Strong connectivity of sensor networks with double antennae

Inspired by the well-known Dipole and Yagi antennae we introduce and study a new theoretical model of directional antennae that we call double antennae. Given a set P of n sensors in the plane equipped with double antennae (with either dipole-like or Yagi-like propagation patterns) of angle ϕ, we study the connectivity and stretch factor   problems, namely finding the minimum range such that there exists an orientation of the double antennae of that range that guarantees strong connectivity or stretch factor of the resulting network. We introduce the new concepts of (2,ϕ)(2,ϕ)-connectivity and ϕ-angular range and use them to characterize the optimality of our algorithms. We prove that the ϕ-angular range is a lower bound on the range required for strong connectivity and show how to compute it in time polynomial in n  . We give an algorithm for orienting the antennae so as to attain strong connectivity using optimal range when ϕ≥3π/4ϕ≥3π/4 and an algorithm that approximates the range to 3 times the optimal range for ϕ≥π/2ϕ≥π/2. For ϕ<π/3ϕ<π/3, we show that the problem is NP-complete to approximate within a factor 3. For ϕ≥π/2ϕ≥π/2, we give an algorithm to orient the antennae so that the resulting connectivity network has a stretch factor of at most 4 compared to the underlying unit disk graph.