Ad-Hoc Network Unicast in Time O(log log n) using Beamforming

We investigate the unicast problem for ad-hoc networks in the plane using MIMO techniques. In particular, we use the multi-node beamforming gain and present a self-synchronizing algorithm for the necessary carrier phase synchronization. First, we consider $n$ nodes in a grid where the transmission power per node is restricted to reach the neighboring node. We extend the idea of multi-hop routing and relay the message by multiple nodes attaining joint beamforming gain with higher reception range. In each round, the message is repeated by relay nodes at dedicated positions after a fixed waiting period. Such simple algorithms can send a message from any node to any other node in time $\mathcal{O}(\log \log n - \log \lambda)$ and with asymptotical energy $\mathcal{O}(\sqrt{n})$, the same energy an optimal multi-hop routing strategy needs using short hops between source and target. Here, $\lambda$ denotes the wavelength of the carrier. For $\lambda \in \Theta(1)$ we prove a tight lower time bound of $\Omega(\log \log n)$. Then, we consider $n$ randomly distributed nodes in a square of area $n$ and we show for a transmission range of $\Theta(\sqrt{\log n})$ and for a wavelength of $\lambda = \Omega(\log^{-1/2}n)$ that the unicast problem can be solved in $\mathcal{O}(\log \log n)$ rounds as well. The corresponding transmission energy increases to $\mathcal{O}(\sqrt{n} \log n)$. Finally, we present simulation results visualizing the nature of our algorithms.