Randomized Incremental Construction for the Hausdorff Voronoi Diagram of point clusters

TL;DR

This paper introduces a randomized incremental construction algorithm for the Hausdorff Voronoi diagram of point clusters, handling both crossing and non-crossing cases efficiently and for the first time with disconnected regions.

Contribution

It extends the RIC framework to compute Hausdorff Voronoi diagrams with disconnected regions and bisectors, improving efficiency for various cluster configurations.

Findings

Expected time complexity for non-crossing clusters: O(n log n + k log n log k)

Expected time complexity for arbitrary clusters: O((m+n log k) log n)

First application of RIC to diagrams with disconnected regions and bisectors

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.