A Randomized Incremental Approach for the Hausdorff Voronoi Diagram of Non-crossing Clusters

In the Hausdorff Voronoi diagram of a set of point-clusters in the plane, the distance between a point t and a cluster P is measured as the maximum distance between t and any point in P while the diagram is defined in a nearest sense. This diagram finds direct applications in VLSI computer-aided design. In this paper, we consider “non-crossing” clusters, for which the combinatorial complexity of the diagram is linear in the total number n of points on the convex hulls of all clusters. We present a randomized incremental construction, based on point-location, to compute the diagram in expected O(n log n) time and expected O(n) space, which considerably improves previous results. Our technique efficiently handles non-standard characteristics of generalized Voronoi diagrams, such as sites of non-constant complexity, sites that are not enclosed in their Voronoi regions, and empty Voronoi regions.