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 first algorithms that compute the geodesic diameter of a given polygonal domain in worst-case time O(n) or O(n(log n + h)). The algorithms are based on our new geometric observations, part of which states as follows: the geodesic diameter of a polygonal domain can be determined by two points in its interior, and in that case there are at least five shortest paths between the two points. ∗Work by S.W. Bae was supported by the Brain Korea 21 Project. Work by Y. Okamoto was supported by Global COE Program “Computationism as a Foundation for the Sciences” and Grant-in-Aid for Scientific Research from Ministry of Education, Science and Culture, Japan, and Japan Society for the Promotion of Science.