In this paper we initiate the study of expander decompositions of a graph $G=(V, E)$ in the streaming model of computation. The goal is to find a partitioning $\mathcal{C}$ of vertices $V$ such that the subgraphs of $G$ induced by the clusters $C \in \mathcal{C}$ are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model. Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of $V$) to within a $(\delta, \epsilon)$-multiplicative/additive error with high probability. The power cut sparsifier uses $\tilde{O}(n/\epsilon\delta)$ space and edges, which we show is asymptotically tight up to polylogarithmic factors in $n$ for constant $\delta$.

翻译：本文首次在流式计算模型中研究了图$G=(V, E)$的扩展子图分解问题。目标是找到顶点集$V$的一个划分$\mathcal{C}$，使得由每个簇$C \in \mathcal{C}$诱导的$G$的子图均为优质扩展子图，同时簇间边的数量较少。经典方法通过递归地对输入图施加平衡稀疏割来构造扩展子图分解。本文给出了在动态流模型中利用小空间实现这种递归稀疏割过程的首个方案。我们提出的核心算法工具是一种新型割稀疏化器——幂割稀疏化器：它能在任意给定顶点诱导子图（或$V$的固定划分中的任意簇）中，以高概率将割值保持至$(\delta, \epsilon)$-乘性/加性误差范围内。该幂割稀疏化器使用$\tilde{O}(n/\epsilon\delta)$空间和边，我们证明对于常数$\delta$，该复杂度在$n$的多对数因子意义下渐近紧确。