Optimal Time-Convex Hull under the Lp Metrics

TL;DR

This paper introduces an optimal algorithm for computing the time-convex hull of a point set under the general Lp metric in the plane with a highway, accounting for faster travel along the highway.

Contribution

It presents the first optimal O(n log n) algorithm for the time-convex hull under the Lp metric with a highway in the plane.

Findings

Algorithm runs in optimal O(n log n) time.

Handles general Lp metrics with 1 ≤ p ≤ ∞.

Efficiently computes shortest time-paths involving the highway.

Abstract

We consider the problem of computing the time-convex hull of a point set under the general $L_p$ metric in the presence of a straight-line highway in the plane. The traveling speed along the highway is assumed to be faster than that off the highway, and the shortest time-path between a distant pair may involve traveling along the highway. The time-convex hull ${TCH}(P)$ of a point set $P$ is the smallest set containing both $P$ and \emph{all} shortest time-paths between any two points in ${TCH}(P)$. In this paper we give an algorithm that computes the time-convex hull under the $L_p$ metric in optimal $O(n\log n)$ time for a given set of $n$ points and a real number $p$ with $1\le p \le \infty$.