All Farthest Neighbors in the Presence of Highways and Obstacles

We consider the problem of computing all farthest neighbors (and the diameter) of a given set of n points in the presence of highways and obstacles in the plane. When traveling on the plane, travelers may use highways for faster movement and must avoid all obstacles. We present an efficient solution to this problem based on knowledge from earlier research on shortest path computation. Our algorithms run in O(nm(logm + log n)) time using O(m + n) space, where the m is the combinatorial complexity of the environment consisting of highways and obstacles.