Voronoi diagrams of moving points in higher dimensional spaces

Abstract

The modeling of realistic dynamic scenes often requires the maintenance of geometric data structures over time. This is the subject of a rising discipline called dynamic computational geometry. In the present work we investigate the behavior of spatial Voronoi diagrams under continuous motions of the underlying sites. Nevertheless, the methodology presented can be applied to many other geometric data structures in computational geometry, as well.

Now, consider a set of n points moving continuously along given trajectories in d-dimensional Euclidean space, d≥ 3. At each instant, the points define a Voronoi diagram which changes continuously except of certain critical instants, so-called topological events.

We classify the appearing events which cause a change in the topology of the Voronoi diagram and present an algorithm for maintaining the Voronoi diagram over time using only O(log n) time per event which is worst-case optimal. In addition, we give an O(n ^d λ _s (n)) upper bound on the number of topological events. Thereby λ _s(n) denotes the maximum length of an (n,s)-Davenport-Schinzel sequence, and s is a constant depending on the underlying trajectories of the moving sites.

Our work generalizes the most recent results by [FuLe 91], [GuMiRo 91] and [Imlm 90] to three and higher dimensions. Application areas include motion planning problems (such as air traffic control) as well as pattern matching problems in static and dynamic scenes.

Work on this paper was partially supported by the Deutsche Forschungsgemeinschaft (DFG) under contract (No 88/10 - 1, 2).

This work was carried out while the second author was at the University of Würzburg.