Abstract:
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 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 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 \Gamma k points fixed), there is an upper bound of O(kn d\Gamma1 s (n) + (n \Gamma k)...
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