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تسجيل مستخدم جديدA New Algorithm for Euclidean Shortest Paths in the Plane
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Haitao Wang
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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 $O(nlog n)$ time and $O(nlog n)$ space, where $n$ is the total number of vertices of all obstacles. Recently, by modifying Hershberger and Suris algorithm, Wang [SODA 2021] reduced the space to $O(n)$ while the runtime of the algorithm is still $O(nlog n)$. In this paper, we present a new algorithm of $O(n+hlog h)$ 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 $Omega(n+hlog h)$ 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.
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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 $O(nlog n)$ time and $O(nlog n)$ space, where $n$ is the total number of vertices of all obstacles. The algorithm is time-optimal because $Omega(nlog n)$ 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 $O(nlog n)$ 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 $O(nlog n)$ 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.
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