Computation in Multicast Networks: Function Alignment and Converse Theorems

Abstract: The classical problem in network coding theory considers communication over multicast networks. Multiple transmitters send independent messages to multiple receivers which decode the same set of messages. In this work, computation over multicast networks is considered: each receiver decodes an identical function of the original messages. For a countably infinite class of two-transmitter two-receiver single-hop linear deterministic networks, the computing capacity is characterized for a linear function (modulo-2 sum) of Bernoulli sources. Inspired by the geometric concept of interference alignment in networks, a new achievable coding scheme called function alignment is introduced. A new converse theorem is established that is tighter than cut-set based and genie-aided bounds. Computation (vs. communication) over multicast networks requires additional analysis to account for multiple receivers sharing a network's computational resources. We also develop a network decomposition theorem which identifies elementary parallel subnetworks that can constitute an original network without loss of optimality. The decomposition theorem provides a conceptually-simpler algebraic proof of achievability that generalizes to $L$-transmitter $L$-receiver networks.