An Optimal Deterministic Algorithm for Geodesic Farthest-Point Voronoi Diagrams in Simple Polygons

TL;DR

This paper introduces an optimal deterministic algorithm for computing geodesic farthest-point Voronoi diagrams in simple polygons, matching the known lower bound and improving upon previous deterministic methods.

Contribution

The paper presents the first deterministic algorithm achieving the optimal $O(n+m ext{log} m)$ time for the problem, resolving a two-decade-old open question.

Findings

The algorithm runs in $O(n+m ext{log} m)$ time, matching the lower bound.

It improves upon previous deterministic algorithms with higher time complexity.

The result confirms the optimality of the approach for this problem.

Abstract

Given a set $S$ of $m$ point sites in a simple polygon $P$ of $n$ vertices, we consider the problem of computing the geodesic farthest-point Voronoi diagram for $S$ in $P$. It is known that the problem has an $\Omega(n+m\log m)$ time lower bound. Previously, a randomized algorithm was proposed [Barba, SoCG 2019] that can solve the problem in $O(n+m\log m)$ expected time. The previous best deterministic algorithms solve the problem in $O(n\log \log n+ m\log m)$ time [Oh, Barba, and Ahn, SoCG 2016] or in $O(n+m\log m+m\log^2 n)$ time [Oh and Ahn, SoCG 2017]. In this paper, we present a deterministic algorithm of $O(n+m\log m)$ time, which is optimal. This answers an open question posed by Mitchell in the Handbook of Computational Geometry two decades ago.