复制模型下通信高效的分布式图聚类与稀疏化
Communication-Efficient Distributed Graph Clustering and Sparsification under Duplication Models
摘要：
在本文中，我们考虑的问题，聚类图节点和稀疏图边缘分布图，当图边缘可能边缘重复观察到在物理上遥远的网站。虽然不同站点之间的边缘重复乍一看似乎是有益的，但实际上它们可能会使聚类和稀疏化更加复杂，因为它们的处理可能需要额外的计算和通信。我们提出了两个完善的通信模型，即消息传递和黑板模型的第一个通信优化算法。具体来说，给定一个图上的$n$节点与边缘观察在$s$网站，我们的算法实现的通信成本$\tilde{O}(ns)$和$\tilde{O}(n+s)$（$\tilde{O}$隐藏一个多对数因子），几乎匹配它们的下界，$Ω(ns)$和$Ω(n+s)$，分别在消息传递和黑板模型。在边分布的假设下，通信成本与非重复模型下的通信成本渐近相同。我们的算法也可以保证聚类质量几乎一样好，集中所有的边缘，然后应用任何标准的聚类算法。此外，我们进行了第一次调查的分布式结构的图空间的黑板模型。我们为乘法和加法扳手提供了几乎匹配的通信下限和上限。例如，在有和没有复制模型的黑板中构造$(2k-1)$ -模型的通信下界分别是$Ω(s+n^{1+1/k}\log s)$和$Ω(s+n^{1+1/k}\max\{1,s^{-1/2-1/(2k)}\log s\})$，它们几乎匹配两个模型的上界$\tilde{O}(s+n^{1+1/k})$。
英文摘要：
In this paper, we consider the problem of clustering graph nodes and sparsifying graph edges over distributed graphs, when graph edges with possibly edge duplicates are observed at physically remote sites. Although edge duplicates across different sites appear to be beneficial at the first glance, in fact they could make the clustering and sparsification more complicated since potentially their processing would need extra computations and communications. We propose the first communication-optimal algorithms for two well-established communication models namely the message passing and the blackboard models. Specifically, given a graph on $n$ nodes with edges observed at $s$ sites, our algorithms achieve communication costs $\tilde{O}(ns)$ and $\tilde{O}(n+s)$ ($\tilde{O}$ hides a polylogarithmic factor), which almost match their lower bounds, $Ω(ns)$ and $Ω(n+s)$, in the message passing and the blackboard models respectively. The communication costs are asymptotically the same as those under non-duplication models, under an assumption on edge distribution. Our algorithms can also guarantee clustering quality nearly as good as that of centralizing all edges and then applying any standard clustering algorithm. Moreover, we perform the first investigation of distributed constructions of graph spanners in the blackboard model. We provide almost matching communication lower and upper bounds for both multiplicative and additive spanners. For example, the communication lower bounds of constructing a $(2k-1)$-spanner in the blackboard with and without duplication models are $Ω(s+n^{1+1/k}\log s)$ and $Ω(s+n^{1+1/k}\max\{1,s^{-1/2-1/(2k)}\log s\})$ respectively, which almost match the upper bound $\tilde{O}(s+n^{1+1/k})$ for both models.
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