Voronoi Diagrams of Moving Points in the Plane

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(n2λs(n)), where λs,(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.

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