Constructing Optimal Highways

TL;DR

This paper studies the problem of constructing highways in the plane to minimize maximum travel time between points, providing efficient algorithms for fixed and arbitrary orientations, and for pairs of highways.

Contribution

It introduces algorithms for finding optimal highways with fixed and arbitrary orientations, and for pairs of highways, improving understanding of city distance optimization.

Findings

Optimal highway with fixed orientation computed in linear time.

Optimal highway with arbitrary orientation computed in O(n^2 log n) time.

Extended to pairs of highways, one horizontal and one vertical.

Abstract

For two points $p$ and $q$ in the plane, a straight line $h$, called a highway, and a real $v>1$, we define the \emph{travel time} (also known as the \emph{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 $v$ 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 $v$, we consider the problem of finding a \emph{highway} that minimizes the maximum travel time over all pairs of points in $S$. If the orientation of the highway is fixed, the optimal highway can be computed in linear time, both for the $L_1$- and the Euclidean metric as the underlying metric. If arbitrary orientations are allowed, then the optimal highway can be computed in $O(n^{2} \log n)$ time. We also consider the problem of computing an optimal pair of highways, one being horizontal, one vertical.