Do Directional Antennas Facilitate in Reducing Interferences?

Abstract

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 $I=O(\sqrt{n})$ for π/2 ≤ α ≤ 3π/2, by applying the Erdös-Szekeres theorem. The corresponding result for α = 2π is $I=\Theta(\sqrt{n})$, 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 $O(\sqrt{n})$ bound in the symmetric model reduces to $O(\sqrt{\Delta})$, where Δ is the maximum degree in UDG(P).

Work by R. Aschner was partially supported by the Lynn and William Frankel Center for Computer Sciences and by the Israel Ministry of Industry, Trade and Labor (consortium CORNET). Work by M.J. Katz was partially supported by the Israel Ministry of Industry, Trade and Labor (consortium CORNET), by grant 1045/10 from the Israel Science Foundation, and by grant 2010074 from the United States – Israel Binational Science Foundation. Work by G. Morgenstern was partially supported by the Lynn and William Frankel Center for Computer Sciences and by the Caesarea Rothschild Institute (CRI).