A New Algorithm for Euclidean Shortest Paths in the Plane

TL;DR

This paper introduces an efficient algorithm for computing obstacle-avoiding shortest paths in the plane, improving previous methods by reducing time and space complexity when a triangulation of free space is provided.

Contribution

The paper presents a new $O(n+h ext{log}h)$ time, $O(n)$ space algorithm for shortest paths, optimal in both metrics, assuming a given triangulation of the free space.

Findings

Achieves optimal time and space complexity for the problem.

Builds a shortest path map for quick query responses.

Improves upon previous algorithms when the number of obstacles is small.

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. Previously, Hershberger and Suri [SIAM J. Comput. 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 [SODA 2021] reduced the space to $O(n)$ while the runtime of the algorithm is still $O(n\log n)$. In this paper, 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, which improves the previous work when $h=o(n)$, is optimal in both time and space as $\Omega(n+h\log h)$ is a lower bound on the runtime. 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.