# Successive Local and Successive Global Omniscience

## TL;DR

This paper introduces the concepts of successive local and global omniscience in cooperative data exchange, providing linear programming solutions and probabilistic bounds for minimum sum-rates in nested user groups.

## Contribution

It generalizes cooperative data exchange to nested groups, deriving linear programming methods and closed-form solutions for minimum sum-rates in large random packet distributions.

## Key findings

- Linear programming characterizes minimum sum-rates for SLO and SGO.
- High probability bounds as packet size grows.
- Closed-form expressions for two nested groups.

## Abstract

This paper considers two generalizations of the cooperative data exchange problem, referred to as the successive local omniscience (SLO) and the successive global omniscience (SGO). The users are divided into $\ell$ nested sub-groups. Each user initially knows a subset of packets in a ground set $X$ of size $k$, and all users wish to learn all packets in $X$. The users exchange their packets by broadcasting coded or uncoded packets. In SLO or SGO, in the $l$th ($1\leq l\leq \ell$) round of transmissions, the $l$th smallest sub-group of users need to learn all packets they collectively hold or all packets in $X$, respectively. The problem is to find the minimum sum-rate (i.e., the total transmission rate by all users) for each round, subject to minimizing the sum-rate for the previous round. To solve this problem, we use a linear-programming approach. For the cases in which the packets are randomly distributed among users, we construct a system of linear equations whose solution characterizes the minimum sum-rate for each round with high probability as $k$ tends to infinity. Moreover, for the special case of two nested groups, we derive closed-form expressions, which hold with high probability as $k$ tends to infinity, for the minimum sum-rate for each round.