Graph Sparsification in the Semi-Streaming Model

Analyzing massive data sets has been one of the key motivations for studying streaming algorithms. In recent years, there has been significant progress in analysing distributions in a streaming setting, but the progress on graph problems has been limited. A main reason for this has been the existence of linear space lower bounds for even simple problems such as determining the connectedness of a graph. However, in many new scenarios that arise from social and other interaction networks, the number of vertices is significantly less than the number of edges. This has led to the formulation of the semi-streaming model where we assume that the space is (near) linear in the number of vertices (but not necessarily the edges), and the edges appear in an arbitrary (and possibly adversarial) order. In this paper we focus on graph sparsification, which is one of the major building blocks in a variety of graph algorithms. There has been a long history of (non-streaming) sampling algorithms that provide sparse graph approximations and it a natural question to ask if the sparsification can be achieved using a small space, and in addition using a single pass over the data? The question is interesting from the standpoint of both theory and practice and we answer the question in the affirmative, by providing a one pass $\tilde{O}(n/\epsilon^{2})$ space algorithm that produces a sparsification that approximates each cut to a $(1+\epsilon)$ factor. We also show that $\Omega(n \log \frac1\epsilon)$ space is necessary for a one pass streaming algorithm to approximate the min-cut, improving upon the $\Omega(n)$ lower bound that arises from lower bounds for testing connectivity.