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

Esfandiari, Hossein; Hajiaghayi, Mohammad Taghi; Liaghat, Vahid; Monemizadeh, Morteza; Onak, Krzysztof

doi:10.1137/1.9781611973730.81

Cited by 46 publications

(73 citation statements)

References 31 publications

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“…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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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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“…For the seemingly easier problem of estimating the maximum matching size (the focus of this paper), the result of [29,36] can be modified to show that computing better than a e/(e − 1)approximation for matching size requires n Ω(1/ log log n) space (see also [37]). It was shown later in [23] that computing better than a 3/2-approximation requires Ω( √ n) bits of space. More recently, this lower bound was extended by [14] to show that computing a (1 + ε)-estimation requires n 1−O(ε) space.…”

Section: Models and Previous Workmentioning

confidence: 99%

“…BHH 0 n,t and Matching Size Estimation. The BHH 0 n,t problem has been used previously in [14,23] to prove lower bounds for estimating matching size in data streams. We now briefly describe this connection.…”

Section: The Boolean Hidden Hypermatching Problemmentioning

confidence: 99%

On Estimating Maximum Matching Size in Graph Streams

Assadi

Khanna

2017

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

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We study the problem of estimating the maximum matching size in graphs whose edges are revealed in a streaming manner. We consider both insertion-only streams, which only contain edge insertions, and dynamic streams that allow both insertions and deletions of the edges, and present new upper and lower bound results for both cases.On the upper bound front, we show that an α-approximate estimate of the matching size can be computed in dynamic streams using O(n 2 /α 4 ) space, and in insertiononly streams using O(n/α 2 )-space. These bounds respectively shave off a factor of α from the space necessary to compute an α-approximate matching (as opposed to only size), thus proving a non-trivial separation between approximate estimation and approximate computation of matchings in data streams.On the lower bound front, we prove that any α-approximation algorithm for estimating matching size in dynamic graph streams requires Ω( √ n/α 2.5 ) bits of space, even if the underlying graph is both sparse and has arboricity bounded by O(α). We further improve our lower bound to Ω(n/α 2 ) in the case of dense graphs. These results establish the first non-trivial streaming lower bounds for superconstant approximation of matching size.Furthermore, we present the first super-linear space lower bound for computing a (1 + ε)-approximation of matching size even in insertion-only streams. In particular, we prove that a (1 + ε)-approximation to matching size requires RS(n) · n 1−O(ε) space; here, RS(n) denotes the maximum number of edge-disjoint induced matchings of size Θ(n) in an n-vertex graph. It is a major open problem with far-reaching implications to determine the value of RS(n), and current results leave open the possibility that RS(n) may be as large as n/ log n. Moreover, using the best known lower bounds for RS(n), our result already rules out any O(n · poly(log n/ε))-space algorithm for (1 + ε)-approximation of matchings. We also show how to avoid the dependency on the parameter RS(n) in proving lower bound for dynamic streams and present a near-optimal lower bound of n 2−O(ε) for (1 + ε)-approximation in this model. Using a well-known connection between matching size and matrix rank, all our lower bounds also hold for the problem of estimating matrix rank. In particular our results imply a near-optimal n 2−O(ε) bit lower bound for (1 + ε)-approximation of matrix ranks for dense matrices in dynamic streams, answering an open question of Li and Woodruff (STOC 2016).

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“…Randomorder streams have been considered for problems including rank selection [33,20,28], frequency moments [3], entropy [21], submodular maximization [32], and graph matching [26,25,13]. Lower bounds that hold even under the assumption of random-order have been developed using multi-party communication complexity [9,7,8,16].…”

Section: Prior Workmentioning

confidence: 99%

Online Facility Location against a t-Bounded Adversary

Lang

2018

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

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In the streaming model, the order of the stream can significantly affect the difficulty of a problem. A tsemirandom stream was introduced as an interpolation between random-order (t = 1) and adversarial-order (t = n) streams where an adversary intercepts a randomorder stream and can delay up to t elements at a time. IITK Sublinear Open Problem #15 asks to find algorithms whose performance degrades smoothly as t increases. We show that the celebrated online facility location algorithm achieves an expected competitive ratio of O( log t log log t ). We present a matching lower bound that any randomized algorithm has an expected competitive ratio of Ω( log t log log t ). We use this result to construct an O(1)-approximate streaming algorithm for k-median clustering that stores O(k log t) points and has O(k log t) worst-case update time. Our technique generalizes to any dissimilarity measure that satisfies a weak triangle inequality, including k-means, M -estimators, and p norms. The special case t = 1 yields an optimal O(k) space algorithm for random-order streams as well as an optimal O(nk) time algorithm in the RAM model, closing a long line of research on this problem. IntroductionOne of the fundamental theoretical questions in the streaming model is to understand how the stream order impacts computation. In the adversarial-order model, results must hold under any order, whereas in the random-order model the order is selected uniformly at random. The order of the stream can strongly affect the resources required to solve a problem. For example, for streams of n integers where the stream may be read sequentially multiple times, determining the median using polylogarithmic space requires Θ(

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

Cited by 46 publications

References 31 publications

Testing Matrix Rank, Optimally

Testing Matrix Rank, Optimally

On Estimating Maximum Matching Size in Graph Streams

Online Facility Location against a t-Bounded Adversary

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