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 all e ∈ (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 + e) times the minimum cost. Our algorithm runs in O (ρ2logρ/e2n3 log (ρn/e)) 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 (ρ2logρ/en3 log (ρn/e)).