Pub Date : 2021-01-01DOI: 10.1137/1.9781611976496.19
Sepehr Assadi, Aditi Dudeja
{"title":"A Simple Semi-Streaming Algorithm for Global Minimum Cuts","authors":"Sepehr Assadi, Aditi Dudeja","doi":"10.1137/1.9781611976496.19","DOIUrl":"https://doi.org/10.1137/1.9781611976496.19","url":null,"abstract":"","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"36 6-7","pages":"172-180"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72471556","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}
Pub Date : 2021-01-01DOI: 10.1137/1.9781611976496.15
William Kuszmaul, C. Leiserson
{"title":"Floors and Ceilings in Divide-and-Conquer Recurrences","authors":"William Kuszmaul, C. Leiserson","doi":"10.1137/1.9781611976496.15","DOIUrl":"https://doi.org/10.1137/1.9781611976496.15","url":null,"abstract":"","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"24 1","pages":"133-141"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78574073","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}
Pub Date : 2021-01-01DOI: 10.1137/1.9781611976496.16
Raphael A Meyer, Cameron Musco, Christopher Musco, David P Woodruff
We study the problem of estimating the trace of a matrix A that can only be accessed through matrix-vector multiplication. We introduce a new randomized algorithm, Hutch++, which computes a (1 ± ε) approximation to tr( A ) for any positive semidefinite (PSD) A using just O(1/ε) matrix-vector products. This improves on the ubiquitous Hutchinson's estimator, which requires O(1/ε2) matrix-vector products. Our approach is based on a simple technique for reducing the variance of Hutchinson's estimator using a low-rank approximation step, and is easy to implement and analyze. Moreover, we prove that, up to a logarithmic factor, the complexity of Hutch++ is optimal amongst all matrix-vector query algorithms, even when queries can be chosen adaptively. We show that it significantly outperforms Hutchinson's method in experiments. While our theory requires A to be positive semidefinite, empirical gains extend to applications involving non-PSD matrices, such as triangle estimation in networks.
{"title":"Hutch++: Optimal Stochastic Trace Estimation.","authors":"Raphael A Meyer, Cameron Musco, Christopher Musco, David P Woodruff","doi":"10.1137/1.9781611976496.16","DOIUrl":"https://doi.org/10.1137/1.9781611976496.16","url":null,"abstract":"<p><p>We study the problem of estimating the trace of a matrix <b><i>A</i></b> that can only be accessed through matrix-vector multiplication. We introduce a new randomized algorithm, Hutch++, which computes a (1 ± <i>ε</i>) approximation to tr( <b><i>A</i></b> ) for any positive semidefinite (PSD) <b><i>A</i></b> using just <i>O</i>(1<i>/ε</i>) matrix-vector products. This improves on the ubiquitous <i>Hutchinson's estimator</i>, which requires <i>O</i>(1<i>/ε</i> <sup>2</sup>) matrix-vector products. Our approach is based on a simple technique for reducing the variance of Hutchinson's estimator using a low-rank approximation step, and is easy to implement and analyze. Moreover, we prove that, up to a logarithmic factor, the complexity of Hutch++ is optimal amongst all matrix-vector query algorithms, even when queries can be chosen adaptively. We show that it significantly outperforms Hutchinson's method in experiments. While our theory requires <b><i>A</i></b> to be positive semidefinite, empirical gains extend to applications involving non-PSD matrices, such as triangle estimation in networks.</p>","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"2021 ","pages":"142-155"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8553228/pdf/nihms-1696966.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39580860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1137/1.9781611976496.4
T. Biedl, A. Lubiw, Anurag Murty Naredla, Peter Dominik Ralbovsky, Graeme Stroud
{"title":"Dispersion for Intervals: A Geometric Approach","authors":"T. Biedl, A. Lubiw, Anurag Murty Naredla, Peter Dominik Ralbovsky, Graeme Stroud","doi":"10.1137/1.9781611976496.4","DOIUrl":"https://doi.org/10.1137/1.9781611976496.4","url":null,"abstract":"","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"25 1","pages":"37-44"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80840553","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}
Pub Date : 2021-01-01DOI: 10.1137/1.9781611976496.10
Manoj Gupta, Shahbaz Khan
{"title":"Simple dynamic algorithms for Maximal Independent Set, Maximum Flow and Maximum Matching","authors":"Manoj Gupta, Shahbaz Khan","doi":"10.1137/1.9781611976496.10","DOIUrl":"https://doi.org/10.1137/1.9781611976496.10","url":null,"abstract":"","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"46 6 1","pages":"86-91"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76200339","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}
Pub Date : 2021-01-01DOI: 10.1137/1.9781611976496.18
Sepehr Assadi, S. C. Liu, R. Tarjan
{"title":"An Auction Algorithm for Bipartite Matching in Streaming and Massively Parallel Computation Models","authors":"Sepehr Assadi, S. C. Liu, R. Tarjan","doi":"10.1137/1.9781611976496.18","DOIUrl":"https://doi.org/10.1137/1.9781611976496.18","url":null,"abstract":"","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"18 1","pages":"165-171"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85164269","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}
Pub Date : 2020-11-12DOI: 10.1137/1.9781611976496.17
Michael Kapralov, Gilbert Maystre, Jakab Tardos
In this paper we revisit the problem of constructing randomized composable coresets for bipartite matching. In this problem the input graph is randomly partitioned across $k$ players, each of which sends a single message to a coordinator, who then must output a good approximation to the maximum matching in the input graph. Assadi and Khanna gave the first such coreset, achieving a $1/9$-approximation by having every player send a maximum matching, i.e. at most $n/2$ words per player. The approximation factor was improved to $1/3$ by Bernstein et al. In this paper, we show that the matching skeleton construction of Goel, Kapralov and Khanna, which is a carefully chosen (fractional) matching, is a randomized composable coreset that achieves a $1/2-o(1)$ approximation using at most $n-1$ words of communication per player. We also show an upper bound of $2/3+o(1)$ on the approximation ratio achieved by this coreset.
