Randomized incremental construction of the Hausdorff Voronoi diagram of non-crossing clusters

TL;DR

This paper introduces efficient randomized incremental algorithms for constructing the Hausdorff Voronoi diagram of non-crossing point clusters, with improved expected and worst-case complexities, applicable in VLSI design.

Contribution

It presents novel randomized algorithms with better expected and worst-case complexities for constructing Hausdorff Voronoi diagrams of non-crossing clusters.

Findings

Expected time complexity: O((n + m(log log(n))^2)log^2(n))

Practical algorithm with expected time O((n + m log(n))log^2(n))

Space complexity: O(n)

Abstract

In the Hausdorff Voronoi diagram of a set of clusters of points in the plane, the distance between a point t and a cluster P is the maximum Euclidean distance between t and a point in P. This diagram has direct applications in VLSI design. We consider so-called "non-crossing" clusters. The complexity of the Hausdorff diagram of m such clusters is linear in the total number n of points in the convex hulls of all clusters. We present randomized incremental constructions for computing efficiently the diagram, improving considerably previous results. Our best complexity algorithm runs in expected time O((n + m(log log(n))^2)log^2(n)) and worst-case space O(n). We also provide a more practical algorithm whose expected running time is O((n + m log(n))log^2(n)) and expected space complexity is O(n). To achieve these bounds, we augment the randomized incremental paradigm for the construction of Voronoi diagrams with the ability to efficiently handle 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.