Computing the Geodesic Centers of a Polygonal Domain

TL;DR

This paper introduces an algorithm to compute the geodesic center of complex polygonal domains with holes, extending previous work from simple polygons to more general shapes with a significantly higher computational complexity.

Contribution

It presents the first algorithm capable of handling general polygonal domains with holes for computing the geodesic center, with a running time of O(n^{12+ε}).

Findings

Handles polygonal domains with holes.

First algorithm for general polygonal domains.

Computational complexity is high, O(n^{12+ε}).

Abstract

We present an algorithm that computes the geodesic center of a given polygonal domain. The running time of our algorithm is $O(n^{12+\epsilon})$ for any $\epsilon>0$, where $n$ is the number of corners of the input polygonal domain. Prior to our work, only the very special case where a simple polygon is given as input has been intensively studied in the 1980s, and an $O(n \log n)$-time algorithm is known by Pollack et al. Our algorithm is the first one that can handle general polygonal domains having one or more polygonal holes.