An Optimal Algorithm for L1 Shortest Paths Among Obstacles in the Plane (Draft)

We present an optimal Θ(n log n) algorithm for determining shortest paths according to the L1 (L∞) metric in the presence of disjoint polygonal obstacles in the plane. Our algorithm requires only linear O(n) space to build a planar subdivision (a Shortest Path Map) with respect to a fixed source point such that the length of a shortest path from the source to any query point can be reported in time O(log n) by locating the query point in the subdivision. An actual shortest path from the source to the query point can be reported in additional time O(k), where k is the number of “turns” in the path. The algorithm uses the continuous Dijkstra methodology of propagating a “wavefront” through the plane. The algorithm can be generalized to find shortest paths according to any “fixed orientation” metric, yielding an O(n log n √ ǫ ) approximation algorithm for finding Euclidean shortest paths among obstacles. The algorithm can further be generalized to the case of multiple sources to build a Voronoi diagram for multiple source points which lie among a collection of obstacles in time θ(N logN), where N is the maximum of the number of sources and the number of obstacle vertices.