Optimal Zero-Error Coding for Computing under Pairwise Shared Side Information

TL;DR

This paper characterizes the optimal zero-error coding rate for computing functions with shared side information, providing a single-letter formula under a specific pairwise shared side information condition, and demonstrates its advantages over existing methods.

Contribution

It introduces the 'pairwise shared side information' condition, enabling a single-letter expression for the optimal rate in zero-error source coding with side information.

Findings

Single-letter expression for optimal rate under the condition

Characterization of the characteristic graph as a disjoint union of OR products

Optimal coding scheme outperforms existing strategies

Abstract

We study the zero-error source coding problem in which an encoder with Side Information (SI) $g(Y)$ transmits source symbols $X$ to a decoder. The decoder has SI $Y$ and wants to recover $f(X,Y)$ where $f,g$ are deterministic. We exhibit a condition on the source distribution and $g$ that we call "pairwise shared side information", such that the optimal rate has a single-letter expression. This condition is satisfied if every pair of source symbols "share" at least one SI symbol for all output of $g$. It has a practical interpretation, as $Y$ models a request made by the encoder on an image $X$, and $g(Y)$ corresponds to the type of request. It also has a graph-theoretical interpretation: under "pairwise shared side information" the characteristic graph can be written as a disjoint union of OR products. In the case where the source distribution is full-support, we provide an analytic expression for the optimal rate. We develop an example under "pairwise shared side information", and we show that the optimal coding scheme outperforms several strategies from the literature.