Voronoi Diagrams of Moving Points

Consider a set of n points in d-dimensional Euclidean space, d 2, each of which is continuously moving along a given individual trajectory. At each instant in time, the points deene a Voronoi diagram. As the points move, the Voronoi diagram changes continuously, but at certain critical instants in time, topological events occur that cause a change in the Voronoi diagram. In this paper, we present a method of maintaining the Voronoi diagram over time, while showing that the number of topological events has an upper bound of O(n d s (n)), where s (n) is the maximum length of a (n; s)-Davenport-Schinzel sequence AgShSh 89, DaSc 65] and s is a constant depending on the motions of the point sites. Our results are a linear-factor improvement over the naive O(n d+2) upper bound on the number of topological events. In addition, we show that if only k points are moving (while leaving the other n ? k points xed), there is an upper bound of O(kn d?1 s (n) + (n ? k) d s (k)) on the number of topological events. We give a numerically stable algorithm for the update of the topological structure of the Voronoi diagram, using only O(log n) time per event (which is worst-case optimal per event).