Duality of a Source Coding Problem and the Semi-Deterministic Broadcast Channel with Rate-Limited Cooperation

TL;DR

This paper explores the duality between a source coding problem with encoder cooperation and a semi-deterministic broadcast channel with decoder cooperation, deriving their capacity and coordination regions and establishing their operational equivalence.

Contribution

It introduces a duality framework linking the Wyner-Ahlswede-Körner source coding problem with a semi-deterministic broadcast channel, deriving their capacity and coordination regions.

Findings

Derived the coordination-capacity region for the WAK problem.

Established the capacity region for the semi-deterministic broadcast channel.

Proved the operational equivalence and duality between the two problems.

Abstract

The Wyner-Ahlswede-K\"orner (WAK) empirical-coordination problem where the encoders cooperate via a finite-capacity one-sided link is considered. The coordination-capacity region is derived by combining several source coding techniques, such as Wyner-Ziv (WZ) coding, binning and superposition coding. Furthermore, a semi-deterministic (SD) broadcast channel (BC) with one-sided decoder cooperation is considered. Duality principles relating the two problems are presented, and the capacity region for the SD-BC setting is derived. The direct part follows from an achievable region for a general BC that is tight for the SD scenario. A converse is established by using telescoping identities. The SD-BC is shown to be operationally equivalent to a class of relay-BCs (RBCs) and the correspondence between their capacity regions is established. The capacity region of the SD-BC is transformed into an equivalent region that is shown to be dual to the admissible region of the WAK problem in the sense that the information measures defining the corner points of both regions coincide. Achievability and converse proofs for the equivalent region are provided. For the converse, we use a probabilistic construction of auxiliary random variables that depends on the distribution induced by the codebook. Several examples illustrate the results.