SIAM journal on computing (Print)最新文献

Improved Cheeger's Inequality and Analysis of Local Graph Partitioning using Vertex Expansion and Expansion Profile 基于顶点展开和扩展轮廓的Cheeger不等式改进及局部图划分分析

SIAM journal on computing (Print) Pub Date : 2015-04-02 DOI: 10.1137/16M1079816

T. C. Kwok, L. Lau, Y. Lee

{"title":"Improved Cheeger's Inequality and Analysis of Local Graph Partitioning using Vertex Expansion and Expansion Profile","authors":"T. C. Kwok, L. Lau, Y. Lee","doi":"10.1137/16M1079816","DOIUrl":"https://doi.org/10.1137/16M1079816","url":null,"abstract":"We prove two generalizations of the Cheeger's inequality. The first generalization relates the second eigenvalue to the edge expansion and the vertex expansion of the graph G, [EQUATION] where pV(G) denotes the robust vertex expansion of G and p(G) denotes the edge expansion of G. The second generalization relates the second eigenvalue to the edge expansion and the expansion profile of G, for all k ≥ 2, [EQUATION] where pk(G) denotes the k-way expansion of G. These show that the spectral partitioning algorithm has better performance guarantees when pV(G) is large (e.g. planted random instances) or pk(G) is large (instances with few disjoint non-expanding sets). Both bounds are tight up to a constant factor. Our approach is based on a method to analyze solutions of Laplacian systems, and this allows us to extend the results to local graph partitioning algorithms. In particular, we show that our approach can be used to analyze personal pagerank vectors, and to give a local graph partitioning algorithm for the small-set expansion problem with performance guarantees similar to the generalizations of Cheeger's inequality. We also present a spectral approach to prove similar results for the truncated random walk algorithm. These show that local graph partitioning algorithms almost match the performance of the spectral partitioning algorithm, with the additional advantages that they apply to the small-set expansion problem and their running time could be sublinear. Our techniques provide common approaches to analyze the spectral partitioning algorithm and local graph partitioning algorithms.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114754456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 17

Capacity of Interactive Communication over Erasure Channels and Channels with Feedback Erasure信道和反馈信道的交互通信能力

SIAM journal on computing (Print) Pub Date : 2015-01-04 DOI: 10.1137/15M1052202

R. Gelles, Bernhard Haeupler

引用次数: 41

Partitioning Well-Clustered Graphs: Spectral Clustering Works! 划分聚类良好的图:谱聚类工作!

SIAM journal on computing (Print) Pub Date : 2014-11-07 DOI: 10.1137/15M1047209

Richard Peng, He Sun, Luca Zanetti

引用次数: 90

A Size-Sensitive Discrepancy Bound for Set Systems of Bounded Primal Shatter Dimension 有界原始破碎维集系统的大小敏感差异界

SIAM journal on computing (Print) Pub Date : 2013-07-30 DOI: 10.1137/140977746

Esther Ezra

{"title":"A Size-Sensitive Discrepancy Bound for Set Systems of Bounded Primal Shatter Dimension","authors":"Esther Ezra","doi":"10.1137/140977746","DOIUrl":"https://doi.org/10.1137/140977746","url":null,"abstract":"Let (X, S) be a set system on an n-point set X. The discrepancy of S is defined as the minimum of the largest deviation from an even split, over all subsets of S e S and two-colorings χ on X. We consider the scenario where, for any subset X' ⊆ X of size m ≤ n and for any parameter 1 ≤ k ≤ m, the number of restrictions of the sets of S to X' of size at most k is only O(md1kd-d1), for fixed integers d > 0 and 1 ≤ d1 ≤ d (this generalizes the standard notion of bounded primal shatter dimension when d1 = d). In this case we show that there exists a coloring χ with discrepancy bound O*(|S|1/2-d1/(2d)n(d1-1)/(2d)), for each S e S, where O*(·) hides a polylogarithmic factor in n. This bound is tight up to a polylogarithmic factor [21, 23] and the corresponding coloring χ can be computed in expected polynomial time using the very recent machinery of Lovett and Meka for constructive discrepancy minimization [20]. Our bound improves and generalizes the bounds obtained from the machinery of Har-Peled and Sharir [15] (and the follow-up work in [27]) for points and halfspaces in d-space for d ≥ 3. Last but not least, we show that our bound yields improved bounds for the size of relative (e, δ)-approximations for set systems of the above kind.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122963703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 17

Circumventing the Price of Anarchy: Leading Dynamics to Good Behavior 规避无政府状态的代价:引导良好行为的动力

