A New Approach to Output-Sensitive Voronoi Diagrams and Delaunay Triangulations

TL;DR

This paper introduces an output-sensitive algorithm for computing Voronoi diagrams in Euclidean space, improving efficiency especially for inputs with polynomial spread and near-linear output size.

Contribution

It presents a novel algorithm that computes Voronoi diagrams with a runtime dependent on output complexity and input spread, using Voronoi refinement and local flips.

Findings

Improves on the state of the art for polynomial spread inputs

Achieves near-linear time for certain input conditions

Uses Voronoi refinement and local flips for efficient computation

Abstract

We describe a new algorithm for computing the Voronoi diagram of a set of $n$ points in constant-dimensional Euclidean space. The running time of our algorithm is $O(f \log n \log \Delta)$ where $f$ is the output complexity of the Voronoi diagram and $\Delta$ is the spread of the input, the ratio of largest to smallest pairwise distances. Despite the simplicity of the algorithm and its analysis, it improves on the state of the art for all inputs with polynomial spread and near-linear output size. The key idea is to first build the Voronoi diagram of a superset of the input points using ideas from Voronoi refinement mesh generation. Then, the extra points are removed in a straightforward way that allows the total work to be bounded in terms of the output complexity, yielding the output sensitive bound. The removal only involves local flips and is inspired by kinetic data structures.