Bottleneck Partial-Matching Voronoi Diagrams and Applications

TL;DR

This paper introduces a novel Voronoi diagram-based approach to optimize the bottleneck distance between point sets under translation, with efficient algorithms and applications in matching and pathfinding.

Contribution

It develops polynomial bounds and algorithms for constructing bottleneck partial-matching Voronoi diagrams, extending to higher dimensions and various applications.

Findings

Polynomial complexity bounds for the diagram

Efficient algorithms for diagram construction

Applications in optimal matching and path computation

Abstract

Given two point sets in the plane, we study the minimization of the bottleneck distance between a point set B and an equally-sized subset of a point set A under translations. We relate this problem to a Voronoi-type diagram and derive polynomial bounds for its complexity that are optimal in the size of A. We devise efficient algorithms for the construction of such a diagram and its lexicographic variant, which generalize to higher dimensions. We use the diagram to find an optimal bottleneck matching under translations, to compute a connecting path of minimum bottleneck cost between two positions of B, and to determine the maximum bottleneck cost in a convex polygon.