Computing external watchman routes on PRAM, BSR, and interconnection network models of parallel computation

An external watchman route in the presence of a polygonal obstacle is a closed path such that each point in the exterior of the polygon is visible to some point along the route. We adapt the merging slopes technique of parallel computational geometry to develop a parallel algorithm for computing a shortest external watchman route in the presence of a convex polygon of n sides. The algorithm runs in O(log n) time using O(n/log n) processors in the CREW-PRAM computational model; this is optimal within a constant factor. The algorithm can be easily adapted to compute a shortest watchman route in O(log n) time on a hypercube with O(n) processors. We also discuss the computation of a shortest external watchman route on star and pancake networks. Finally, a constant time algorithm for solving the merging slopes problem on a BSR with n processors is described. This leads to algorithms with the same time and processor count for solving the external watchman route problem, for computing Minkowski sum, critical support lines, and separability of two convex polygons, for finding the maximum distance between two convex polygons, and for computing the smallest enclosing box, diameter, and width of a convex polygon.