Voronoi Diagrams for a Moderate-Sized Point-Set in a Simple Polygon

TL;DR

This paper introduces improved algorithms for computing geodesic Voronoi diagrams within simple polygons, achieving optimal performance for certain cases and addressing open questions in computational geometry.

Contribution

It presents new algorithms for geodesic Voronoi diagrams in simple polygons that are more efficient and optimal for specific sizes of site sets, partially answering a longstanding open problem.

Findings

Algorithms for geodesic nearest-point and farthest-point Voronoi diagrams are optimal for m <= n/ polylog n.

New methods improve upon previous algorithms for higher-order diagrams.

Partially resolves a question from the Handbook of Computational Geometry.

Abstract

Given a set of sites in a simple polygon, a geodesic Voronoi diagram of the sites partitions the polygon into regions based on distances to sites under the geodesic metric. We present algorithms for computing the geodesic nearest-point, higher-order and farthest-point Voronoi diagrams of m point sites in a simple n-gon, which improve the best known ones for m <= n/ polylog n. Moreover, the algorithms for the geodesic nearest-point and farthest-point Voronoi diagrams are optimal for m <= n/ polylog n. This partially answers a question posed by Mitchell in the Handbook of Computational Geometry.