On the energy benefit of compute-and-forward for multiple unicasts

Compute-and-forward (CF) is a technique which exploits broadcast and superposition in wireless networks. In this paper, the CF energy benefit is studied for networks with unicast sessions and modeled by connected graphs. This benefit is defined as the ratio of the minimum energy consumption by traditional routing techniques, not using broadcast and superposition features, and the corresponding CF consumption. It is shown to be upper bounded by min(d̄, K, 12√K), where d̄ and K are the average hop-count distance and the number of sessions, respectively. Also, it can be concluded that the energy benefit of network coding (NC) is also upper bounded by the same value, which is a new scaling law of the energy benefit for NC as a function of K.