Voronoi diagrams for moving disks and applications

Abstract

In this paper we discuss the kinetic maintenance of the Euclidean Voronoi diagram and its dual, the Delaunay triangulation, for a set of moving disks. The most important aspect in our approach is that we can maintain the Voronoi diagram even in the case of intersecting disks. We achieve that by augmenting the Delaunay triangulation with some edges associated with the disks that do not contribute to the Voronoi diagram. Using the augmented Delaunay triangulation of the set of disks as the underlying structure, we discuss how to maintain, as the disks move, (1) the closest pair, (2) the connectivity of the set of disks and (3) in the case of non-intersecting disks, the near neighbors of a disk.