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Title: Approximate shortest paths in anisotropic regions
Authors: Cheng, Siu-Wing
Na, Hyeon-Suk
Vigneron, Antoine
Wang, Yajun
Keywords: Shortest path
Approximation algorithm
Computational geometry
Issue Date: 2008
Citation: SIAM journal on computing, v.38, no. 3, p. 802-824
Abstract: Our goal is to find an approximate shortest path for a point robot moving in a planar subdivision with n vertices. Let ρ ≥ 1 be a real number. Distances in each face of this subdivision are measured by a convex distance function whose unit disk is contained in a concentric unit Euclidean disk, and contains a concentric Euclidean disk with radius 1/ρ. Different convex distance functions may be used for different faces, and obstacles are allowed. These convex distance functions may be asymmetric. For any ε ∈ (0, 1) and for any two points vs and vd, we give an algorithm that finds a path from vs to vd whose cost is at most (1 + ε) times the optimal. Our algorithm runs in O (ρ2 log ρ/ε2 n3log (ρn/ε)) time. This bound does not depend on any other parameters; in particular it does not depend on the minimum angle in the subdivision. We give applications to two special cases that have been considered before: the weighted region problem and motion planning in the presence of uniform flows. For the weighted region problem with weights in [1, ρ] ∪ {∞}, the time bound of our algorithm improves to O (ρ log ρ/ε n3log (ρn/ε)).
Rights: Copyright © SIAM. This paper is made available with permission of the Society for Industrial and Applied Mathematics for limited noncommerical distribution only.
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