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. As the points move, their 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, at a cost of O(logn) per event, while showing that the number of topological events has an upper bound of 0(nXB(n)), where X,{n) is the (nearly linear) maximum length of a (n,s)-DavenportSchinzel sequence, and s is a constant depending on the motions of the point sites. 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 0(kn~\3(n) + (n — fc) A3(fc)) on the number of topological events.

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