Robust construction of the additively-weighted voronoi diagram via topology-oriented incremental algorithm

Abstract

Voronoi diagrams tessellate the space where each cell corresponds to an associated generator under an a priori defined distance and have been extensively used to solve geometric problems of various disciplines. Additively-weighted Voronoi diagrams, also called the Voronoi diagram of disks and spheres, have many critical applications and a few algorithms are known. However, algorithmic robustness remains a major hurdle to use these Voronoi diagrams in practice. There are two important yet different approaches to design robust algorithms: the exact-computation and topology-oriented approaches. The former uses high-precision arithmetic and guarantees the correctness mathematically with the cost of a significant use of computational resources. The latter focuses on topological properties to keep consistency using logical computation rather than numerical computation. In this paper, we present a robust and efficient algorithm for computing the Voronoi diagram of disks using a topology-oriented incremental method. The algorithm is rather simple as it primarily checks topological changes only during each disk is incrementally inserted into a previously constructed Voronoi diagram of some other disks.