Establishing Strong Connectivity using Optimal Radius Half-Disk Antennas

Given a set S of points in the plane representing wireless devices, each point equipped with a directional antenna of radius r and aperture angle α ≥ 180◦, our goal is to find orientations and a minimum r for these antennas such that the induced communication graph is strongly connected. We show that r = √ 3 suffices to establish strong connectivity, assuming that the longest edge in the Euclidean minimum spanning tree for S is 1. This result is optimal in the sense that r = √ 3 is necessary in the worst-case for α ∈ [180◦, 240◦). In contrast, r = 2 is sometimes necessary when α < 180◦.