Conference Paper

# Direction assignment in wireless networks.

Conference: Proceedings of the 22nd Annual Canadian Conference on Computational Geometry, Winnipeg, Manitoba, Canada, August 9-11, 2010

Source: DBLP

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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; · 0.57 Impact Factor -
##### Conference Paper: Switching to Directional Antennas with Constant Increase in Radius and Hop Distance.

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**ABSTRACT:**For any angle α π, we show that any connected communication graph that is induced by a set P of n transceivers using omni-directional antennas of radius 1, can be replaced by a strongly connected communication graph, in which each transceiver in P is equipped with a directional antenna of angle α and radius rdirr_{\mbox{\tiny dir}}, for some constant rdir = rdir(a)r_{\mbox{\tiny dir}} = r_{\mbox{\tiny dir}}(\alpha). Moreover, the new communication graph is a c-spanner of the original graph, for some constant c = c(α), with respect to number of hops.Algorithms and Data Structures - 12th International Symposium, WADS 2011, New York, NY, USA, August 15-17, 2011. Proceedings; 01/2011 - [Show abstract] [Hide abstract]

**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).

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