Randomized incremental construction for the Hausdorff Voronoi diagram revisited and extended

Abstract

The Hausdorff Voronoi diagram of clusters of points in the plane is a generalization of Voronoi diagrams based on the Hausdorff distance function. Its combinatorial complexity is $O(n+m)$, where n is the total number of points and $m$ is the number of crossings between the input clusters ($m=O(n^2)$); the number of clusters is k. We present efficient algorithms to construct this diagram following the randomized incremental construction (RIC) framework (Clarkson and Shor in Discrete Comput Geom 4:387–421, 1989; Clarkson et al. in Comput Geom Theory Appl 3(4):185–212, 1993). Our algorithm for non-crossing clusters ($m=0$) runs in expected $O(n\log {n} + k\log n \log k)$ time and deterministic O(n) space. The algorithm for arbitrary clusters runs in expected $O((m+n\log {k})\log {n})$ time and $O(m+n\log {k})$ space. The two algorithms can be combined in a crossing-oblivious scheme within the same bounds. We show how to apply the RIC framework to handle non-standard characteristics of generalized Voronoi diagrams, including sites (and bisectors) of non-constant complexity, sites that are not enclosed in their Voronoi regions, empty Voronoi regions, and finally, disconnected bisectors and disconnected Voronoi regions. The Hausdorff Voronoi diagram finds direct applications in VLSI CAD.