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 via the randomized incremental construction (RIC) framework [Clarkson et al. 89,93]. For non-crossing clusters (\(m=0\)), our algorithm runs in expected \(O(n\log {n} + k\log n \log k)\) time and deterministic O(n) space. For arbitrary clusters the algorithm 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 efficiently 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 Voronoi regions. The diagram finds direct applications in VLSI CAD.