Abstract. 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 linear-time 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. 1

Abstract. 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 set of points if it minimizes the maximum travel time between any two points of the set. We give a simple linear-time 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 points on a line with respect to a given walkway. Then we describe a 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. Finally, we consider several variants of the transportation model, including escalators, used for traveling between two floors of a building, and elevators, which are strictly vertical. 1

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) worst-case 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

Given a large weighted graph G = (V, E) and a subset U of V, we define several graphs with vertex set U in which two vertices are adjacent if they satisfy some prescribed proximity rule. These rules use the shortest path distance in G and generalize the proximity rules that generate some of the most common proximity graphs in Euclidean spaces. We prove basic properties of the defined graphs and provide algorithms for their computation. 1

We consider a variation of Voronoi diagram, or time-based Voronoi diagram, for a set S of points in the presence of transportation lines or highways in the plane. A shortest time-distance path from a query point to any given point in S is a path that takes the least travelling time. The travelling speeds and hence travelling times of the subpaths along the highways and in the plane are different. M. Abellanas et al. [1] gave a simple algorithm that runs in O(n log n) time, for computing the time-based Voronoi diagram for a set of n points in the presence of one highway in the plane. We consider a generalization of this problem to the case when there are two or more highways. We give a characterization of this problem and present an O(n log n) time algorithm for the problem where there are two highways. The algorithm can be easily extended to multiple highways if a certain intersection condition of highways holds. 1

Abstract. We study a class of geometric optimization problems closely related to the 2-center 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