Voronoi diagrams of moving points in the plane

Abstract

Consider a set of n points in the Euclidean plane each of which is continuously moving along a given trajectory. At each instant in time, the points define 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 Delaunay diagram. In this paper, we present a method of maintaining the Voronoi diagram over time, while showing that the number of topological events has a nearly cubic upper bound of O(n2λ3(n)), where λ3,(n) is the maximum length of an (n, s)-Davenport-Schinzel sequence and s is a constant depending on the motions of the point sites. In the special case of points moving at constant speed along straight lines, we get s = 4, implying an upper bound of O(n32α(n)), where α(n) is the extremely slowly-growing inverse of Ackermann’s function. Our results are a linear-factor improvement over the naive quartic 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 fixed), there is an upper bound of O(kn λs(n) + (n - k)2 λs,(k)) on the number of topological events, which is nearly quadratic if k is constant. 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).