The Energy Complexity of BFS in Radio Networks

We consider a model of energy complexity in Radio Networks in which transmitting or listening on the channel costs one unit of energy and computation is free. This simplified model captures key aspects of battery-powered sensors: that battery life is most influenced by transceiver usage, and that at low transmission powers, the actual cost of transmitting and listening are very similar. The energy complexity of tasks in single-hop networks is well understood. Recent work of Chang et al. considered energy complexity in multi-hop networks and showed that $\mathsf{Broadcast}$ admits an energy-efficient protocol, by which we mean each of the $n$ nodes in the network spends $O(\text{polylog}(n))$ energy. This work left open the strange possibility that all natural problems in multi-hop networks might admit such an energy-efficient solution. In this paper we prove that the landscape of energy complexity is rich enough to support a multitude of problem complexities. Whereas $\mathsf{Broadcast}$ can be solved by an energy-efficient protocol, exact computation of $\mathsf{Diameter}$ cannot, requiring $\Omega(n)$ energy. Our main result is that $\mathsf{Breadth First Search}$ has sub-polynomial energy complexity at most $2^{O(\sqrt{\log n\log\log n})}=n^{o(1)}$; whether it admits an efficient $O(\text{polylog}(n))$-energy protocol is an open problem. Our main algorithm involves recursively solving a generalized BFS problem on a cluster graph introduced by Miller, Peng, and Xu. In this application, we make crucial use of a close relationship between distances in this cluster graph, and distances in the original network. This relationship is new and may be of independent interest.