Zero-Error Function Computation through a Bidirectional Relay

TL;DR

This paper characterizes the rate region for zero-error function computation in a three-node wireless network with a relay, using graph coloring techniques, and provides bounds and conditions for the relay to compute the function.

Contribution

It introduces a new rate region characterization for zero-error function computation with a relay, using probabilistic graph coloring and provides inner and outer bounds.

Findings

Rate region characterized via graph coloring.

Inner and outer bounds meet in simple cases.

Sufficient conditions for relay to compute the function.

Abstract

We consider zero error function computation in a three node wireless network. Nodes A and B observe $X$ and $Y$ respectively, and want to compute a function $f(X,Y)$ with zero error. 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 $R_C$ to help them compute $f(X,Y)$ with zero error. We allow block coding, and study the region of rate-triples $(R_A,R_B,R_C)$ that are feasible. The rate region is characterized in terms of graph coloring of some suitably defined probabilistic graphs. We give single letter inner and outer bounds which meet for some simple examples. We provide a sufficient condition on the joint distribution $p_{XY}$ under which the relay can also compute $f(X,Y)$ if A and B can compute it with zero error.