Part I: Improving Computational Efficiency of Communication for Omniscience.

Communication for omniscience (CO) refers to the problem where the users in a finite set $V$ observe a discrete multiple random source and want to exchange data over broadcast channels to reach omniscience, the state where everyone recovers the entire source. This paper studies how to improve the computational complexity for the problem of minimizing the sum-rate for attaining omniscience in $V$. While the existing algorithms rely on the submodular function minimization (SFM) techniques and complete in $O(|V|^2 \cdot \text{SFM}(|V|)$ time, we prove the strict strong map property of the nesting SFM problem. We propose a parametric (PAR) algorithm that utilizes the parametric SFM techniques and reduce the the complexity to $O(|V| \cdot \text{SFM}(|V|)$. The output of the PAR algorithm is in fact the segmented Dilworth truncation of the residual entropy for all minimum sum-rate estimates $\alpha$, which characterizes the principal sequence of partitions (PSP) and solves some related problems: It not only determines the secret capacity, a dual problem to CO, and the network strength of a graph, but also outlines the hierarchical solution to a combinatorial clustering problem. \end{abstract}