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Computation in Multicast Networks: Function Alignment and Converse Theorems
, 2012
"... 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 iden ..."
Abstract

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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 twotransmitter tworeceiver singlehop linear deterministic networks, the computing capacity is characterized for a linear function (modulo2 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 cutset based and genieaided bounds. Computation (vs. communication) over multicast networks requires additional analysis to account for multiple receivers sharing a networkâ€™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 conceptuallys impler algebraic proof of achievability that generalizes to Ltransmitter Lreceiver networks.