Geodesic farthest-point Voronoi diagram in linear time

TL;DR

This paper introduces a randomized algorithm that efficiently computes the geodesic farthest-point Voronoi diagram in a simple polygon, achieving linear expected time for vertex sites and extending to general sites, solving an open problem.

Contribution

The paper presents the first linear-time randomized algorithm for geodesic farthest-point Voronoi diagrams in polygons, addressing an open problem in computational geometry.

Findings

Expected $O(n + m)$ time for vertex sites

Extended to $O(n + m \log m)$ time for arbitrary sites

Solves an open problem in the field

Abstract

Let $P$ be a simple polygon with $n$ vertices. For any two points in $P$, the geodesic distance between them is the length of the shortest path that connects them among all paths contained in $P$. Given a set $S$ of $m$ sites being a subset of the vertices of $P$, we present a randomized algorithm to compute the geodesic farthest-point Voronoi diagram of $S$ in $P$ running in expected $O(n + m)$ time. That is, a partition of $P$ into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to the geodesic distance. In particular, this algorithm can be extended to run in expected $O(n + m\log m)$ time when $S$ is an arbitrary set of $m$ sites contained in $P$, thereby solving the open problem posed by Mitchell in Chapter 27 of the Handbook of Computational Geometry.