On the Geodesic Centers of Polygonal Domains

TL;DR

This paper investigates the computation of Euclidean geodesic centers in polygonal domains, providing new theoretical bounds and an improved algorithm with reduced complexity from previous methods.

Contribution

It introduces a necessary condition for geodesic centers, bounds their total number, and presents an improved algorithm with complexity $O(n^{11}\log n)$ for computing all centers.

Findings

Number of geodesic centers is bounded by $O(n^{10})$

Proposed algorithm runs in $O(n^{11}\log n)$ time

New $ ext{π}$-range property aids in analysis

Abstract

In this paper, we study the problem of computing Euclidean geodesic centers of a polygonal domain $\mathcal{P}$ with a total of $n$ vertices. We discover many interesting observations. We give a necessary condition for a point being a geodesic center. We show that there is at most one geodesic center among all points of $\mathcal{P}$ that have topologically-equivalent shortest path maps. This implies that the total number of geodesic centers is bounded by the combinatorial size of the shortest path map equivalence decomposition of $\mathcal{P}$, which is known to be $O(n^{10})$. One key observation is a $\pi$-range property on shortest path lengths when points are moving. With these observations, we propose an algorithm that computes all geodesic centers in $O(n^{11}\log n)$ time. Previously, an algorithm of $O(n^{12+\epsilon})$ time was known for this problem, for any $\epsilon>0$.