Sparse Cut Projections in Graph Streams

Abstract

Finding sparse cuts is an important tool for analyzing large graphs that arise in practice, such as the web graph, online social communities, and VLSI circuits. When dealing with such graphs having billions of nodes, it is often hard to visualize global partitions. While studies on sparse cuts have traditionally looked at cuts with respect to all the nodes in the graph, some recent works analyze graph properties projected onto a small subset of vertices that may be of interest in a given context, e.g., relevant documents to a query in a search engine. In this paper, we study how sparse cuts in a graph partition a certain subset of nodes. We call this partition a cut projection. We study the problem of finding cut projections in the streaming model that is appropriate in this context as the input graph is too large to store in main memory. Specifically, for a d-regular graph G on n nodes with a cut of conductance Φ and constant balance, we show how to partition a randomly chosen set of k nodes in $\tilde{O}(\frac{1}{\sqrt{\alpha\Phi}})$ passes over the graph stream and space $\tilde{O}(n\alpha + \frac{n^{3/4}k^{1/4}}{\sqrt{\alpha}\Phi^{19/4}})$, for any choice of α ≤ 1. The resulting partition is the projection of a cut of conductance of at most $\tilde{O}(\sqrt{\Phi})$. We note that for k < nα ⁶Φ^O(1), this can be done in $\tilde{O}(1/\sqrt{\alpha\Phi})$ passes and space $\tilde{O}(n\alpha)$ that is sublinear in the number of nodes.