The Geodesic Diameter of Polygonal Domains

This paper studies the geodesic diameter of polygonal domains having h holes and n corners. For simple polygons (i.e., h = 0), it is known that the geodesic diameter is determined by a pair of corners of a given polygon and can be computed in linear time. For general polygonal domains with h >= 1, however, no algorithm for computing the geodesic diameter was known prior to this paper. In this paper, we present the first algorithm that computes the geodesic diameter of a given polygonal domain in worst-case time O(n(7.73)) or O(n(7)(log n+h)). Among other results, we show the following geometric observation: the geodesic diameter can be determined by two points in its interior. In such a case, there are at least five shortest paths between the points.