Introduction to algorithms
Ray shooting amidst convex polygons in 2D
Journal of Algorithms
Voronoi Diagram for services neighboring a highway
Information Processing Letters
Time Convex Hull with a Highway
ISVD '07 Proceedings of the 4th International Symposium on Voronoi Diagrams in Science and Engineering
CATS '07 Proceedings of the thirteenth Australasian symposium on Theory of computing - Volume 65
Optimal location of transportation devices
Computational Geometry: Theory and Applications
Shortest paths and voronoi diagrams with transportation networks under general distances
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Optimal time-convex hull under the Lp metrics
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
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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 hullH(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 @Q(nlogn) worst-case time algorithm to find the highway hull under the L"1 metric, as well as an O(nlog^2n) 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.