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All Farthest Neighbors in the Presence of Highways and Obstacles ⋆
"... Abstract. We consider the problem of computing all farthest neighbors (and the diameter) of a given set of n points in the presence of highways and obstacles in the plane. When traveling on the plane, travelers may use highways for faster movement and must avoid all obstacles. We present an efficien ..."
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Abstract. We consider the problem of computing all farthest neighbors (and the diameter) of a given set of n points in the presence of highways and obstacles in the plane. When traveling on the plane, travelers may use highways for faster movement and must avoid all obstacles. We present an efficient solution to this problem based on knowledge from earlier research on shortest path computation. Our algorithms run in O(nm(log m + log 2 n)) time using O(m + n) space, where the m is the combinatorial complexity of the environment consisting of highways and obstacles. 1
Optimal Location of Transportation Devices
, 2007
"... We consider algorithms for finding the optimal location of a simple transportation device, that we call a moving walkway, consisting of a pair of points in the plane between which the travel speed is high. More specifically, one can travel from one endpoint of the walkway to the other at speed v> ..."
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We consider algorithms for finding the optimal location of a simple transportation device, that we call a moving walkway, consisting of a pair of points in the plane between which the travel speed is high. More specifically, one can travel from one endpoint of the walkway to the other at speed v> 1, but can only travel at unit speed between any other pair of points. The travel time between any two points in the plane is the minimum between the actual geometric distance, and the time needed to go from one point to the other using the walkway. A location for a walkway is said to be optimal with respect to a given finite set of points if it minimizes the maximum travel time between any two points of the set. We give a simple lineartime algorithm for finding an optimal location in the case where the points are on a line. We also give an Ω(n log n) lower bound for the problem of computing the travel time diameter of a set of n points on a line with respect to a given walkway. Then we describe an O(n log n) algorithm for locating a walkway with the additional restriction that the walkway must be horizontal. This algorithm is based on a recent generic method for solving quasiconvex programs with implicitly defined constraints. It is used in a (1+ε)approximation algorithm for optimal location of a walkway of arbitrary orientation.
Bichromatic 2center of pairs of points
"... Abstract. We study a class of geometric optimization problems closely related to the 2center problem: Given a set S of n pairs of points, assign to each point a color (“red ” or “blue”) so that each pair’s points are assigned different colors and a function of the radii of the minimum enclosing bal ..."
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Abstract. We study a class of geometric optimization problems closely related to the 2center problem: Given a set S of n pairs of points, assign to each point a color (“red ” or “blue”) so that each pair’s points are assigned different colors and a function of the radii of the minimum enclosing balls of the red points and the blue points, respectively, is optimized. In particular, we consider the problems of minimizing the maximum and minimizing the sum of the two radii. For each case, minmax and minsum, we consider distances measured in the L2 and in the L ∞ metrics. Our problems are motivated by a facility location problem in transportation system design, in which we are given origin/destination pairs of points for desired travel, and our goal is to locate an optimal road/flight segment in order to minimize the travel to/from the endpoints of the segment. 1
Moving walkways, escalators, and elevators
 CoRR
"... Abstract. We study a simple geometric model of transportation facility that consists of two points between which the travel speed is high. This elementary definition can model shuttle services, tunnels, bridges, teleportation devices, escalators or moving walkways. The travel time between a pair of ..."
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Abstract. We study a simple geometric model of transportation facility that consists of two points between which the travel speed is high. This elementary definition can model shuttle services, tunnels, bridges, teleportation devices, escalators or moving walkways. The travel time between a pair of points is defined as a time distance, in such a way that a customer uses the transportation facility only if it is helpful. We give algorithms for finding the optimal location of such a transportation facility, where optimality is defined with respect to the maximum travel time between two points in a given set. 1
Travel Time Distances Induced by Transportation Networks and General Underlying Distances
, 2013
"... This paper considers a generalization of travel time distances by taking general underlying distance functions into account. We suggest a reasonable set of axioms defining a certain class of distance functions that can be facilitated with transportation networks. It turns out to be able to build an ..."
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This paper considers a generalization of travel time distances by taking general underlying distance functions into account. We suggest a reasonable set of axioms defining a certain class of distance functions that can be facilitated with transportation networks. It turns out to be able to build an abstract framework for computing shortest path maps and Voronoi diagrams with respect to the induced travel time distance under such a general setting. We apply our framework in convex distance functions as a concrete example, resulting in efficient algorithms that compute the traveltime Voronoi diagram for a set of given sites. More specifically, the Voronoi diagram with respect to the traveltime distance induced by a convex distance based on a kgon can be computed in O(m(n+m)(k log(n+m) +m)) time and O(km(n+m)) space, where n is the number of Voronoi sites and m is the complexity of the given transportation network.
Highway Hull Revisited
"... Abstract. 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 ..."
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Abstract. 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 Θ(n log n) worstcase time algorithm to find the highway hull under the L1 metric, as well as an O(n log 2 n) time algorithm for the L2 metric which improves the best known result of O(n 2) [15, 20]. 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. 1
An Improved Algorithm for Inserting a Highway in a City Metric Based on Minimization of Quasiconvex Functions
"... We introduce an improved algorithm to locate a segment highway such that the maximum city distance between a given set of n points is minimized (where the city distance is measured with speed v> 1 on a highway and 1 in the underlying metric elsewhere). We consider that such highway is built in a ..."
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We introduce an improved algorithm to locate a segment highway such that the maximum city distance between a given set of n points is minimized (where the city distance is measured with speed v> 1 on a highway and 1 in the underlying metric elsewhere). We consider that such highway is built in a complex transportation system with H other highways and obstacles. The algorithm runs in O(n 3 H 3 (log 2 n + log H)) time using O(nH) space improving the previous O(n 4 H 4) time and space bound. 1