# Direction Assignment in Wireless Networks

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Boaz Ben-Moshe, Jul 30, 2015 Available from:-
- "In other related work, Nijnatten [5] also considers the problem of finding suitable orientations of α-antennas to form a strongly connected graph, but in his variant of the problem he allows a different r for each antenna and minimizes the overall power consumption of the network. Ben-Moshe et al. [3] consider 90 • -antennas but restrict the orientations to one of the four standard quadrant directions. Their algorithm constructs a connected bidirectional communication graph that requires a radius value r equal to at most twice optimal (in the restricted orientations setting). "

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**ABSTRACT:**Let S be a set of points in the plane, such that the unit disk graph with vertex set S 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 t-spanner of the unit disk graph for S (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 bounded below by 120° and 90°. We show that, in the case of 120° angles, a radius of 5 suffices to establish a hop 4-spanner; and in the case of 90° angles, a radius of 7 suffices to establish a hop 5-spanner.Discrete Mathematics Algorithms and Applications 10/2013; 05(03). DOI:10.1142/S1793830913500080 -
- "Carmi et al. [10] also observe that for α < π/3 it is not always possible to orient the antennas such that the induced SCG is connected. Ben-Moshe et al. [4] investigate the problem of orienting quadrant antennas with only four possible orientations (π/4, 3π/4, 5π/4, and 7π/4), and vertical half-strip antennas with only two possible orientations (up and down). Both problems are studied under the symmetric model. "

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**ABSTRACT:**Let P be a set of points in the plane, representing directional antennas of angle a and range r. The coverage area of the antenna at point p is a circular sector of angle a and radius r, whose orientation is adjustable. For a given orientation assignment, the induced symmetric communication graph (SCG) of P is the undirected graph that contains the edge (u,v) iff v lies in u's sector and vice versa. In this paper we ask what is the smallest angle a for which there exists an integer n=n(a), such that for any set P of n antennas of angle a and unbounded range, one can orient the antennas so that the induced SCG is connected, and the union of the corresponding wedges is the entire plane. We show that the answer to this problem is a=\pi/2, for which n=4. Moreover, we prove that if Q_1 and Q_2 are 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 Q_1 \cup Q_2 is connected. This latter result enables us to apply the construction locally, and to solve the following two further problems. In the first problem, we are given a connected unit disk graph (UDG), corresponding to a set P of omni-directional antennas of range 1, and the goal is to replace these 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 UDG, w.r.t. hop distance. In our solution r = 14\sqrt{2} and the spanning ratio is 8. In the second problem, 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 r_p, such that the resulting SCG is connected, and \sum_{p \in P} r_p^\beta is minimized, where \beta \ge 1 is a constant. We present an O(1)-approximation algorithm.Computational Geometry 08/2011; 46(9). DOI:10.1016/j.comgeo.2013.06.003 · 0.57 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper addresses the problem of finding an orientation and a minimum radius for directional antennas of a fixed angle placed at the points of a planar set S, that induce a strongly connected communication graph. We consider problem instances in which antenna angles are fixed at 90 and 180 degrees, and establish upper and lower bounds for the minimum radius necessary to guarantee strong connectivity. In the case of 90-degree angles, we establish a lower bound of 2 and an upper bound of 7. In the case of 180-degree angles, we establish a lower bound of sqrt(3) and an upper bound of 1+sqrt(3). Underlying our results is the assumption that the unit disk graph for S is connected. Comment: 8 pages, 10 figures