Quickest Path Queries on Transportation Network

This paper considers the problem of finding a quickest path between two points in the Euclidean plane in the presence of a transportation network. A transportation network consists of a planar network where each road (edge) has an individual speed. A traveller may enter and exit the network at any point on the roads. Along any road the traveller moves with a fixed speed depending on the road, and outside the network the traveller moves at unit speed in any direction. We give an exact algorithm for the basic version of the problem: given a transportation network of total complexity n in the Euclidean plane, a source point s and a destination point t, and the quickest path between s and t. We also show how the transportation network can be preprocessed in time O(n^2 log n) into a data structure of size O(n^2) such that (1 + \epsilon)-approximate cheapest path cost queries between any two points in the plane can be answered in time O(1\epsilon^4 log n).