Zero-error Function Computation on a Directed Acyclic Network

TL;DR

This paper investigates the average communication rates needed for zero-error function computation over a simple directed acyclic network, providing systematic outer bounds using entropy properties.

Contribution

It introduces a systematic method to derive outer bounds on the rate region for arbitrary functions in DAG networks, focusing on the average case rather than worst-case scenarios.

Findings

Derived outer bounds for the rate region using entropy lower bounds.

Applied bounds to specific demand functions to evaluate their effectiveness.

Focused on average communication rates, contrasting with existing worst-case bounds.

Abstract

We study the rate region of variable-length source-network codes that are used to compute a function of messages observed over a network. The particular network considered here is the simplest instance of a directed acyclic graph (DAG) that is not a tree. Existing work on zero-error function computation in DAG networks provides bounds on the \textit{computation capacity}, which is a measure of the amount of communication required per edge in the worst case. This work focuses on the average case: an achievable rate tuple describes the expected amount of communication required on each edge, where the expectation is over the probability mass function of the source messages. We describe a systematic procedure to obtain outer bounds to the rate region for computing an arbitrary demand function at the terminal. Our bounding technique works by lower bounding the entropy of the descriptions observed by the terminal conditioned on the function value and by utilizing the Schur-concave property of the entropy function. We evaluate these bounds for certain example demand functions.