Randomized Incremental Construction for the Hausdorff Voronoi Diagram of point clusters

Abstract

This paper applies the randomized incremental construction (RIC) framework to computing the Hausdorff Voronoi diagram of a family of k clusters of points in the plane. The total number of points is n. The diagram is a generalization of Voronoi diagrams based on the Hausdorff distance function. The combinatorial complexity of the Hausdorff Voronoi diagram is O(n+m), where m is the total number of crossings between pairs of clusters. For non-crossing clusters (m=0), our algorithm works in expected O(n log n + k log n log k) time and deterministic O(n) space. For arbitrary clusters (m=O(n^2)), the algorithm runs in expected O((m+n log k) log n) time and O(m +n log k) space. When clusters cross, bisectors are disconnected curves resulting in disconnected Voronoi regions that challenge the incremental construction. This paper applies the RIC paradigm to a Voronoi diagram with disconnected regions and disconnected bisectors, for the first time.