Approximating matching size from random streams

Kapralov, Michael; Khanna, Sanjeev; Sudan, Madhu

doi:10.1137/1.9781611973402.55

Cited by 71 publications

(93 citation statements)

References 12 publications

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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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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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“…Very recently it has been shown that it is sometimes possible to approximate the cost of the solution without even having enough space to load the vertex set of the graph into memory (e.g. [20,13,11]). Our work contributes to the study of streaming algorithms by providing a tight impossibility result for non-trivially approximating MAX-CUT value in o(n) space.…”

Section: Introductionmentioning

confidence: 99%

(1 + Ω(1))-Αpproximation to MAX-CUT Requires Linear Space

Kapralov¹,

Khanna²,

Sudan³

et al. 2017

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

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We consider the problem of estimating the value of MAX-CUT in a graph in the streaming model of computation. We show that there exists a constant * > 0 such that any randomized streaming algorithm that computes a (1 + * )-approximation to MAX-CUT requires Ω(n) space on an n vertex graph. By contrast, there are algorithms that produce a (1 + )-approximation in space O(n/ 2 ) for every > 0. Our result is the first linear space lower bound for the task of approximating the max cut value and partially answers an open question from the literature [2]. The prior state of the art ruled out (2 − )-approximation inÕ( √ n) space or (1 + )-approximation in n 1−O( ) space, for any > 0.Previous lower bounds for the MAX-CUT problem relied, in essence, on a lower bound on the communication complexity of the following task: Several players are each given some edges of a graph and they wish to determine if the union of these edges is -close to forming a bipartite graph, using one-way communication. The previous works proved a lower bound of Ω( √ n) for this task when = 1/2, and n 1−O( ) for every > 0, even when one of the players is given a candidate bipartition of the graph and the graph is promised to be bipartite with respect to this partition or -far from bipartite. This added information was essential in enabling the previous analyses but also yields a weak bound since, with this extra information, there is an n 1−O( ) communication protocol for this problem. In this work, we give an Ω(n) lower bound on the communication complexity of the original problem (without the extra information) for = Ω(1) in the three-player setting. Obtaining this Ω(n) lower bound on the communication complexity is the main technical result in this paper. We achieve it by a

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“…Surprisingly, very little is known about this problem in the streaming model which is one of the most fundamental models of the area of algorithms for big data. The only known result is a recent algorithm by Kapralov, Khanna, and Sudan [18], which computes an estimate within a factor of O(polylog(n)) in the randomorder streaming model using O(polylog(n)) space. In the random-order model, the input stream is assumed to be chosen uniformly at random from the set of all possible permutations of the edges.…”

Section: Introductionmentioning

confidence: 99%

“…A number of dynamic algorithms for matchings [16,29,3,26,12] have applied various partitioning techniques. The polylog(n)-approximation algorithm for random-order streams [18] employs a combination of both partitioning and exploration. Local exploration is applied by sublinear-time algorithms for the maximum matching size.…”

mentioning

confidence: 99%

“…Equipped with additional information, such a decomposition allows for easy access to a large matching, which is a result of combining together large matchings involving each layer. Kapralov, Khanna, and Sudan [18] show that one can get O(polylog(n))-approximation to the size of a maximum matching in the random-order model, using O(polylog(n)) space. The intuition behind their approach can be interpreted as a combination of the local exploration and partitioning approaches.…”

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confidence: 99%

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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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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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Approximating matching size from random streams

Cited by 71 publications

References 12 publications

Online Facility Location against a t-Bounded Adversary

Online Facility Location against a t-Bounded Adversary

(1 + Ω(1))-Αpproximation to MAX-CUT Requires Linear Space

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

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