Approximate Shortest Paths in Anisotropic Regions

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 v s and v d , we give an algorithm that finds a path from v s to v d whose cost is at most (1 + ε) times the optimal. Our algorithm runs in O ρ 2 log ρ ε 2 n 3 log ρn εtime. 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 n 3 log ρn ε .

Optimal path finding in direction, location, and time dependent environments

“…It is important to acknowledge that earlier work on 'convex distance functions' [9,11,12] have stated some results similar to Lemma 2.1 and Theorem 2.3. However, none of the found literature provides a rigorous proof in its entirety, compelling us to include our proofs developed independently of the cited literature.…”

“…Asymmetric direction-dependence is occasionally considered in literature [12,61], however the introduced anisotropy makes a strong assumption of the distance function convexity which we relax in our analysis. The path-finding problems in a locationdependent environment examine a presence of polygonal obstacles [3,31,37,42,44] and uniform-weighted regions [11,47,67]. On the other hand, all the problems studied in the field of computational geometry are predominantly static, and timedependence is not considered in these settings.…”

“…In the most recent work on anisotropic movement, Cheng et al [11] generalize the problem studied by Reif and Sun, and look at the shortest path finding in anisotropic regions where the direction-dependency of the speed is not restricted to the effect of the uniform flow. Nevertheless, Cheng et al still limit their research to the speed function with a very specific structure, referred to as a 'convex distance function' (first discussed by Chew and Drysdale [12]).…”

Algorithms for Approximate Shortest Path Queries on Weighted Polyhedral Surfaces

“…During the write-up of the journal version of [6,7], the results in [19,20] with respect to the weighted region problem in planar domains have appeared. This work is inspired by our work and that of Reif and Sun [38], and the authors were able to remove the dependence on the geometric parameters at the expense of increasing the dependence on n as well as the ratio of the weights (max weight to min weight).…”

Shortest Gently Descending Paths

“…The weights in this problem capture, for example, the effect the gravity and friction on a vehicle moving on a slope. All the papers on this problem use the Steiner point approach [7,10,15,17]. As we mentioned before, these algorithms for anisotropic paths assume that every face f is totally traversable, i.e., there is a feasible path from any point to any other point in f .…”