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Symmetric Connectivity with Directional Antennas

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=π/2, for which n=4. Moreover, we prove that if Q_1 and Q_2 are quadruplets of antennas of angle π/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 π/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 π/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^βis minimized, where β\ge 1 is a constant. We present an O(1)-approximation algorithm.