DoF of a Cooperative X-Channel with an Application to Distributed Computing

We consider a cooperative X-channel with K transmitters (TXs) and K receivers (Rxs) where Txs and Rxs are gathered into groups of size r respectively. Txs belonging to the same group cooperate to jointly transmit a message to each of the K − r Rxs in all other groups, and each Rx individually decodes all its intended messages. By introducing a new interference alignment (IA) scheme, we prove that when K/r is an integer the Sum Degrees of Freedom (Sum-DoF) of this channel is lower bounded by 2r if K/r ∈ {2, 3} and by $\frac{{K(K - r) - {r^2}}}{{2K - 3r}}$ if K/r ≥ 4. We also prove that the Sum-DoF is upper bounded by $\frac{{{\text{K}}({\text{K}} - {\text{r}})}}{{2{\text{K}} - 3{\text{r}}}}$. The proposed IA scheme finds application in a wireless distributed MapReduce framework, where it improves the normalized data delivery time (NDT) compared to the state of the art.