Two of the most widely used approaches to obtain polynomial-time approximation schemes (PTASs) on planar graphs are the Lipton-Tarjan separator-based approach and Baker’s approach. In 2005, Demaine and Hajiaghayi strengthened both approaches using bidimensionality and obtained efficient polynomial-time approximation schemes (EPTASs) for several problems, including Connected Dominating Set and Feedback Vertex Set. In this work, we unify the two strengthened approaches to combine the best of both worlds. We develop a framework allowing the design of EPTAS on classes of graphs with the subquadratic grid minor (SQGM) property. Roughly speaking, a class of graphs has the SQGM property if, for every graph G from the class, the fact that G contains no t× t grid as a minor guarantees that the treewidth of G is subquadratic in t. For example, the class of planar graphs and, more generally, classes of graphs excluding some fixed graph as a minor, have the SQGM property. At the heart of our framework is a decomposition lemma stating that for “most” bidimensional problems on a graph class G with the SQGM property, there is a polynomial-time algorithm that, given a graph G ε G as input and an ε > 0, outputs a vertex set X of size ε ċ OPT such that the treewidth of G - X is f(ε). Here, OPT is the objective function value of the problem in question and f is a function depending only on ε. This allows us to obtain EPTASs on (apex)-minor-free graphs for all problems covered by the previous framework as well as for a wide range of packing problems, partial covering problems and problems that are neither closed under taking minors nor contractions. To the best of our knowledge, for many of these problems—including Cycle Packing, F-Packing, F-Deletion, Max Leaf Spanning Tree, or Partial r-Dominating Set —no EPTASs, even on planar graphs, were previously known. We also prove novel excluded grid theorems in unit disk and map graphs without large cliques. Using these theorems, we show that these classes of graphs have the SQGM property. Based on the developed framework, we design EPTASs and subexponential time parameterized algorithms for various classes of problems on unit disk and map graphs.
{"title":"Excluded Grid Minors and Efficient Polynomial-Time Approximation Schemes","authors":"F. Fomin, D. Lokshtanov, Saket Saurabh","doi":"10.1145/3154833","DOIUrl":"https://doi.org/10.1145/3154833","url":null,"abstract":"Two of the most widely used approaches to obtain polynomial-time approximation schemes (PTASs) on planar graphs are the Lipton-Tarjan separator-based approach and Baker’s approach. In 2005, Demaine and Hajiaghayi strengthened both approaches using bidimensionality and obtained efficient polynomial-time approximation schemes (EPTASs) for several problems, including Connected Dominating Set and Feedback Vertex Set. In this work, we unify the two strengthened approaches to combine the best of both worlds. We develop a framework allowing the design of EPTAS on classes of graphs with the subquadratic grid minor (SQGM) property. Roughly speaking, a class of graphs has the SQGM property if, for every graph G from the class, the fact that G contains no t× t grid as a minor guarantees that the treewidth of G is subquadratic in t. For example, the class of planar graphs and, more generally, classes of graphs excluding some fixed graph as a minor, have the SQGM property. At the heart of our framework is a decomposition lemma stating that for “most” bidimensional problems on a graph class G with the SQGM property, there is a polynomial-time algorithm that, given a graph G ε G as input and an ε > 0, outputs a vertex set X of size ε ċ OPT such that the treewidth of G - X is f(ε). Here, OPT is the objective function value of the problem in question and f is a function depending only on ε. This allows us to obtain EPTASs on (apex)-minor-free graphs for all problems covered by the previous framework as well as for a wide range of packing problems, partial covering problems and problems that are neither closed under taking minors nor contractions. To the best of our knowledge, for many of these problems—including Cycle Packing, F-Packing, F-Deletion, Max Leaf Spanning Tree, or Partial r-Dominating Set —no EPTASs, even on planar graphs, were previously known. We also prove novel excluded grid theorems in unit disk and map graphs without large cliques. Using these theorems, we show that these classes of graphs have the SQGM property. Based on the developed framework, we design EPTASs and subexponential time parameterized algorithms for various classes of problems on unit disk and map graphs.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82563343","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}
H. Attiya, Alexey Gotsman, Sandeep Hans, N. Rinetzky
Transactional memory (TM) facilitates the development of concurrent applications by letting a programmer designate certain code blocks as atomic. The common approach to stating TM correctness is through a consistency condition that restricts the possible TM executions. Unfortunately, existing consistency conditions fall short of formalizing the intuitive semantics of atomic blocks through which programmers use a TM. To close this gap, we formalize programmer expectations as observational refinement between TM implementations. This states that properties of a program using a concrete TM implementation can be established by analyzing its behavior with an abstract TM, serving as a specification of the concrete one. We show that a variant of Transactional Memory Specification (TMS), a TM consistency condition, is equivalent to observational refinement for a programming language where local variables are rolled back upon a transaction abort. We thereby establish that TMS is the weakest acceptable condition for this case. We then propose a new consistency condition, called Strong Transactional Memory Specification (STMS), and show that it is equivalent to observational refinement for a language where local variables are not rolled back upon aborts. Finally, we show that under certain natural assumptions on TM implementations, STMS is equivalent to a variant of a well-known condition of opacity. Our results suggest a new approach to evaluating TM consistency conditions and enable TM implementors and language designers to make better-informed decisions.
