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.

Research supported in part by the Swiss National Science Foundation, projects SNF 20GG21-134355 (ESF EUROCORES EuroGIGA/VORONOI) and SNF 200021E-154387. E. K. was also supported partially by F.R.S.-FNRS and the SNF P2TIP2-168563 under the Early PostDoc Mobility program.

Research performed mainly while E. K. was at the Università della Svizzera italiana (USI).