Querying approximate shortest paths in anisotropic regions

We present a data structure for answering approximate shortest path queries in a planar subdivision from a fixed source. Let p ≤ 1 be a real number. Distances in each face of this subdivision are measured by a possibly asymmetric convex distance function whose unit disk is contained in a concentric unit Euclidean disk and contains a concentric Euclidean disk with radius 1/p. Different convex distance functions may be used for different faces, and obstacles are allowed. Let e be any number strictly between 0 and 1. Our data structure returns a (1 + e) approximation of the shortest path cost from the fixed source to a query destination in 0(log) time. Afterwards, a (1 + ε)-approximate shortest path can be reported in 0(log n) time plus the complexity of the path. The data structure uses 0(log)space and can be built in 0((log))² time- Our time and space bounds do not depend on any other parameter; in particular, they do not depend on any geometric parameter of the subdivision such as the minimum angle.