Sparsification Algorithm for Cut Problems on Semi-streaming Model

The emergence of social networks and other interaction networks have brought to fore the questions of processing massive graphs. The (semi) streaming model, where we assume that the space is (near) linear in the number of vertices (but not necessarily the edges) is an useful and efficient model for processing large graphs. In many of these graphs the numbers of vertices are significantly less than the number of edges, and hence attract the semi-streaming model. We focus on the problem of graph sparsification in a single pass, that is, constructing a small space representation of the graph such that we can estimate the size of any cut. Graph sparsification is one of the major building blocks which is used in a variety of algorithms, and there has been a long history of (non-streaming) sampling that provide sparse approximations. Thus the space requirement for graph sparsification is a natural question. Since Ω(n) space is necessary for a one pass streaming algorithm to determine if a graph is connected, it gives an Ω(n) lower bound for any sparsification algorithms which approximates cuts multiplicatively. We show an essentially tight upper bound, that is, using Õ(n/ǫ) space we can create a sparsification in a single pass which approximates each cut to a (1± ǫ) factor. Department of Computer and Information Sciences, University of Pennsylvania, Philadelphia PA 19104-6389. Email: [email protected] Department of Computer and Information Sciences, University of Pennsylvania, Philadelphia PA 19104-6389. Email: [email protected]