The Geodesic $2$-center Problem in a Simple Polygon

TL;DR

This paper presents an exact algorithm for the geodesic 2-center problem in simple polygons, efficiently computing a set of two points minimizing the maximum geodesic distance within the polygon.

Contribution

It introduces the first exact algorithm for the geodesic 2-center problem with a time complexity of O(n^2 log^2 n), advancing computational geometry methods.

Findings

Algorithm computes geodesic 2-center in O(n^2 log^2 n) time

Provides a precise solution for the geodesic 2-center problem in simple polygons

Enhances understanding of geodesic center problems in computational geometry

Abstract

The geodesic $k$-center problem in a simple polygon with $n$ vertices consists in the following. Find a set $S$ of $k$ points in the polygon that minimizes the maximum geodesic distance from any point of the polygon to its closest point in $S$. In this paper, we focus on the case where $k=2$ and present an exact algorithm that returns a geodesic $2$-center in $O(n^2\log^2 n)$ time.