A semidynamic construction of higher-order voronoi diagrams and its randomized analysis

Thek-Delaunay tree extends the Delaunay tree introduced in 1], and 2]. It is a hierarchical data structure that allows the semidynamic construction of the higher-order Voronoi diagrams of a finite set ofn points in any dimension. In this paper we prove that a randomized construction of thek-Delaunay tree, and thus of all the orderk Voronoi diagrams, can be done inO(n logn+k ³n) expected time and O(k²n) expected storage in the plane, which is asymptotically optimal for fixedk. Our algorithm extends tod-dimensional space with expected time complexityO(k ^(d+1)/2+1 n ^(d+1)/2) and space complexityO(k ^(d+1)/2 n ^(d+1)/2). The algorithm is simple and experimental results are given.This work has been supported in part by the ESPRIT Basic Research Action No. 3075 (ALCOM).