On the Geodesic Centers of Polygonal Domains

In this paper, we study the problem of computing Euclidean geodesic centers of a polygonal domain 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 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 P , which is known to be O(n). 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 log n) time. Previously, an algorithm of O(n) time was known for this problem, for any ǫ > 0.