Maintaining proximity in higher dimensional spaces

Abstract

Dynamic computational geometry is concerned with the maintenance of geometric data structures over time. In the present work we investigate the behavior of spatial nearestneighbor Voronoi diagrams and planar higher-order Voronoi diagrams under continuous motions of the underlying sites. Nevertheless, the methodology presented is interesting in itself and can be applied to many other geometric data structures in computational geometry, as well.

In this paper, we 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. At first, we classify the events which appear in the d-dimensional nearest-neighbor Voronoi diagram and outline an O(n ^d λ _s (n)) upper bound on the number of topological events. Thereby λ _s (n) denotes the maximum length of a (n, s)-Davenport-Schinzel sequence, and s is a constant depending on the underlying trajectories of the moving sites.

Next, we apply our methodology to planar order-m Voronoi diagrams, m- 1,..., n−1. The basic result in this context is that topological events affect at most three successive orders of Voronoi diagrams. Additionally, we give a tight O(n ⁴) upper bound on the number of topological events which appear in all order-m Voronoi diagrams.

Both cases allow simple and numerically stable algorithms for maintaining the corresponding Voronoi diagram(s) over time using only O(log n) time per event, which is worstcase optimal. This work generalizes the most recent results by [FuLe 91], [GuMiRo 91] and [ImIm 90] to three and higher dimensions as well as higher orders.

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 first author was at the University of Würzburg.