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Spanning Properties of Graphs Induced by Directional Antennas
"... Let S be a set of points in the plane, whose unit disk graph is connected. We address the problem of finding orientations and a minimum radius for directional antennas of a fixed cone angle placed at the points of S, such that the induced communication graph G[S] is a hop tspanner (meaning that G[ ..."
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Let S be a set of points in the plane, whose unit disk graph is connected. We address the problem of finding orientations and a minimum radius for directional antennas of a fixed cone angle placed at the points of S, such that the induced communication graph G[S] is a hop tspanner (meaning that G[S] is strongly connected, and contains a directed path with at most t edges between any pair of points within unit distance). We consider problem instances in which antenna angles are fixed at 120 ◦ and 90◦. We show that, in the case of 120 ◦ angles, a radius of 5 suffices to establish a hop 5spanner; in the case of 90◦ angles, a radius of 7 suffices to establish a hop 6spanner; and for any angle strictly less than 180◦, a radius of 2 is necessary to establish strong connectivity.
Symmetric Connectivity with Directional Antennas∗
, 2013
"... Let P be a set of points in the plane, representing transceivers equipped with a directional antenna of angle α and range r. The coverage area of the antenna at point p is a circular sector of angle α and radius r, whose orientation can be adjusted. For a given assignment of orientations, the induce ..."
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Let P be a set of points in the plane, representing transceivers equipped with a directional antenna of angle α and range r. The coverage area of the antenna at point p is a circular sector of angle α and radius r, whose orientation can be adjusted. For a given assignment of orientations, the induced symmetric communication graph (SCG) of P is the undirected graph, in which two vertices (i.e., points) u and v are connected by an edge if and only if v lies in u’s sector and vice versa. In this paper we ask what is the smallest angle α for which there exists an integer n = n(α), such that for any set P of n antennas of angle α and unbounded range, one can orient the antennas so that (i) the induced SCG is connected, and (ii) the union of the corresponding wedges is the entire plane. We show (by construction) that the answer to this problem is α = pi/2, for which n = 4. Moreover, we prove that if Q1 and Q2 are two quadruplets of antennas of angle pi/2 and unbounded range, separated by a line, to which one applies the above construction, independently, then the induced SCG of Q1 ∪Q2 is connected. This latter result enables us to apply the construction locally, and to solve the following two further problems. In the first problem (replacing omnidirectional antennas with directional antennas), we are given a connected unit disk graph, corresponding to a set P of omnidirectional antennas of range 1, and the goal is to replace the omnidirectional antennas by directional antennas of angle pi/2 and range r = O(1) and to orient them, such that the induced SCG is connected, and, moreover, is an O(1)spanner of the unit disk graph, w.r.t. hop distance. In our solution r = 14 2 and the spanning ratio is 8. In the second problem (orientation and power assignment), we are given a set P of directional antennas of angle pi/2 and adjustable range. The goal is to assign to each antenna p, an orientation and a range rp, such that the resulting SCG is (i) connected, and (ii) p∈P r
THE TRAVELING SALESMAN PROBLEM UNDER SQUARED EUCLIDEAN DISTANCES
, 2010
"... Let P be a set of points in R d, and let α � 1 be a real number. We define the distance between two points p,q ∈ P as pq  α, where pq  denotes the standard Euclidean distance between p and q. We denote the traveling salesman problem under this distance function by Tsp(d,α). We design a 5appro ..."
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Let P be a set of points in R d, and let α � 1 be a real number. We define the distance between two points p,q ∈ P as pq  α, where pq  denotes the standard Euclidean distance between p and q. We denote the traveling salesman problem under this distance function by Tsp(d,α). We design a 5approximation algorithm for Tsp(2,2) and generalize this result to obtain an approximation factor of 3 α−1 + √ 6 α /3 for d = 2 and all α � 2. We also study the variant RevTsp of the problem where the traveling salesman is allowed to revisit points. We present a polynomialtime approximation scheme for RevTsp(2, α) with α � 2, and we show that RevTsp(d,α) is apxhard if d � 3 and α> 1. The apxhardness proof carries over to Tsp(d, α) for the same parameter ranges.