Optimal time-convex hull for a straight-line highway in Lp-metrics

We consider the problem of computing the time-convex hull of a point set under the general LpLp 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)TCH(P) of a point set P is the smallest set containing both P and all   shortest time-paths between any two points in TCH(P)TCH(P). In this paper we give an algorithm that computes the time-convex hull under the LpLp metric in optimal O(nlog⁡n)O(nlog⁡n) time for a given set of n points and a real number p   with 1≤p≤∞1≤p≤∞.