In this paper we study the problem of finding $(\epsilon, \phi)$-expander decompositions of a graph in the streaming model, in particular for dynamic streams of edge insertions and deletions. The goal is to partition the vertex set so that every component induces a $\phi$-expander, while the number of inter-cluster edges is only an $\epsilon$ fraction of the total volume. It was recently shown that there exists a simple algorithm to construct a $(O(\phi \log n), \phi)$-expander decomposition of an $n$-vertex graph using $\widetilde{O}(n/\phi^2)$ bits of space [Filtser, Kapralov, Makarov, ITCS'23]. This result calls for understanding the extent to which a dependence in space on the sparsity parameter $\phi$ is inherent. We move towards answering this question on two fronts. We prove that a $(O(\phi \log n), \phi)$-expander decomposition can be found using $\widetilde{O}(n)$ space, for every $\phi$. At the core of our result is the first streaming algorithm for computing boundary-linked expander decompositions, a recently introduced strengthening of the classical notion [Goranci et al., SODA'21]. The key advantage is that a classical sparsifier [Fung et al., STOC'11], with size independent of $\phi$, preserves the cuts inside the clusters of a boundary-linked expander decomposition within a multiplicative error. Notable algorithmic applications use sequences of expander decompositions, in particular one often repeatedly computes a decomposition of the subgraph induced by the inter-cluster edges (e.g., the seminal work of Spielman and Teng on spectral sparsifiers [Spielman, Teng, SIAM Journal of Computing 40(4)], or the recent maximum flow breakthrough [Chen et al., FOCS'22], among others). We prove that any streaming algorithm that computes a sequence of $(O(\phi \log n), \phi)$-expander decompositions requires ${\widetilde{\Omega}}(n/\phi)$ bits of space, even in insertion only streams.

翻译：本文研究在流式模型中寻找图的$(\epsilon, \phi)$-扩展图分解问题，特别针对边插入与删除的动态流。该问题的目标是对顶点集进行划分，使得每个连通分量都诱导出一个$\phi$-扩展图，而簇间边的数量仅占总体积的$\epsilon$比例。最近的研究表明，存在一种简单算法能够使用$\widetilde{O}(n/\phi^2)$比特的空间为$n$顶点图构造$(O(\phi \log n), \phi)$-扩展图分解[Filtser, Kapralov, Makarov, ITCS'23]。这一结果促使我们深入探究空间复杂度对稀疏性参数$\phi$的依赖在何种程度上是本质性的。我们从两个前沿方向推进这一问题的解答：我们证明对于任意$\phi$，均可使用$\widetilde{O}(n)$空间找到$(O(\phi \log n), \phi)$-扩展图分解。我们成果的核心是首个用于计算边界关联扩展图分解的流式算法，这是对经典概念的近期强化形式[Goranci et al., SODA'21]。其关键优势在于，具有与$\phi$无关的规模的经典稀疏化器[Fung et al., STOC'11]，能够在乘法误差范围内保持边界关联扩展图分解各簇内部割的精度。值得注意的算法应用常涉及扩展图分解序列，特别是需要反复计算由簇间边诱导的子图分解（例如Spielman和Teng关于谱稀疏化的开创性工作[Spielman, Teng, SIAM Journal of Computing 40(4)]，或近期最大流研究的突破性进展[Chen et al., FOCS'22]等）。我们证明任何计算$(O(\phi \log n), \phi)$-扩展图分解序列的流式算法都需要${\widetilde{\Omega}}(n/\phi)$比特空间，该结论即使在仅含插入操作的流中依然成立。