three-Dimensional Voronoi Diagrams

Abstract

While Voronoi diagrams are used in a wide range of applications, leading algorithms (e.g., Fortune's algorithm) are limited to two-dimensional Voronoi diagrams. Problematically, many of the space-dividing applications of Voronoi diagrams exist in three-dimensional spaces rather than two-dimensional spaces. While two-dimensional Voronoi diagrams have been used in cases where three-dimensional space can be simplified to two-dimensional space with an acceptable loss of precision, such simplification is not always feasible. In this paper we extend existing work on divide-and-conquer algorithms for computing two-dimensional discretized Voronoi diagrams by introducing and comparing two novel algorithms for calculating three-dimensional discretized Voronoi diagrams. A comparison of the two algorithms is presented for a range of both space sizes and number of sites.