Symmetric connectivity with directional antennas

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 uʼs sector and vice versa. In this paper we ask what is the smallest angle α   for which there exists an integer n=n(α)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 question is α=π/2α=π/2, for which n=4n=4. Moreover, we prove that if Q1Q1 and Q2Q2 are two quadruplets of antennas of angle π/2π/2 and unbounded range, separated by a line, to which one applies the above construction, independently, then the induced SCG of Q1∪Q2Q1∪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 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 π/2π/2 and range r=O(1)r=O(1) and to orient them, such that the induced SCG is connected, and, moreover, is an O(1)O(1)-spanner of the unit disk graph, w.r.t. hop distance. In our solution r=142 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 π/2π/2 and adjustable range. The goal is to assign to each antenna p  , an orientation and a range rprp, such that the resulting SCG is (i) connected, and (ii) ∑p∈Prpβ is minimized, where β⩾1β⩾1 is a constant. For this problem, we devise an O(1)O(1)-approximation algorithm.