Optimal Construction for Time-Convex Hull with Two Orthogonal Highways in the L1-metric

TL;DR

This paper presents an optimal algorithm for constructing the time-convex hull in the presence of two orthogonal highways under L1 and L2 metrics, achieving O(n log n) time complexity.

Contribution

It introduces the first optimal O(n log n) time algorithm for the time-convex hull problem with two orthogonal highways in both L1 and L2 metrics.

Findings

Achieved O(n log n) time complexity for arbitrary highway speeds in L1-metric.

Achieved O(n log n) time complexity for infinite highway speed in L2-metric.

Provided a practical algorithm for time-convex hull construction with highways.

Abstract

We consider the time-convex hull problem in the presence of two orthogonal highways H. In this problem, the travelling speed on the highway is faster than off the highway, and the time-convex hull of a point set P is the closure of P with respect to the inclusion of shortest time-paths. In this paper, we provide the algorithm for constructing the time-convex hull with two orthogonal highways. We reach the optimal result of O(n log n) time for arbitrary highway speed in the L1-metric. For the L2-metric with infinite highway speed, we hit the goal of O(n log n) time as well.