Fast query structures in anisotropic media

We study the minimum cost path problem in an environment where the cost is direction-dependent (anisotropic). This problem arise in sailing, robotics, aircraft navigation, and routing of autonomous vehicles, where the cost is affected by the direction of waves, winds or slope of the terrain. We present an approximation algorithm to find a minimum cost path for a point robot moving in a planar subdivision, where each face is assigned a translational flow that reflects the cost of travelling within this face. In addition we consider the nearest neighbor problem in an anisotropic media that is given a planar subdivision together with a set Z of m sites, a query point q, find the nearest site in Z from q. Our main contribution is a data structure that given a subdivision with translational flows returns a (1 + ε)-approximate minimum cost path in the subdivision between any two query points in the plane.