Function Computation Through a Bidirectional Relay

We consider a function computation problem in a three-node wireless network. Nodes A and B observe two correlated sources X and Y, respectively, and want to compute a function f (X, Y). To achieve this, nodes A and B send messages to a relay node C at rates RA and RB, respectively. The relay C then broadcasts a message to A and B at rate RC. We allow block coding and study the achievable region of rate triples under both zero-error and E-error. As a preparation, we first consider a broadcast network from the relay to A and B. A and B have side information X and Y, respectively. The relay node C observes both X and Y and broadcasts an encoded message to A and B. We want to obtain the optimal broadcast rate such that A and B can recover the function f (X, Y) from the received message and their individual side information X and Y, respectively. For this problem, we show equivalence between E-error and zero-error computations-this gives a rate characterization for zero-error computation. As a corollary, this also gives a rate characterization for the relay network under zero error for a class of functions called component-wise one-to-one functions when the support set of pXY is full. For the relay network, the zero-error rate region for arbitrary functions is characterized in terms of graph coloring of some suitably defined probabilistic graphs. We then give a single-letter inner bound to this rate region. Furthermore, we extend the graph theoretic ideas to address the E-error problem and obtain a single-letter inner bound.