On Computation Rates for Arithmetic Sum

TL;DR

This paper investigates the computation rate for the sum function over a directed acyclic network, revealing complexities in upper bounds and proposing schemes with in-network compression for improved computation efficiency.

Contribution

It introduces new bounds on the computation rate for sum over non-tree networks and develops variable length coding schemes with in-network compression.

Findings

Upper bounds on computation rate are complex and require entropy analysis.

Achievable schemes using variable length codes improve computation efficiency.

Lower bounds involve analyzing the entropy of clumpy distributions.

Abstract

For zero-error function computation over directed acyclic networks, existing upper and lower bounds on the computation capacity are known to be loose. In this work we consider the problem of computing the arithmetic sum over a specific directed acyclic network that is not a tree. We assume the sources to be i.i.d. Bernoulli with parameter $1/2$. Even in this simple setting, we demonstrate that upper bounding the computation rate is quite nontrivial. In particular, it requires us to consider variable length network codes and relate the upper bound to equivalently lower bounding the entropy of descriptions observed by the terminal conditioned on the function value. This lower bound is obtained by further lower bounding the entropy of a so-called \textit{clumpy distribution}. We also demonstrate an achievable scheme that uses variable length network codes and in-network compression.