Do directional antennas facilitate in reducing interferences?

The coverage area of a directional antenna located at point p is a circular sector of angle α, whose orientation and radius can be adjusted. The interference at p, denoted I(p), is the number of antennas that cover p, and the interference of a communication graph G = (P,E) is I(G) = max {I(p) : p ∈P}. In this paper we address the question in its title. That is, we study several variants of the following problem: What is the minimum interference I, such that for any set P of n points in the plane, representing transceivers equipped with a directional antenna of angle α, one can assign orientations and ranges to the points in P, so that the induced communication graph G is either connected or strongly connected and I(G) ≤ I. In the asymmetric model (i.e., G is required to be strongly connected), we prove that I = O(1) for α < 2π/3, in contrast with I = Θ(logn) for α = 2π, proved by Korman [12]. In the symmetric model (i.e., G is required to be connected), the situation is less clear. We show that I = Θ(n) for α < π/2, and prove that for π/2 ≤ α ≤ 3π/2, by applying the Erdös-Szekeres theorem. The corresponding result for α = 2π is , proved by Halldórsson and Tokuyama [10]. As in [12] and [10] who deal with the case α = 2π, in both models, we assign ranges that are bounded by some constant c, assuming that UDG(P) (i.e., the unit disk graph over P) is connected. Moreover, the bound in the symmetric model reduces to , where Δ is the maximum degree in UDG(P).