{"title":"Communication Efficient Coresets for Maximum Matching","authors":"Michael Kapralov, Gilbert Maystre, Jakab Tardos","doi":"10.1137/1.9781611976496.17","DOIUrl":"https://doi.org/10.1137/1.9781611976496.17","url":null,"abstract":"In this paper we revisit the problem of constructing randomized composable coresets for bipartite matching. In this problem the input graph is randomly partitioned across $k$ players, each of which sends a single message to a coordinator, who then must output a good approximation to the maximum matching in the input graph. Assadi and Khanna gave the first such coreset, achieving a $1/9$-approximation by having every player send a maximum matching, i.e. at most $n/2$ words per player. The approximation factor was improved to $1/3$ by Bernstein et al. \u0000In this paper, we show that the matching skeleton construction of Goel, Kapralov and Khanna, which is a carefully chosen (fractional) matching, is a randomized composable coreset that achieves a $1/2-o(1)$ approximation using at most $n-1$ words of communication per player. We also show an upper bound of $2/3+o(1)$ on the approximation ratio achieved by this coreset.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"18 1","pages":"156-164"},"PeriodicalIF":0.0,"publicationDate":"2020-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75275872","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}
Pub Date : 2020-10-29DOI: 10.1137/1.9781611976496.7
David R Karger, David P. Williamson
In this note, we revisit the recursive random contraction algorithm of Karger and Stein for finding a minimum cut in a graph. Our revisit is occasioned by a paper of Fox, Panigrahi, and Zhang which gives an extension of the Karger-Stein algorithm to minimum cuts and minimum $k$-cuts in hypergraphs. When specialized to the case of graphs, the algorithm is somewhat different than the original Karger-Stein algorithm. We show that the analysis becomes particularly clean in this case: we can prove that the probability that a fixed minimum cut in an $n$ node graph is returned by the algorithm is bounded below by $1/(2H_n-2)$, where $H_n$ is the $n$th harmonic number. We also consider other similar variants of the algorithm, and show that no such algorithm can achieve an asymptotically better probability of finding a fixed minimum cut.
{"title":"Recursive Random Contraction Revisited","authors":"David R Karger, David P. Williamson","doi":"10.1137/1.9781611976496.7","DOIUrl":"https://doi.org/10.1137/1.9781611976496.7","url":null,"abstract":"In this note, we revisit the recursive random contraction algorithm of Karger and Stein for finding a minimum cut in a graph. Our revisit is occasioned by a paper of Fox, Panigrahi, and Zhang which gives an extension of the Karger-Stein algorithm to minimum cuts and minimum $k$-cuts in hypergraphs. When specialized to the case of graphs, the algorithm is somewhat different than the original Karger-Stein algorithm. We show that the analysis becomes particularly clean in this case: we can prove that the probability that a fixed minimum cut in an $n$ node graph is returned by the algorithm is bounded below by $1/(2H_n-2)$, where $H_n$ is the $n$th harmonic number. We also consider other similar variants of the algorithm, and show that no such algorithm can achieve an asymptotically better probability of finding a fixed minimum cut.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"19 1","pages":"68-73"},"PeriodicalIF":0.0,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74735796","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}
Pub Date : 2020-10-26DOI: 10.1137/1.9781611976496.1
J. V. D. Brand
Many algorithms use data structures that maintain properties of matrices undergoing some changes. The applications are wide-ranging and include for example matchings, shortest paths, linear programming, semi-definite programming, convex hull and volume computation. Given the wide range of applications, the exact property these data structures must maintain varies from one application to another, forcing algorithm designers to invent them from scratch or modify existing ones. Thus it is not surprising that these data structures and their proofs are usually tailor-made for their specific application and that maintaining more complicated properties results in more complicated proofs. In this paper we present a unifying framework that captures a wide range of these data structures. The simplicity of this framework allows us to give short proofs for many existing data structures regardless of how complicated the to be maintained property is. We also show how the framework can be used to speed up existing iterative algorithms, such as the simplex algorithm. More formally, consider any rational function $f(A_1,...,A_d)$ with input matrices $A_1,...,A_d$. We show that the task of maintaining $f(A_1,...,A_d)$ under updates to $A_1,...,A_d$ can be reduced to the much simpler problem of maintaining some matrix inverse $M^{-1}$ under updates to $M$. The latter is a well studied problem called dynamic matrix inverse. By applying our reduction and using known algorithms for dynamic matrix inverse we can obtain fast data structures and iterative algorithms for much more general problems.
