Voronoi Diagrams in the Hilbert Metric

TL;DR

This paper investigates the properties of Voronoi diagrams under the Hilbert metric within convex polygons, providing algorithms with efficient expected and worst-case complexities for their computation.

Contribution

It introduces two algorithms, one randomized and one deterministic, for computing Voronoi diagrams in the Hilbert metric for convex polygons, with proven time and space complexities.

Findings

Expected algorithm runs in O(m n + n (log n)(log m n)) time.

Deterministic algorithm runs in O(m n log n) time.

Worst-case complexity of the Voronoi diagram is Θ(m n).

Abstract

The Hilbert metric is a distance function defined for points lying within a convex body. It generalizes the Cayley-Klein model of hyperbolic geometry to any convex set, and it has numerous applications in the analysis and processing of convex bodies. In this paper, we study the geometric and combinatorial properties of the Voronoi diagram of a set of point sites under the Hilbert metric. Given any convex polygon $K$ bounded by $m$ sides, we present two algorithms (one randomized and one deterministic) for computing the Voronoi diagram of an $n$-element point set in the Hilbert metric induced by $K$. Our randomized algorithm runs in $O(m n + n (\log n)(\log m n))$ expected time, and our deterministic algorithm runs in time $O(m n \log n)$. Both algorithms use $O(m n)$ space. We show that the worst-case combinatorial complexity of the Voronoi diagram is $\Theta(m n)$.