Highway Hull Revisited

TL;DR

This paper introduces efficient algorithms for computing the highway hull in the plane under L_1 and L_2 metrics, improving computational complexity and defining the useful region for highway inclusion.

Contribution

It presents the first Theta(n log n) algorithm for highway hulls under L_1 and an improved O(n log^2 n) algorithm for L_2, along with the concept of the useful region.

Findings

Theta(n log n) algorithm for L_1 highway hulls

O(n log^2 n) algorithm for L_2 highway hulls

Definition and construction of the useful region

Abstract

A highway H is a line in the plane on which one can travel at a greater speed than in the remaining plane. One can choose to enter and exit H at any point. The highway time distance between a pair of points is the minimum time required to move from one point to the other, with optional use of H. The highway hull HH(S,H) of a point set S is the minimal set containing S as well as the shortest paths between all pairs of points in HH(S,H), using the highway time distance. We provide a Theta(n log n) worst-case time algorithm to find the highway hull under the L_1 metric, as well as an O(n log^2 n) time algorithm for the L_2 metric which improves the best known result of O(n^2). We also define and construct the useful region of the plane: the region that a highway must intersect in order that the shortest path between at least one pair of points uses the highway.