{"title":"Characterizing Transactional Memory Consistency Conditions Using Observational Refinement","authors":"H. Attiya, Alexey Gotsman, Sandeep Hans, N. Rinetzky","doi":"10.1145/3131360","DOIUrl":"https://doi.org/10.1145/3131360","url":null,"abstract":"Transactional memory (TM) facilitates the development of concurrent applications by letting a programmer designate certain code blocks as atomic. The common approach to stating TM correctness is through a consistency condition that restricts the possible TM executions. Unfortunately, existing consistency conditions fall short of formalizing the intuitive semantics of atomic blocks through which programmers use a TM. To close this gap, we formalize programmer expectations as observational refinement between TM implementations. This states that properties of a program using a concrete TM implementation can be established by analyzing its behavior with an abstract TM, serving as a specification of the concrete one. We show that a variant of Transactional Memory Specification (TMS), a TM consistency condition, is equivalent to observational refinement for a programming language where local variables are rolled back upon a transaction abort. We thereby establish that TMS is the weakest acceptable condition for this case. We then propose a new consistency condition, called Strong Transactional Memory Specification (STMS), and show that it is equivalent to observational refinement for a language where local variables are not rolled back upon aborts. Finally, we show that under certain natural assumptions on TM implementations, STMS is equivalent to a variant of a well-known condition of opacity. Our results suggest a new approach to evaluating TM consistency conditions and enable TM implementors and language designers to make better-informed decisions.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82043303","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}
M. Braverman, K. Efremenko, R. Gelles, Bernhard Haeupler
We study coding schemes for multiparty interactive communication over synchronous networks that suffer from stochastic noise, where each bit is independently flipped with probability ε. We analyze the minimal overhead that must be added by the coding scheme to succeed in performing the computation despite the noise. Our main result is a lower bound on the communication of any noise-resilient protocol over a synchronous star network with n parties (where all parties communicate in every round). Specifically, we show a task that can be solved by communicating T bits over the noise-free network, but for which any protocol with success probability of 1-o(1) must communicate at least Ω (T /log n log log n) bits when the channels are noisy. By a 1994 result of Rajagopalan and Schulman, the slowdown we prove is the highest one can obtain on any topology, up to a log log n factor. We complete our lower bound with a matching coding scheme that achieves the same overhead; thus, the capacity of (synchronous) star networks is Θ (log log n/log n). Our bounds prove that, despite several previous coding schemes with rate Ω (1) for certain topologies, no coding scheme with constant rate Ω (1) exists for arbitrary n-party noisy networks.
{"title":"Constant-Rate Coding for Multiparty Interactive Communication Is Impossible","authors":"M. Braverman, K. Efremenko, R. Gelles, Bernhard Haeupler","doi":"10.1145/3050218","DOIUrl":"https://doi.org/10.1145/3050218","url":null,"abstract":"We study coding schemes for multiparty interactive communication over synchronous networks that suffer from stochastic noise, where each bit is independently flipped with probability ε. We analyze the minimal overhead that must be added by the coding scheme to succeed in performing the computation despite the noise. Our main result is a lower bound on the communication of any noise-resilient protocol over a synchronous star network with n parties (where all parties communicate in every round). Specifically, we show a task that can be solved by communicating T bits over the noise-free network, but for which any protocol with success probability of 1-o(1) must communicate at least Ω (T /log n log log n) bits when the channels are noisy. By a 1994 result of Rajagopalan and Schulman, the slowdown we prove is the highest one can obtain on any topology, up to a log log n factor. We complete our lower bound with a matching coding scheme that achieves the same overhead; thus, the capacity of (synchronous) star networks is Θ (log log n/log n). Our bounds prove that, despite several previous coding schemes with rate Ω (1) for certain topologies, no coding scheme with constant rate Ω (1) exists for arbitrary n-party noisy networks.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75696382","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}
The Invited Article section of this issue consists of the article “Embeddability in the 3-Sphere Is Decidable” by Jiri Matousek, Eric Sedgwick, Martin Tancer, and Uli Wagner, which won the best paper award at the 30th Annual Symposium on Computational Geometry (SoCG’14). We would like to thank the SoGT’14 Program Committee for their help in selecting this invited article, and thank editor Jean-Daniel Boissonnat for handling the article.
