A New Algorithm for Euclidean Shortest Paths in the Plane

Given a set of pairwise disjoint polygonal obstacles in the plane, finding an obstacle-avoiding Euclidean shortest path between two points is a classical problem in computational geometry and has been studied extensively. Previously, Hershberger and Suri (in SIAM Journal on Computing, 1999) gave an algorithm of O(n log n) time and O(n log n) space, where n is the total number of vertices of all obstacles. Recently, by modifying Hershberger and Suri’s algorithm, Wang (in SODA’21) reduced the space to O(n) while the runtime of the algorithm is still O(n log n). In this article, we present a new algorithm of O(n+h log h) time and O(n) space, provided that a triangulation of the free space is given, where h is the number of obstacles. The algorithm is better than the previous work when h is relatively small. Our algorithm builds a shortest path map for a source point s so that given any query point t, the shortest path length from s to t can be computed in O(log n) time and a shortest s-t path can be produced in additional time linear in the number of edges of the path.