连续局部和连续全局全知
Successive Local and Successive Global Omniscience
摘要：
本文考虑了两个推广的合作数据交换问题，被称为连续的本地全知（SLO）和连续的全球全知（SGO）。用户被划分为$\ell$嵌套子组。每个用户最初知道大小为$k$的基集$X$中的分组子集，并且所有用户都希望获知$X$中的所有分组。用户通过广播编码或未编码的分组来交换他们的分组。在SLO或SGO中，在第$l$（$1\leq l\leq \ell$）轮传输中，第$l$最小用户子组需要分别获知它们共同持有的所有分组或$X$中的所有分组。问题是找到最小和速率（即，所有用户的总传输速率），以最小化前一轮的总速率。为了解决这个问题，我们使用线性规划方法。对于数据包随机分布在用户之间的情况下，我们构造了一个线性方程组，其解表征了$k$趋于无穷大的概率为每一轮的最小和速率。此外，对于两个嵌套群的特殊情况下，我们得到了封闭形式的表达式，其中持有的概率很高，因为$k$趋于无穷大，为每轮的最小和率。
英文摘要：
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.
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