Computing the L1 Geodesic Diameter and Center of a Polygonal Domain

For a polygonal domain with h holes and a total of n vertices, we present algorithms that compute the L1 geodesic diameter in O(n2 + h4) time and the L1 geodesic center in O((n4 + n2h4)α(n)) time, respectively, where α(·) denotes the inverse Ackermann function. No algorithms were known for these problems before. For the Euclidean counterpart, the best algorithms compute the geodesic diameter in O(n7.73) or O(n7(h+ logn)) time, and compute the geodesic center in O(n11 logn) time. Therefore, our algorithms are significantly faster than the algorithms for the Euclidean problems. Our algorithms are based on several interesting observations on L1 shortest paths in polygonal domains.