Shortest Paths Among Obstacles in the Plane Revisited

Abstract

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. The previous best algorithm was given by Hershberger and Suri [FOCS 1993, SIAM J. Comput. 1999] and the algorithm runs in $O(n\\log n)$ time and $O(n\\log n)$ space, where $n$ is the total number of vertices of all obstacles. The algorithm is time-optimal because $\\Omega(n\\log n)$ is a lower bound. It has been an open problem for over two decades whether the space can be reduced to $O(n)$. In this paper, we settle it by solving the problem in $O(n\\log n)$ time and $O(n)$ space, which is optimal in both time and space; we achieve this by modifying the algorithm of Hershberger and Suri. Like their original algorithm, our new algorithm can build a shortest path map for a source point $s$ in $O(n\\log n)$ time and $O(n)$ space, such that given any query point $t$, the length of a shortest path from $s$ to $t$ can be computed in $O(\\log n)$ time and a shortest path can be produced in additional time linear in the number of edges of the path.