SIAM journal on computing (Print) Pub Date : 2013-01-29 DOI: 10.1137/110821317

Maria-Florina Balcan, Avrim Blum, Y. Mansour

引用次数: 33

Smoothed Analysis of the Successive Shortest Path Algorithm 连续最短路径算法的平滑分析

SIAM journal on computing (Print) Pub Date : 2013-01-06 DOI: 10.1137/140989893

Tobias Brunsch, Kamiel Cornelissen, B. Manthey, Heiko Röglin, Clemens Rösner

引用次数: 23

Skew Bisubmodularity and Valued CSPs 偏双模性与有值csp

SIAM journal on computing (Print) Pub Date : 2013-01-06 DOI: 10.1137/120893549

Anna Huber, A. Krokhin, Robert Powell

引用次数: 49

An Almost Optimal Algorithm for Computing Nonnegative Rank 一种计算非负秩的几乎最优算法

SIAM journal on computing (Print) Pub Date : 2013-01-06 DOI: 10.1137/140990139

Ankur Moitra

引用次数: 58

The Fast Cauchy Transform and Faster Robust Linear Regression 快速柯西变换与快速鲁棒线性回归

SIAM journal on computing (Print) Pub Date : 2012-07-19 DOI: 10.1137/140963698

K. Clarkson, P. Drineas, M. Magdon-Ismail, Michael W. Mahoney, Xiangrui Meng, David P. Woodruff

{"title":"The Fast Cauchy Transform and Faster Robust Linear Regression","authors":"K. Clarkson, P. Drineas, M. Magdon-Ismail, Michael W. Mahoney, Xiangrui Meng, David P. Woodruff","doi":"10.1137/140963698","DOIUrl":"https://doi.org/10.1137/140963698","url":null,"abstract":"@ are a coreset for the problem, consisting of sampled and rescaled rows of A and b; and s is independent of n and polynomial in d. Our results improve on the best previous algorithms when n G d, for all p e [1, ∞) except p = 2; in particular, they improve the O(nd1.376+) running time of Sohler and Woodruff (STOC, 2011) for p = 1, that uses asymptotically fast matrix multiplication, and the O(nd5 log n) time of Dasgupta et al. (SICOMP, 2009) for general p, that uses ellipsoidal rounding. We also provide a suite of improved results for finding well-conditioned bases via ellipsoidal rounding, illustrating tradeoffs between running time and conditioning quality, including a one-pass conditioning algorithm for general ep problems. \u0000 \u0000To complement this theory, we provide a detailed empirical evaluation of implementations of our algorithms for p = 1, comparing them with several related algorithms. Among other things, our empirical results clearly show that, in the asymptotic regime, the theory is a very good guide to the practical performance of these algorithms. Our algorithms use our faster constructions of well-conditioned bases for ep spaces and, for p = 1, a fast subspace embedding of independent interest that we call the Fast Cauchy Transform: a matrix Π:","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133603494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 94

Restricted Isometry of Fourier Matrices and List Decodability of Random Linear Codes 傅立叶矩阵的限制等距和随机线性码的列表可译码性

SIAM journal on computing (Print) Pub Date : 2012-07-04 DOI: 10.1137/120896773

Mahdi Cheraghchi, V. Guruswami, A. Velingker

{"title":"Restricted Isometry of Fourier Matrices and List Decodability of Random Linear Codes","authors":"Mahdi Cheraghchi, V. Guruswami, A. Velingker","doi":"10.1137/120896773","DOIUrl":"https://doi.org/10.1137/120896773","url":null,"abstract":"@q, with probability arbitrarily close to 1, is list decodable at radius 1--1/q -- e with list size L = O(1/e2) and rate R = Ωq(e2/(log3(1/e))). Up to the polylogarithmic factor in 1/e and constant factors depending on q, this matches the lower bound L = Ωq(1/e2) for the list size and upper bound R = Oq(e2) for the rate. Previously only existence (and not abundance) of such codes was known for the special case q = 2 (Guruswami, Hastad, Sudan and Zuckerman, 2002).In order to obtain our result, we employ a relaxed version of the well known Johnson bound on list decoding that translates the average Hamming distance between codewords to list decoding guarantees. We furthermore prove that the desired average-distance guarantees hold for a code provided that a natural complex matrix encoding the codewords satisfies the Restricted Isometry Property with respect to the Euclidean norm (RIP-2). For the case of random binary linear codes, this matrix coincides with a random submatrix of the Hadamard-Walsh transform matrix that is well studied in the compressed sensing literature.Finally we improve the analysis of Rudelson and Vershynin (2008) on the number of random frequency samples required for exact reconstruction of k-sparse signals of length N. Specifically we improve the number of samples from O(k log (N) log2 (k)(log k+log log N)) to O(k log(N) log3(k)). The proof involves bounding the expected supremum of a related Gaussian process by using an improved analysis of the metric defined by the process. This improvement is crucial for our application in list decoding.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"235 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132586041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 116