• DOI:10.1137/1.9781611976465.51
• Corpus ID: 224706389

Shortest Paths Among Obstacles in the Plane Revisited

@article{Wang2020ShortestPA,
  title={Shortest Paths Among Obstacles in the Plane Revisited},
  author={Haitao Wang},
  journal={ArXiv},
  year={2020},
  volume={abs/2010.09115}
}

• Haitao Wang
• Published 18 October 2020
• Computer Science
• ArXiv

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.