Determining the separation of simple polygons

Given simple polygons P and Q, their separation, denoted by (P;Q), is de ned to be the minimum distance between their boundaries. We present a parallel algorithm for nding a closest pair among all pairs (p; q), p 2 P and q 2 Q. The algorithm runs in O(logn) time using O(n) processors on a CREW PRAM, where n = jP j + jQj. This algorithm is time-optimal and improves by a factor of O(logn) on the time complexity of previous parallel methods. The algorithm can be implemented serially in (n) time, which gives the rst optimal sequential algorithmfor determining the separationof simple polygons. Our results are obtained by providing a uni ed treatment of the separation and the closest visible vertex problems for simple polygons.