On the Streaming Complexity of Expander Decomposition

TL;DR

It is proved 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.

Abstract

In this paper we study the problem of finding $(ε, φ)$-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 $φ$-expander, while the number of inter-cluster edges is only an $ε$ fraction of the total volume. It was recently shown that there exists a simple algorithm to construct a $(O(φ\log n), φ)$-expander decomposition of an $n$-vertex graph using $\widetilde{O}(n/φ^2)$ 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 $φ$ is inherent. We move towards answering this question on two fronts. We prove that a $(O(φ\log n), φ)$-expander decomposition can be found using $\widetilde{O}(n)$ space, for every $φ$. 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 $φ$, 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 $(O(φ\log n), φ)$-expander decompositions requires ${\widetildeΩ}(n/φ)$ bits of space, even in insertion only streams.

Figures (7)

• Figure 1: Illustration of the graph we use for proving the lower bound. Thick bullets represent important vertices, thick lines represent important edges, dotted lines represent edges connecting the important vertices to $T$.
• Figure 2: Illustration of the ideal expander decomposition of the graph. Thick bullets represent important vertices, smaller bullets represent ordinary vertices, thick lines represent important edges, dotted lines represent the rest of the edges. Grey areas represent the clusters in the decomposition.
• Figure 3: Sequence of nested cuts.
• Figure 4: Illustration of the graph we use for proving the lower bound. Thick bullets represent important vertices, thick lines represent important edges, dotted lines represent edges connecting the important vertices to $T$.
• Figure 5: Illustration of the ideal expander decomposition of the graph. Thick bullets represent important vertices, smaller bullets represent ordinary vertices, thick lines represent important edges, dotted lines represent the rest of the edges. Grey areas represent the clusters in the decomposition.

Theorems & Definitions (147)

• Theorem 1: ED algorithm -- exponential time decoding
• Theorem 2: ED algorithm -- polynomial time decoding
• Theorem 3: RED lower bound
• Definition 1.1: Expander decomposition
• Theorem 2.1: Exponential time decoding BLD
• Theorem 1: ED algorithm -- exponential time decoding
• proof
• Theorem 2.2: Polynomial time decoding BLD
• Theorem 2: ED algorithm -- polynomial time decoding
• proof