Voronoi Graph -- Improved raycasting and integration schemes for high dimensional Voronoi diagrams

TL;DR

This paper introduces an improved, exact vertex search method and a depth-first traversal algorithm for high-dimensional Voronoi diagrams, enhancing computational efficiency and enabling Monte Carlo volume approximations.

Contribution

It presents a novel exact vertex search technique and a depth-first traversal algorithm for high-dimensional Voronoi diagrams, outperforming previous methods in speed and accuracy.

Findings

Faster exact vertex search method than previous bisection approach

Comparable or improved computation times compared to qHull

Effective Monte Carlo approximation for volume and boundary integrals

Abstract

The computation of Voronoi Diagrams, or their dual Delauney triangulations is difficult in high dimensions. In a recent publication Polianskii and Pokorny propose an iterative randomized algorithm facilitating the approximation of Voronoi tesselations in high dimensions. In this paper, we provide an improved vertex search method that is not only exact but even faster than the bisection method that was previously recommended. Building on this we also provide a depth-first graph-traversal algorithm which allows us to compute the entire Voronoi diagram. This enables us to compare the outcomes with those of classical algorithms like qHull, which we either match or marginally beat in terms of computation time. We furthermore show how the raycasting algorithm naturally lends to a Monte Carlo approximation for the volume and boundary integrals of the Voronoi cells, both of which are of importance for finite Volume methods. We compare the Monte-Carlo methods to the exact polygonal integration, as well as a hybrid approximation scheme.