Constructing Two-Dimensional Voronoi Diagrams via Divide-and-Conquer of Envelopes in Space

TL;DR

This paper introduces a flexible divide-and-conquer framework for efficiently computing various types of two-dimensional Voronoi diagrams, including on parametric surfaces, with exact results even in degenerate cases.

Contribution

The authors develop a general envelope-based divide-and-conquer approach for Voronoi diagrams that is adaptable to different site types, distance functions, and surfaces, with near-optimal expected running time.

Findings

Framework supports diagrams on parametric surfaces in R^3.

Expected running time is near-optimal with randomization.

Implemented algorithms for multiple Voronoi diagram types and applications.

Abstract

We present a general framework for computing two-dimensional Voronoi diagrams of different classes of sites under various distance functions. The framework is sufficiently general to support diagrams embedded on a family of two-dimensional parametric surfaces in $R^3$. The computation of the diagrams is carried out through the construction of envelopes of surfaces in 3-space provided by CGAL (the Computational Geometry Algorithm Library). The construction of the envelopes follows a divide-and-conquer approach. A straightforward application of the divide-and-conquer approach for computing Voronoi diagrams yields algorithms that are inefficient in the worst case. We prove that through randomization the expected running time becomes near-optimal in the worst case. We show how to employ our framework to realize various types of Voronoi diagrams with different properties by providing implementations for a vast collection of commonly used Voronoi diagrams. We also show how to apply the new framework and other existing tools from CGAL to compute minimum-width annuli of sets of disks, which requires the computation of two Voronoi diagrams of two different types, and of the overlay of the two diagrams. We do not assume general position. Namely, we handle degenerate input, and produce exact results.