Shortest Paths in the Plane with Obstacle Violations

We study the problem of finding shortest paths in the plane among h convex obstacles, where the path is allowed to pass through (violate) up to k obstacles, for k ≤ h. Equivalently, the problem is to find shortest paths that become obstacle-free if k obstacles are removed from the input. Given a fixed source point s, we show how to construct a map, called a shortest k-path map, so that all destinations in the same region of the map have the same combinatorial shortest path passing through at most k obstacles. We prove a tight bound of Θ(kn) on the size of this map, and show that it can be computed in O(k2n logn) time, where n is the total number of obstacle vertices. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems