Approximate shortest paths in anisotropic regions*

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 ε{lunate} (0, 1) and for any two points v_s and v_s 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(ρ² log ρ/ε² n³ log (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 logρ/ε n³ log (ρn/ε)).