Querying Approximate Shortest Paths in Anisotropic Regions

Abstract

We present a data structure for answering approximate shortest path queries in a planar subdivision from a fixed source. Let $\rho\geqslant1$ 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 $\varepsilon$ be any number strictly between 0 and 1. Our data structure returns a $(1+\varepsilon)$ approximation of the shortest path cost from the fixed source to a query destination in $O(\log\frac{\rho n}{\varepsilon})$ time. Afterwards, a $(1+\varepsilon)$-approximate shortest path can be reported in $O(\log n)$ time plus the complexity of the path. The data structure uses $O(\frac{\rho^2n^3}{\varepsilon^2}\log\frac{\rho n}{\varepsilon})$ space and can be built in $O(\frac{\rho^2n^3}{\varepsilon^2}(\log\frac{\rho n}{\varepsilon})^2)$ 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.