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 rho >= 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/rho. 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 + epsilon) approximation of the shortest path cost from the fixed source to a query destination in O(log rho n/epsilon) time. Afterwards, a (1 + epsilon)- approximate shortest path can be reported in O(log n) time plus the complexity of the path. The data structure uses O(rho(2)n(3)/epsilon(2) log rho n/epsilon) space and can be built in O(rho(2)n(3)/epsilon(2) (log rho n/epsilon)(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 angleclos

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 ?? ??? 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/??. Different convex distance functions may be used for differentfaces, and obstacles are allowed. Let ?? be any number strictly between 0and 1. Our data structure returns a (1+??)approximation of the shortest path cost from the fixed source to a querydestination in O(log??n/??) time. Afterwards, a(1+??)-approximate shortest path can be reported in time linear in itscomplexity. The data structure uses O(??2 n4/??2 log ??n/??) space and can be built in O((??2 n4)/(??2)(log ??n/??)2) time. Our time and space bounds do not depend onany geometric parameter of the subdivision such as the minimum angle