Randomized Incremental Construction of Abstract Voronoi Diagrams

Abstract Voronoi diagrams were introduced by R . Klein [Kle89b, Kle88a, Kle88b] as an axiomatic basis ofVoronoi diagrams. We show how to construct abstract Voronoi diagrams in time O(nlogn) by a randomized algorithm, which is based on Clarkson and Shor's randomized incremental construction technique [CS89]. The new algorithm has the following advantages over previous algorithms: • It can handle a much wider dass of abstract Voronoi diagrams than the algorithms presented in [Kle89b, MM091J . • It can be adapted to a concrete kind ofVoronoi diagram by providing a single basic operation, namely the construction of a Voronoi diagram of five sites. Moreover, all geometrie decisions are confined to the basic operation, and using this operation, abstract Voronoi diagrams can be constructed in a purely combinatorial manner.