连通网络中的分布式数据摘要
Distributed Data Summarization in Well-Connected Networks
摘要：
我们研究分布式算法的数据摘要中的一些基本问题。给定一个$n$节点的通信图$G$，每个节点最初都可以持有一个值，我们专注于计算$\sum_{i=1}^N g(f_i)$，其中$f_i$是值$i$的出现次数，$g$是某个固定函数。这包括重要的统计数据，如不同元素的数量，频率矩和数据的经验熵。在CONGEST模型中，一个简单的自适应流下界表明，它需要$\tildeΩ(D+ n)$轮，其中$D$是图的直径，以准确计算其中的一些统计数据。然而，这些下限并不适用于良好连通的图。我们给出了一个算法，计算$\sum_{i=1}^{N} g(f_i)$准确地在$τ_G \cdot 2^{O(\sqrt{\log n})}$轮，其中$τ_G$是混合时间的$G$。这在计算$k$最频繁的元素中也有应用。我们证明了GOSSIP模型和CONGEST模型在良好连通图中具有很高的相似性。特别是，我们表明，GOSSIP模型的每一轮可以模拟几乎完美的$\tilde{O}(τ_G $轮的CONGEST模型。为此，我们开发了一种新的算法的GOSSIP模型，$1\pm ε$近似$p$ -th频率矩$F_p = \sum_{i=1}^N f_i^p$在$\tilde{O}(ε^{-2} n^{1-k/p})$轮，$p \geq2$，当不同的元素$F_0$的数量是最多$O\left(n^{1/(k-1)}\right)$。这个结果可以转换回CONGEST模型，其中因子$\tilde{O}(τ_G)$在轮数中爆炸。
英文摘要：
We study distributed algorithms for some fundamental problems in data summarization. Given a communication graph $G$ of $n$ nodes each of which may hold a value initially, we focus on computing $\sum_{i=1}^N g(f_i)$, where $f_i$ is the number of occurrences of value $i$ and $g$ is some fixed function. This includes important statistics such as the number of distinct elements, frequency moments, and the empirical entropy of the data. In the CONGEST model, a simple adaptation from streaming lower bounds shows that it requires $\tildeΩ(D+ n)$ rounds, where $D$ is the diameter of the graph, to compute some of these statistics exactly. However, these lower bounds do not hold for graphs that are well-connected. We give an algorithm that computes $\sum_{i=1}^{N} g(f_i)$ exactly in $τ_G \cdot 2^{O(\sqrt{\log n})}$ rounds where $τ_G$ is the mixing time of $G$. This also has applications in computing the top $k$ most frequent elements. We demonstrate that there is a high similarity between the GOSSIP model and the CONGEST model in well-connected graphs. In particular, we show that each round of the GOSSIP model can be simulated almost-perfectly in $\tilde{O}(τ_G $ rounds of the CONGEST model. To this end, we develop a new algorithm for the GOSSIP model that $1\pm ε$ approximates the $p$-th frequency moment $F_p = \sum_{i=1}^N f_i^p$ in $\tilde{O}(ε^{-2} n^{1-k/p})$ rounds, for $p \geq2$, when the number of distinct elements $F_0$ is at most $O\left(n^{1/(k-1)}\right)$. This result can be translated back to the CONGEST model with a factor $\tilde{O}(τ_G)$ blow-up in the number of rounds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
