Reduction of the Search Space in the Edge-Tracing Algorithm for the Voronoi Diagram of 3D Balls

Abstract

Voronoi diagram for 3D balls can be applicable to various fields in science and engineering. The edge-tracing algorithm constructs the Voronoi diagram in O(mn) time in the worst-case where m and n are the numbers of edges and balls, respectively. The computation time of the algorithm is dominated by finding the end vertex of a given edge since all edges in the Voronoi diagram should be traced essentially. In this paper, we define the feasible region which a ball to define the end vertex of a given edge should intersect. Then, balls which do not intersect the feasible region are filtered out before finding an end vertex since they cannot define an end vertex. Therefore, we improve the runtime-performance of the edge-tracing algorithm via the feasible region.