Highway Hull Revisited

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 H(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 H(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)) [F. Hurtado, B. Palop, V. SacristAn, Diagramas de Voronoi con distancias temporales, in: Actas de los VIII Encuentros de Geometra Computacional, 1999, pp. 279-288 (in Spanish); B. Palop, Algorithmic problems on proximity and location under metric constraints, PhD thesis, Universitat Politecnica de Catalunya, 2003]. 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. (C) 2009 Elsevier B.V. All rights reserved.