Function Computation Through a Bidirectional Relay

Abstract

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 $R_{A}$ and $R_{B}$ , respectively. The relay C then broadcasts a message to A and B at rate $R_{C}$ . We allow block coding and study the achievable region of rate triples under both zero-error and $\epsilon $ -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 $\epsilon $ -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 $p_{XY}$ 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 $\epsilon $ -error problem and obtain a single-letter inner bound.