Symmetric connectivity with directional antennas

Let P be a set of points in the plane, representing transceivers equipped with a directional antenna of angle alpha and range r. The coverage area of the antenna at point p is a circular sector of angle alpha 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 alpha for which there exists an integer n = n(alpha), such that for any set P of n antennas of angle alpha 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 question is alpha = pi/2, for which n = 4. Moreover, we prove that if Q(1) and Q(2) 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 Q(1) boolean OR 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 (replacing omni-directional antennas with directional antennas), we are given a connected unit disk graph, corresponding to a set P of omni-directional antennas of range 1, and the goal is to replace the omni-directional 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 root 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 r(p), such that the resulting SCG is (i) connected, and (ii) Sigma(p is an element of P) r(p)(beta) is minimized, where beta >= 1 is a constant. For this problem, we devise an O (1)-approximation algorithm. (C) 2013 Elsevier B.V. All rights reserved.