Sparse Cut Projections on Graph Streams

Finding sparse cuts form an important tool for analyzing/partitioning large graphs that arise in practice, such as the web graph, online social communities, and VLSI circuits. A well developed framework for working with such large graphs is the streaming model wherein the input is assumed to be on disk, and any algorithm is allowed to make few passes over the input that may be too large to store in main memory. While there are several works on computing approximate sparsest cuts, few consider the problem in a streaming model. In this work, we present an approach for finding cuts of small conductance on graph streams where the graph is presented as a stream of edges. We also show that this problem can be solved more efficiently if we are only required to partition a small set of k nodes with respect to a sparse cut. Specifically, for a d-regular graph G of balance b, and for any Φ that is at least the conductance of G, we show how to compute a cut with conductance at most Õ( √ Φ) in Õ( √ 1 Φα ) passes over the graph stream and using space Õ(min{nα+ 1 b ( nα dΦ + n d √ αΦ5/2 ) , (nα+ 1 b n dαΦ ) √ 1 Φα+ 1 Φ}), for any choice of α ≤ 1. Further, we can partition a randomly chosen set of k nodes in Õ( 1 √ αΦ ) passes over the graph stream and space Õ(nα+ (min{1/b, k})n3/4k1/4 √ αΦ19/4 ), for any choice of α ≤ 1. The resulting partition is the projection of a cut of conductance of at most Õ( √ Φ). We note that for graphs of constant balance b, for k < nαΦ, this can be done in Õ(1/ √ αΦ) passes Õ(nα) space.