Graph Sparsification via Refinement Sampling [article]

Ashish Goel and Michael Kapralov and Sanjeev Khanna
<span title="2010-04-27">2010</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
A graph G'(V,E') is an -sparsification of G for some >0, if every (weighted) cut in G' is within (1±) of the corresponding cut in G. A celebrated result of Benczur and Karger shows that for every undirected graph G, an -sparsification with O(n n/^2) edges can be constructed in O(m^2n) time. Applications to modern massive data sets often constrain algorithms to use computation models that restrict random access to the input. The semi-streaming model, in which the algorithm is constrained to use
more &raquo; ... (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( n) time per edge, uses O(^2 n) space per node, and outputs an -sparsifier with O(n^3 n/^2) edges. At a slightly increased space and time complexity, we can reduce the sparsifier size to O(n n/^2) edges matching the Benczur-Karger result, while improving upon the Benczur-Karger runtime for m=ω(n^3 n). Finally, we show that an -sparsifier with O(n n/^2) edges can be constructed in two passes over the data and O(m) time whenever m =Ω(n^1+δ) for some constant δ>0. As a by-product of our approach, we also obtain an O(m n+n n) time streaming algorithm to compute a sparse k-connectivity certificate of a graph.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="">arXiv:1004.4915v1</a> <a target="_blank" rel="external noopener" href="">fatcat:wn7uc7e2svglbakiozlllcjtsi</a> </span>
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