Querying approximate shortest paths in anisotropic regions

We present a data structure for answering approximate shortest path queries ina planar subdivision from a fixed source. Let ρ ≥ 1 be a real number.Distances in each face of this subdivision are measured by a possiblyasymmetric convex distance function whose unit disk is contained in aconcentric unit Euclidean disk, and contains a concentric Euclidean disk withradius 1/ρ. Different convex distance functions may be used for differentfaces, and obstacles are allowed. Let ε be any number strictly between 0and 1. Our data structure returns a (1+ε)approximation of the shortest path cost from the fixed source to a querydestination in O(logρn/ε) time. Afterwards, a(1+ε)-approximate shortest path can be reported in time linear in itscomplexity. The data structure uses O(ρ<sup>2</sup> n<sup>4</sup>/ε<sup>2</sup> log ρn/ε) space and can be built in O((ρ<sup>2</sup> n<sup>4</sup>)/(ε<sup>2</sup>)(log ρn/ε)<sup>2</sup>) time. Our time and space bounds do not depend onany geometric parameter of the subdivision such as the minimum angle.

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