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

Abstract

Listen

Translate article

Given in the plane 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 solves 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 that takes $$O(n+m\log m)$$ time, which is optimal. This answers affirmatively an open question posed by Mitchell in the Handbook of Computational Geometry two decades ago.