On the Geodesic Centers of Polygonal Domains

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 $π$-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 $O(n^{12+ε})$ time was known for this problem, for any $ε>0$.