Computing L1 Shortest Paths among Polygonal Obstacles in the Plane

Given a point s and a set of h pairwise disjoint polygonal obstacles of totally n vertices in the plane, we present a new algorithm for building an L1 shortest path map of size O(n) in O(T ) time and O(n) space such that for any query point t, the length of the L1 shortest obstacleavoiding path from s to t can be reported in O(log n) time and the actual shortest path can be found in additional time proportional to the number of edges of the path, where T is the time for triangulating the free space. It is currently known that T = O(n+ h log h) for an arbitrarily small constant ǫ > 0. If the triangulation can be done optimally (i.e., T = O(n+ h log h)), then our algorithm is optimal. Previously, the best algorithm computes such an L1 shortest path map in O(n log n) time and O(n) space. Our techniques can be extended to obtain improved results for other related problems, e.g., computing the L1 geodesic Voronoi diagram for a set of point sites in a polygonal domain, finding shortest paths with fixed orientations, finding approximate Euclidean shortest paths, etc.