Graph Sparsification via Refinement Sampling

A graph G(V,E) is an ǫ-sparsification of G for some ǫ > 0, if every (weighted) cut in G is within (1 ± ǫ) of the corresponding cut in G. A celebrated result of Benczúr and Karger shows that for every undirected graph G, an ǫ-sparsification with O(n log n/ǫ) edges can be constructed in O(m log n) time. The notion of cut-preserving graph sparsification has played an important role in speeding up algorithms for several fundamental network design and routing problems. Applications to modern massive data sets often constrain algorithms to use computation models that restrict random access to the input. The semistreaming model, in which the algorithm is constrained to use Õ(n) space, has been shown to be a good abstraction for analyzing graph algorithms in applications to large data sets. Recently, a semi-streaming algorithm for graph sparsification was presented by Anh and Guha; the total running time of their implementation is Ω(mn), too large for applications where both space and time are important. In this paper, we introduce a new technique for graph sparsification, namely refinement sampling, that gives an Õ(m) time semi-streaming algorithm for graph sparsification. Specifically, we show that refinement sampling can be used to design a one-pass streaming algorithm for sparsification that takes O(log logn) time per edge, uses O(log n) space per node, and outputs an ǫ-sparsifier with O(n log n/ǫ) edges. At a slightly increased space and time complexity, we can reduce the sparsifier size to O(n logn/ǫ) edges matching the Benczúr-Karger result, while improving upon the Benczúr-Karger runtime for m = ω(n log n). Finally, we show that an ǫ-sparsifier with O(n log n/ǫ) edges can be constructed in two passes over the data and O(m) time whenever m = Ω(n) for some constant δ > 0. As a by-product of our approach, we also obtain an O(m log logn + n logn) time streaming algorithm to compute a sparse k-connectivity certificate of a graph. Departments of Management Science and Engineering and (by courtesy) Computer Science, Stanford University. Email: [email protected]. Research supported in part by NSF award IIS-0904325. Institute for Computational and Mathematical Engineering, Stanford University. Email: [email protected]. Research supported by a Stanford Graduate Fellowship. Department of Computer and Information Science, University of Pennsylvania, Philadelphia PA. Email: [email protected]. Supported in part by NSF Awards CCF-0635084 and IIS-0904314.