Fitting Voronoi Diagrams to Planar Tesselations

TL;DR

This paper presents an algorithm to find a minimal set of points whose Voronoi diagram matches a given planar tessellation, with a linear number of points relative to the tessellation size under certain angle constraints.

Contribution

The paper introduces a new algorithm for the Generalized Inverse Voronoi Problem that efficiently constructs fitting Voronoi diagrams with a linear number of points.

Findings

Algorithm solves GIVP with linear point count

Works under constant smallest angle assumption

Efficiently fits Voronoi diagrams to planar tessellations

Abstract

Given a tesselation of the plane, defined by a planar straight-line graph $G$, we want to find a minimal set $S$ of points in the plane, such that the Voronoi diagram associated with $S$ "fits" \ $G$. This is the Generalized Inverse Voronoi Problem (GIVP), defined in \cite{Trin07} and rediscovered recently in \cite{Baner12}. Here we give an algorithm that solves this problem with a number of points that is linear in the size of $G$, assuming that the smallest angle in $G$ is constant.