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 all 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 minimum cost. 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 log ρ/ϵ n³ log (ρn ϵ)).