Expander Decomposition in Dynamic Streams

TL;DR

This paper introduces a novel approach to compute expander decompositions in dynamic graph streams using a new type of cut sparsifier, enabling efficient partitioning with good expansion properties in small space.

Contribution

It presents the first recursive sparsest cut algorithm for expander decomposition in the dynamic streaming model utilizing a new power cut sparsifier.

Findings

Power cut sparsifier preserves cuts within specified error bounds.

Algorithm uses $ ilde{O}(n/ ext{epsilon} ext{delta})$ space and edges.

Space complexity is asymptotically tight for constant delta.

Abstract

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 $\delta$.