2012
DOI: 10.48550/arxiv.1203.4900
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Single pass sparsification in the streaming model with edge deletions

Abstract: In this paper we give a construction of cut sparsifiers of Benczúr and Karger in the dynamic streaming setting in a single pass over the data stream. Previous constructions either required multiple passes or were unable to handle edge deletions. We use Õ(1/ǫ 2 ) time for each stream update and Õ(n/ǫ 2 ) time to construct a sparsifier. Our ǫ-sparsifiers have O(n log 3 n/ǫ 2 ) edges. The main tools behind our result are an application of sketching techniques of Ahn et al. [SODA'12] to estimate edge connectivity… Show more

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Cited by 5 publications
(7 citation statements)
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“…We remark that the previous work on the weighted matching problem in the streaming model [8], as well as our current work, assumes that the weight of each edge remains the same during the stream. Other works on weighted graph streams make the same assumption [2,3,22,27]. The reason behind this assumption is that-as shown in this paper, if this assumption is lifted, we can derive a lower bound on the space complexity of the k-matching problem that is at least linear in the size of the graph, and hence, can be much larger than the desirable space complexity.…”
Section: Related Workmentioning
confidence: 84%
“…We remark that the previous work on the weighted matching problem in the streaming model [8], as well as our current work, assumes that the weight of each edge remains the same during the stream. Other works on weighted graph streams make the same assumption [2,3,22,27]. The reason behind this assumption is that-as shown in this paper, if this assumption is lifted, we can derive a lower bound on the space complexity of the k-matching problem that is at least linear in the size of the graph, and hence, can be much larger than the desirable space complexity.…”
Section: Related Workmentioning
confidence: 84%
“…They present an algorithm for computing a cut sparsifier of G, which is a strictly weaker, but still useful, approximation than a spectral sparsifier [BK96]. Their work was improved in [AGM12b] and [GKP12], which use a linear compression of size Õ( n ǫ 2 ) to compute a cut sparsifier. The more challenging problem of computing a spectral sparsifier from a linear sketch was addressed in [AGM13], who give an Õ( n 5/3 ǫ 2 ) space solution.…”
Section: Prior Workmentioning
confidence: 99%
“…Cuts in graphs are a fundamental object of study, and play a central role in the study of graph algorithms. Consequently, the problem of sparsifying a graph while approximately preserving its cut structure has been extensively studied (see, for instance, [17,6,18,25,1,2,13,5,3,21,15,4,16], and references therein). A cut-preserving sparsifier not only reduces the space requirement for any computation, but it can also reduce the time complexity of solving many fundamental cut, flow, and matching problems as one can now run the algorithms on the sparsifier which may contain far fewer edges.…”
Section: Introductionmentioning
confidence: 99%