The Streaming Complexity of Cycle Counting, Sorting By Reversals, and Other Problems

We consider the problem of estimating the size of a maximum matching when the edges are revealed in a streaming fashion. When the input graph is planar, we present a simple and elegant streaming algorithm that with high probability estimates the size of a maximum matching within a constant factor usingÕ(n 2/3 ) space, where n is the number of vertices. The approach generalizes to the family of graphs that have bounded arboricity, which include graphs with an excluded constant-size minor. To the best of our knowledge, this is the first result for estimating the size of a maximum matching in the adversarial-order streaming model (as opposed to the random-order streaming model) in o(n) space. We circumvent the barriers inherent in the adversarial-order model by exploiting several structural properties of planar graphs, and more generally, graphs with bounded arboricity. We further reduce the required memory size toÕ( √ n) for three restricted settings: (i) when the input graph is a forest; (ii) when we have 2-passes and the input graph has bounded arboricity; and (iii) when the edges arrive in random order and the input graph has bounded arboricity.Finally, we design a reduction from the Boolean Hidden Matching Problem to show that there is no randomized streaming algorithm that estimates the size of the maximum matching to within a factor better than 3/2 and uses only o(n 1/2 ) bits of space. Using the same reduction, we show that there is no deterministic algorithm that computes this kind of estimate in o(n) bits of space. The lower bounds hold even for graphs that are collections of paths of constant length.

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“…In this section, we show streaming lower bounds by reducing from the Boolean Hidden Matching Problem [2,10,33], which we refer to as BHM.…”

Section: Hardness Resultsmentioning

confidence: 99%

Streaming Algorithms for Estimating the Matching Size in Planar Graphs and Beyond

Esfandiari¹,

Hajiaghayi²,

Liaghat³

et al. 2014

Proceedings of the Twenty-Sixth 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%

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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“…The Boolean Hidden Matching problem and the related Boolean Hidden Hypermatching problem of Verbin and Yu [VY11] have been very influential in streaming lower bounds: streaming problems that have recently been shown to admit reductions from Boolean Hidden ( In this paper we develop several new Fourier analytic ideas that go beyond the Boolean Hidden Matching problem in several directions:…”

Section: Our Techniquesmentioning

confidence: 99%

The Sketching Complexity of Graph and Hypergraph Counting

Kallaugher¹,

Kapralov²,

Price³

2018

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

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Subgraph counting is a fundamental primitive in graph processing, with applications in social network analysis (e.g., estimating the clustering coefficient of a graph), database processing and other areas. The space complexity of subgraph counting has been studied extensively in the literature, but many natural settings are still not well understood. In this paper we revisit the subgraph (and hypergraph) counting problem in the sketching model, where the algorithm's state as it processes a stream of updates to the graph is a linear function of the stream. This model has recently received a lot of attention in the literature, and has become a standard model for solving dynamic graph streaming problems.In this paper we give a tight bound on the sketching complexity of counting the number of occurrences of a small subgraph H in a bounded degree graph G presented as a stream of edge updates. Specifically, we show that the space complexity of the problem is governed by the fractional vertex cover number of the graph H. Our subgraph counting algorithm implements a natural vertex sampling approach, with sampling probabilities governed by the vertex cover of H. Our main technical contribution lies in a new set of Fourier analytic tools that we develop to analyze multiplayer communication protocols in the simultaneous communication model, allowing us to prove a tight lower bound. We believe that our techniques are likely to find applications in other settings. Besides giving tight bounds for all graphs H, both our algorithm and lower bounds extend to the hypergraph setting, albeit with some loss in space complexity.

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The Streaming Complexity of Cycle Counting, Sorting By Reversals, and Other Problems

Cited by 37 publications

References 23 publications

Streaming Algorithms for Estimating the Matching Size in Planar Graphs and Beyond

Streaming Algorithms for Estimating the Matching Size in Planar Graphs and Beyond

Testing Matrix Rank, Optimally

The Sketching Complexity of Graph and Hypergraph Counting

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