Constructing Optimal Highways

Abstract

For two points p and q in the plane, a (unbounded) line h, called a highway, and a real ν > 1, we define the travel time (also known as the city distance) from p and q to be the time needed to traverse a quickest path from p to q, where the distance is measured with speed ν on h and with speed 1 in the underlying metric elsewhere. Given a set S of n points in the plane and a highway speed ν, we consider the problem of finding an axis-parallel line, the highway, that minimizes the maximum travel time over all pairs of points in S. We achieve a linear-time algorithm both for the L1- and the Euclidean metric as the underlying metric. We also consider the problem of computing an optimal pair of highways, one being horizontal, one vertical. © 2007, Australian Computer Society, Inc.