Maintaining Voronoi Diagrams in Parallel

We are given a set of n points moving continuously along given trajectories in d-dimensional Euclid-ean space. At each instant, these sites deene a Voro-noi diagram which changes continuously over time except of certain critical instances, so-called topological events 4]. In this paper, we present an algorithm for maintaining the Voronoi diagram in parallel over time using only O(1) time per event on a CREW PRAM with O(n d d 2 e) processors which is worst-case optimal. This work generalizes the most recent single processor algorithms by 17, 20, 28] to PRAMs. Additionally, the presented technique provides the rst eecient algorithm for computing static higher dimensional Voronoi diagrams in parallel. Finally, we present a new upper bound on the number of topological events which may appear during the ow of the sites. This result tightens a long open gap between the upper and lower worst-case bounds actually providing matching bounds in some cases and is interesting in its own right.