Time Convex Hull with a Highway

We consider the problem of computing the time convex hull of a set of points in the presence of a straight-line highway in the plane. The traveling speed in the plane is assumed to be much slower than that along the highway. The shortest time path between two arbitrary points is either the straight-line segment connecting these two points or a path that passes through the highway. The time convex hull, CH_t(P), of a set P of n points is the smallest set containing P such that all the shortest time paths between any two points lie in CH_t(P). In this paper we give a Theta(n log n) time algorithm for solving the time convex hull problem for a set of n points in the presence of a highway.