Sublinear Estimation of Weighted Matchings in Dynamic Data Streams

Bury, Marc; Schwiegelshohn, Chris

doi:10.1007/978-3-662-48350-3_23

Cited by 39 publications

(64 citation statements)

References 33 publications

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“…However, for sparse networks it seems to be reasonable to require that a space efficient algorithm uses space o(n). In this case, if we only aim at slightly sublinear space, one can still approximate some network parameters [BS15,HP16]. However, if we would like our space requirement to be, say, log O(1) n then it seems that only very few network parameters can be approximated in the classical worstcase order model.…”

Section: Introductionmentioning

confidence: 99%

Estimating Graph Parameters from Random Order Streams

Pan¹,

Sohler²

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Section: Introductionmentioning

confidence: 99%

Estimating Graph Parameters from Random Order Streams

Pan¹,

Sohler²

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Denote by Mx the length n/p boolean vector For our purpose, it is more convenient to focus on a special case of Boolean Hidden Hypermatching problem, namely, BHH 0 n,p where the vector w = 0 n/p (p is an even integer) and Bob's task is to output YES if Mx = 0 n/p and output NO if Mx = 1 n/p . It is known that we can reduce any instance of BHH n,p to an instance of BHH 0 2n,p deterministically without any communication between Alice and Bob [11,31,45], by the following reduction.…”

Section: H1mentioning

confidence: 99%

“…Previous papers [2,11,17] used the BHH 0 n,p problem to prove lower bounds for estimating matching size in the data stream: given an instance (x, M) in BHH 0 n,p (Denote by D BHH the hard distribution of BHH 0 n,p ), we create a graph G(V ∪ W, E) with |V | = |W | = n via the following algorithm.…”

Section: H1mentioning

confidence: 99%

Testing Matrix Rank, Optimally

Balcan¹,

Li²,

Woodruff³

et al. 2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We show that for the problem of testing if a matrix A ∈ F n×n has rank at most d, or requires changing an -fraction of entries to have rank at most d, there is a non-adaptive query algorithm making O(d 2 / ) queries. Our algorithm works for any field F. This improves upon the previous O(d 2 / 2 ) bound (Krauthgamer and Sasson, SODA '03), and bypasses an Ω(d 2 / 2 ) lower bound of (Li, Wang, and Woodruff, KDD '14) which holds if the algorithm is required to read a submatrix. Our algorithm is the first such algorithm which does not read a submatrix, and instead reads a carefully selected non-adaptive pattern of entries in rows and columns of A. We complement our algorithm with a matching Ω(d 2 / ) query complexity lower bound for non-adaptive testers over any field. We also give tight bounds of Θ(d 2 ) queries in the sensing model for which query access comes in the form of X i , A := tr(X i A); perhaps surprisingly these bounds do not depend on .Testing rank is only one of many tasks in determining if a matrix has low intrinsic dimensionality. We next develop a novel property testing framework for testing numerical properties of a real-valued matrix A more generally, which includes the stable rank, Schatten-p norms, and SVD entropy. Specifically, we propose a bounded entry model, where A is required to have entries bounded by 1 in absolute value. Such a model provides a meaningful framework for testing numerical quantities and avoids trivialities caused by single entries being arbitrarily large. It is also well-motivated by recommendation systems. We give upper and lower bounds for a wide range of problems in this model, and discuss connections to the sensing model above. We obtain several results for estimating the operator norm that may be of independent interest. For example, we show that if the stable rank is constant, A F = Ω(n), and the singular value gap σ 1 (A)/σ 2 (A) = (1/ ) γ for any constant γ > 0, then the operator norm can be estimated up to a (1 ± )-factor non-adaptively by querying O(1/ 2 ) entries. This should be contrasted to adaptive methods such as the power method, or previous non-adaptive sampling schemes based on matrix Bernstein inequalities which read a 1/ 2 × 1/ 2 submatrix and thus make Ω(1/ 4 ) queries. Similar to our non-adaptive algorithm for testing rank, our scheme instead reads a carefully selected pattern of entries.

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“…While a 0.5-approximation for unweighted matching in the semi-streaming setting is trivial, such an approximation for weighted matching appears nontrivial. There is a sequence of works improving the approximation factor of weighted matching in the semi-streaming setting [19,29,51], and just recently Paz and Schwartzman provide a semi-streaming (0.5 − ǫ)-approximation algorithm.…”

Section: Other Related Workmentioning

confidence: 99%

Metric Sublinear Algorithms via Linear Sampling

Esfandiari

Mitzenmacher

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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In this work we provide a new technique to design fast approximation algorithms for graph problems where the points of the graph lie in a metric space. Specifically, we present a sampling approach for such metric graphs that, using a sublinear number of edge weight queries, provides a linear sampling, where each edge is (roughly speaking) sampled proportionally to its weight.For several natural problems, such as densest subgraph and max cut among others, we show that by sparsifying the graph using this sampling process, we can run a suitable approximation algorithm on the sparsified graph and the result remains a good approximation for the original problem. Our results have several interesting implications, such as providing the first sublinear time approximation algorithm for densest subgraph in a metric space, and improving the running time of estimating the average distance.

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Sublinear Estimation of Weighted Matchings in Dynamic Data Streams

Cited by 39 publications

References 33 publications

Estimating Graph Parameters from Random Order Streams

Estimating Graph Parameters from Random Order Streams

Testing Matrix Rank, Optimally

Metric Sublinear Algorithms via Linear Sampling

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