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

Alexander Sikorski, Martin Heida

TL;DR

High-dimensional Voronoi diagram computation is notoriously challenging due to combinatorial explosion. This paper introduces an exact, faster incircle RayCast procedure and a depth-first exhaustive graph traversal (VoroniGraph) to compute the full diagram, achieving competitive runtime with qHull. It then offers three integration schemes—Monte Carlo, Leibniz-rule based exact polygonal integration, and a heuristic Monte-Carlo approach—to estimate volumes and surface integrals over Voronoi cells, enabling finite-volume computations in high dimensions. Together, these advances yield a flexible, scalable toolkit for exact and approximate Voronoi computations and their numerical integration across dimensions, with practical impact for simulations and geometric processing.

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.

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

TL;DR

High-dimensional Voronoi diagram computation is notoriously challenging due to combinatorial explosion. This paper introduces an exact, faster incircle RayCast procedure and a depth-first exhaustive graph traversal (VoroniGraph) to compute the full diagram, achieving competitive runtime with qHull. It then offers three integration schemes—Monte Carlo, Leibniz-rule based exact polygonal integration, and a heuristic Monte-Carlo approach—to estimate volumes and surface integrals over Voronoi cells, enabling finite-volume computations in high dimensions. Together, these advances yield a flexible, scalable toolkit for exact and approximate Voronoi computations and their numerical integration across dimensions, with practical impact for simulations and geometric processing.

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.
Paper Structure (23 sections, 1 theorem, 24 equations, 10 figures, 4 tables, 5 algorithms)

This paper contains 23 sections, 1 theorem, 24 equations, 10 figures, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Voronoi diagrams via raycasting
  3. Notation
  4. The incircle RayCast procedure
  5. The VoronoiGraph algorithm
  6. Remark on the computational complexity
  7. Integration
  8. Monte-Carlo Integration
  9. Leibnitz Integration
  10. The Leibnitz rule
  11. Integration of a piecewise linearly interpolated function
  12. The integration algorithm
  13. Heuristic Monte-Carlo integration
  14. Computations and evaluation of performance
  15. Performance of the incircle RayCast
  16. ...and 8 more sections

Key Result

Lemma 3.1

Given a complete Voronoi diagram $\mathcal{M}$ the algorithm ($i,\mathcal{M},f$) yields for each cell $C_i$ the exact volume, as well as the exact area of all interfaces with its neighbors. Furthermore, it provides the exact integral of a function $F$ which is the linear interpolation of $f$ on the vertices, the cell centers $x_i$ and the midpoints $r_m$ calc .

Figures (10)

  • Figure 1: Illustration of a voronoi diagram with labeled generators, two vertices and their connecting edge. The circles illustrate the equidistance of the generators to the respective vertices. The edge is equidistant to its generators as well as perpendicular to the plane spanned by them.
  • Figure 2: Illustration of the incircle RayCast search in 2D. From left to right: (a) We start with an initial guess $r$ (equidistant to the current generators) and a search direction (perpendicular to the generator's plane) $u$ (b) Nearest neighbour search (on the right half-plane) leads to a possible generator $x$. The projection then leads to the vertex candidate $r'$ (c) Nearest neighbour search finds another possible generator, leading to another candidate. (d) Nearest neighbour search returns no other generator, confirming the incircle criterion.
  • Figure 3: Schematic of the construction of the center of a regular 3--simplex from the regular 2--simplex center using Pythagoras' theorem.
  • Figure 4: Comparison of the compute time of the Voronoi diagram in dimensions $d=2$ (lower three curves) and $d=6$ (upper three curves) over varying number of input nodes $n=(100,300,1\,000,...,300\,000)$.
  • Figure 5: Compute time of the volume and integral calculations using the Polygon (P), Monte-Carlo ( MC ) and heuristic MC (HMC) methods. The number of samples (#rays, #t) for MC are, from left to right: (1 000, 2), (10 000, 2) and (10 000, 10).
  • ...and 5 more figures

Theorems & Definitions (1)

  • Lemma 3.1