{"title":"Unifying Matrix Data Structures: Simplifying and Speeding up Iterative Algorithms","authors":"J. V. D. Brand","doi":"10.1137/1.9781611976496.1","DOIUrl":"https://doi.org/10.1137/1.9781611976496.1","url":null,"abstract":"Many algorithms use data structures that maintain properties of matrices undergoing some changes. The applications are wide-ranging and include for example matchings, shortest paths, linear programming, semi-definite programming, convex hull and volume computation. Given the wide range of applications, the exact property these data structures must maintain varies from one application to another, forcing algorithm designers to invent them from scratch or modify existing ones. Thus it is not surprising that these data structures and their proofs are usually tailor-made for their specific application and that maintaining more complicated properties results in more complicated proofs. \u0000In this paper we present a unifying framework that captures a wide range of these data structures. The simplicity of this framework allows us to give short proofs for many existing data structures regardless of how complicated the to be maintained property is. We also show how the framework can be used to speed up existing iterative algorithms, such as the simplex algorithm. \u0000More formally, consider any rational function $f(A_1,...,A_d)$ with input matrices $A_1,...,A_d$. We show that the task of maintaining $f(A_1,...,A_d)$ under updates to $A_1,...,A_d$ can be reduced to the much simpler problem of maintaining some matrix inverse $M^{-1}$ under updates to $M$. The latter is a well studied problem called dynamic matrix inverse. By applying our reduction and using known algorithms for dynamic matrix inverse we can obtain fast data structures and iterative algorithms for much more general problems.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"42 1","pages":"1-13"},"PeriodicalIF":0.0,"publicationDate":"2020-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89881041","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}
Pub Date : 2020-10-07DOI: 10.1137/1.9781611976496.11
Yixin Cao
Corneil, Olariu, and Stewart [SODA 1998; SIAM Journal on Discrete Mathematics 2009] presented a recognition algorithm for interval graphs by six graph searches. Li and Wu [Discrete Mathematics & Theoretical Computer Science 2014] simplified it to only four. The great simplicity of the latter algorithm is however eclipsed by the complicated and long proofs. The main purpose of this paper is to present a new and significantly short proof for Li and Wu's algorithm, as well as a simpler implementation. We also give a self-contained simpler interpretation of the recognition algorithm of Corneil [Discrete Applied Mathematics 2004] for unit interval graphs, based on three sweeps of graph searches. Moreover, we show that two sweeps are already sufficient. Toward the proofs of the main results, we make several new structural observations that might be of independent interests.
Corneil, Olariu, and Stewart [SODA 1998;SIAM Journal on Discrete Mathematics 2009]提出了一种基于六次图搜索的区间图识别算法。Li和Wu[离散数学&理论计算机科学2014]将其简化为只有四个。然而，后一种算法的简单性被复杂而冗长的证明所掩盖。本文的主要目的是为Li和Wu的算法提供一个新的和显著简短的证明，以及一个更简单的实现。我们还基于图搜索的三次扫描，对单位间隔图的Corneil[离散应用数学2004]的识别算法给出了一个独立的更简单的解释。此外，我们证明两次扫描已经足够了。为了证明主要结果，我们做了几个新的结构观察，可能是独立的兴趣。
{"title":"Recognizing (Unit) Interval Graphs by Zigzag Graph Searches","authors":"Yixin Cao","doi":"10.1137/1.9781611976496.11","DOIUrl":"https://doi.org/10.1137/1.9781611976496.11","url":null,"abstract":"Corneil, Olariu, and Stewart [SODA 1998; SIAM Journal on Discrete Mathematics 2009] presented a recognition algorithm for interval graphs by six graph searches. Li and Wu [Discrete Mathematics & Theoretical Computer Science 2014] simplified it to only four. The great simplicity of the latter algorithm is however eclipsed by the complicated and long proofs. The main purpose of this paper is to present a new and significantly short proof for Li and Wu's algorithm, as well as a simpler implementation. We also give a self-contained simpler interpretation of the recognition algorithm of Corneil [Discrete Applied Mathematics 2004] for unit interval graphs, based on three sweeps of graph searches. Moreover, we show that two sweeps are already sufficient. Toward the proofs of the main results, we make several new structural observations that might be of independent interests.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"9 1","pages":"92-106"},"PeriodicalIF":0.0,"publicationDate":"2020-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78446744","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}
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