Part II: A Practical Approach for Successive Omniscience

TL;DR

This paper introduces a practical multi-stage approach for successive omniscience, optimizing the sequence of local omniscience in user subsets to achieve global omniscience efficiently.

Contribution

It proposes a multi-stage successive omniscience method using a parametric algorithm to identify optimal subsets for local omniscience, improving upon previous single-stage approaches.

Findings

The multi-stage approach achieves the minimum sum-rate for global omniscience.

A nesting sequence of subsets guides the successive local omniscience process.

Derived rate vectors optimize individual user transmissions at each stage.

Abstract

In Part I, we studied the communication for omniscience (CO) problem and proposed a parametric (PAR) algorithm to determine the minimum sum-rate at which a set of users indexed by a finite set $V$ attain omniscience. The omniscience in CO refers to the status that each user in $V$ recovers the observations of a multiple random source. It is called the global omniscience in this paper in contrast to the study of the successive omniscience (SO), where the local omniscience is attained subsequently in user subsets. By inputting a lower bound on the minimum sum-rate for CO, we apply the PAR algorithm to search a complimentary subset $X_* \subsetneq V$ such that if the local omniscience in $X_*$ is reached first, the global omniscience whereafter can still be attained with the minimum sum-rate. We further utilize the outputs of the PAR algorithm to outline a multi-stage SO approach that is characterized by $K \leq |V| - 1$ complimentary subsets $X_*^{(k)}, \forall k \in \{1,\dotsc,K\}$ forming a nesting sequence $X_*^{(1)} \subsetneq \dotsc \subsetneq X_*^{(K)} = V$. Starting from stage $k = 1$, the local omniscience in $X_*^{(k)}$ is attained at each stage $k$ until the final global omniscience in $X_*^{(K)} = V$. A $|X_*{(k)}|$-dimensional local omniscience achievable rate vector is also derived for each stage $k$ designating individual users transmitting rates. The sum-rate of this rate vector in the last stage $K$ coincides with the minimized sum-rate for the global omniscience.