# 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… 2 Citations ## Figures from this paper This paper 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(logn) time and a shortest s-t path can be produced in additional time linear in the number of edges of the path. • Computer Science 2022 Sixth IEEE International Conference on Robotic Computing (IRC) • 2022 Two objectives contribute to the increased efficiency of the presented approach compared to classical automatic lawn mowing techniques. ## References SHOWING 1-10 OF 31 REFERENCES • Computer Science, Mathematics SIAM J. Comput. • 1999 The algorithm is based on an efficient implementation of wavefront propagation among polygonal obstacles, and it actually computes a planar map encoding shortest paths from a fixed source point to all other points of the plane; the map can be used to answer single-source shortest path queries in O(log n) time. • Computer Science SoCG '13 • 2013 An algorithm is proposed that computes an approximate shortest path map, a data structure with O(n log n) size, that allows it to report the (approximate) length of a shortest path from a fixed source point to any query point in the plane in O(log n) time. • Mathematics, Computer Science SCG '87 • 1987 This paper shows how to preprocess the polygon so that, given two query points, the length of the shortest path inside thePolygon can be found in time in time proportional to the number of turns it makes. It is shown that the complexity of the kth shortest path map is O(kh + kn), which is tight, and a simple visibility-based algorithm to compute the kTH shortest path between two points in O(km(m+ kn) log kn) time is presented. The shortest path query time for Sharir and Schorr's algorithm is shown to be O(k + log n) where k is the number faces traversed by the path and the preprocessed output requires O(n squared) space. This paper presents an O(n + m log m)-time algorithm for computing the geodesic nearest-point Voronoi diagram of m points in a simple n-gon that matches the best known lower bound of Ω(n) as well as improving the previously best known algorithms. A construction algorithm is given that is nearly optimal in the sense that if a single Voronoi vertex can be computed in O ( log n ) time, then the construction time will become the optimal O ( n + m log m ) . • Computer Science SIAM J. Comput. • 1986 A substantial refinement of the technique of Lee and Preparata for locating a point in$\mathcal{S}\$ based on separating chains is exhibited, which can be implemented in a simple and practical way, and is extensible to subdivisions with edges more general than straight-line segments.
• Mathematics, Computer Science
Networks
• 1984
The goal is to find interesting cases for which the solution can be obtained without the explicit construction of the entire visibility graph, which solve the problems by constructing the shortest-path tree from the source to all the vertices of the obstacles and to the destination.