Touring a sequence of polygons

Given a sequence of k polygons in the plane, a start point s, and a target point, t, we seek a shortest path that starts at s, visits in order each of the polygons, and ends at t. If the polygons are disjoint and convex, we give an algorithm running in time O(kn log(n/k)), where n is the total number of vertices specifying the polygons. We also extend our results to a case in which the convex polygons are arbitrarily intersecting and the subpath between any two consecutive polygons is constrained to lie within a simply connected region; the algorithm uses O(nk² log n) time. Our methods are simple and allow shortest path queries from s to a query point t to be answered in time O(k log n + m), where m is the combinatorial path length. We show that for nonconvex polygons this "touring polygons" problem is NP-hard. The touring polygons problem is a strict generalization of some classic problems in computational geometry, including the safari problem, the zoo-keeper problem, and the watchman route problem in a simple polygon. Our new results give an order of magnitude improvement in the running times of the safari problem and the watchman route problem: We solve the safari problem in O(n² log n) time and the watchman route problem (through a fixed point s) in time O(n³ log n), compared with the previous time bounds of O(n³) and O(n⁴), respectively.