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 reduces the complexity to $O(|V| \cdot \text {SFM} (|V|)$ . We propose efficient solutions to the successive omniscience (SO): attaining omniscience successively in user subsets. We first focus on how to determine a complimentary subset $X_{*}\subsetneq V$ in the existing two-stage SO such that if the local omniscience in $X_{*}$ is reached first, the global omniscience whereafter can still be attained with the minimum sum-rate. It is shown that such a subset can be extracted at one of the iterations of the PAR algorithm. We then propose a novel multi-stage SO strategy: a nesting sequence of complimentary user subsets $X_{*}^{(1)} \subsetneq \dotsc \subsetneq X_{*}^{(K)} = V$ , the omniscience in which is attained progressively by the monotonic rate vectors $\mathbf {r}_{V}^{(1)} \leq \dotsc \leq \mathbf {r} _{V}^{(K)}$ . We propose algorithms to obtain this $K$ -stage SO from the returned results by the PAR algorithm. The run time of these algorithms is the same as the PAR algorithm.

中文翻译：

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提高全知和连续全知的通信计算效率

全知通信 (CO) 是指有限集合中的用户 $V$ 观察离散的多个随机源，并希望通过广播频道交换数据以达到全知，即每个人都恢复整个源的状态。本文研究了如何提高计算复杂度的最小化和率的问题，以获得全知。 $V$ . 而现有的算法依赖于子模块函数最小化 (SFM) 技术并在 $O(|V|^{2} \cdot \text {SFM} (|V|)$ 时间，我们证明了嵌套 SFM 问题的严格强映射性质。我们提出了一种参数 (PAR) 算法，该算法利用参数 SFM 技术并将复杂性降低到 $O(|V| \cdot \text {SFM} (|V|)$ . 我们提出了连续全知（SO）的有效解决方案：在用户子集中连续获得全知。我们首先关注如何确定一个互补子集 $X_{*}\subsetneq V$ 在现有的两阶段 SO 中，如果局部无所不知 $X_{*}$ 首先达到，然后仍然可以以最小的总和率达到全局无所不知。结果表明，可以在 PAR 算法的迭代之一中提取这样的子集。然后，我们提出了一种新颖的多阶段 SO 策略：互补用户子集的嵌套序列 $X_{*}^{(1)} \subsetneq \dotsc \subsetneq X_{*}^{(K)} = V$ ，通过单调速率向量逐步获得的无所不知 $\mathbf {r}_{V}^{(1)} \leq \dotsc \leq \mathbf {r} _{V}^{(K)}$ . 我们提出算法来获得这个 $K$ -stage SO 从 PAR 算法返回的结果。这些算法的运行时间与PAR算法相同。