{"title":"Invited Article Foreword","authors":"É. Tardos","doi":"10.1145/3159447","DOIUrl":"https://doi.org/10.1145/3159447","url":null,"abstract":"The Invited Article section of this issue consists of the article “Embeddability in the 3-Sphere Is Decidable” by Jiri Matousek, Eric Sedgwick, Martin Tancer, and Uli Wagner, which won the best paper award at the 30th Annual Symposium on Computational Geometry (SoCG’14). We would like to thank the SoGT’14 Program Committee for their help in selecting this invited article, and thank editor Jean-Daniel Boissonnat for handling the article.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90757581","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}
Differential-algebraic equations (DAEs) are widely used for the modeling of dynamical systems. The difficulty in numerically solving a DAE is measured by its differentiation index. For highly accurate simulation of dynamical systems, it is important to convert high-index DAEs into low-index DAEs. Most of the existing simulation software packages for dynamical systems are equipped with an index-reduction algorithm given by Mattsson and Söderlind. Unfortunately, this algorithm fails if there are numerical cancellations. These numerical cancellations are often caused by accurate constants in structural equations. Distinguishing those accurate constants from generic parameters that represent physical quantities, Murota and Iri introduced the notion of a mixed matrix as a mathematical tool for faithful model description in a structural approach to systems analysis. For DAEs described with the use of mixed matrices, efficient algorithms to compute the index have been developed by exploiting matroid theory. This article presents an index-reduction algorithm for linear DAEs whose coefficient matrices are mixed matrices, i.e., linear DAEs containing physical quantities as parameters. Our algorithm detects numerical cancellations between accurate constants and transforms a DAE into an equivalent DAE to which Mattsson–Söderlind’s index-reduction algorithm is applicable. Our algorithm is based on the combinatorial relaxation approach, which is a framework to solve a linear algebraic problem by iteratively relaxing it into an efficiently solvable combinatorial optimization problem. The algorithm does not rely on symbolic manipulations but on fast combinatorial algorithms on graphs and matroids. Our algorithm is proved to work for any linear DAEs whose coefficient matrices are mixed matrices. Furthermore, we provide an improved algorithm under an assumption based on dimensional analysis of dynamical systems. Through numerical experiments, it is confirmed that our algorithms run sufficiently fast for large-scale DAEs and output DAEs such that physical meanings of coefficients are easy to interpret. Our algorithms can also be applied to nonlinear DAEs by regarding nonlinear terms as parameters.
{"title":"Index Reduction for Differential-algebraic Equations with Mixed Matrices","authors":"S. Iwata, Taihei Oki, Mizuyo Takamatsu","doi":"10.1145/3341499","DOIUrl":"https://doi.org/10.1145/3341499","url":null,"abstract":"Differential-algebraic equations (DAEs) are widely used for the modeling of dynamical systems. The difficulty in numerically solving a DAE is measured by its differentiation index. For highly accurate simulation of dynamical systems, it is important to convert high-index DAEs into low-index DAEs. Most of the existing simulation software packages for dynamical systems are equipped with an index-reduction algorithm given by Mattsson and Söderlind. Unfortunately, this algorithm fails if there are numerical cancellations. These numerical cancellations are often caused by accurate constants in structural equations. Distinguishing those accurate constants from generic parameters that represent physical quantities, Murota and Iri introduced the notion of a mixed matrix as a mathematical tool for faithful model description in a structural approach to systems analysis. For DAEs described with the use of mixed matrices, efficient algorithms to compute the index have been developed by exploiting matroid theory. This article presents an index-reduction algorithm for linear DAEs whose coefficient matrices are mixed matrices, i.e., linear DAEs containing physical quantities as parameters. Our algorithm detects numerical cancellations between accurate constants and transforms a DAE into an equivalent DAE to which Mattsson–Söderlind’s index-reduction algorithm is applicable. Our algorithm is based on the combinatorial relaxation approach, which is a framework to solve a linear algebraic problem by iteratively relaxing it into an efficiently solvable combinatorial optimization problem. The algorithm does not rely on symbolic manipulations but on fast combinatorial algorithms on graphs and matroids. Our algorithm is proved to work for any linear DAEs whose coefficient matrices are mixed matrices. Furthermore, we provide an improved algorithm under an assumption based on dimensional analysis of dynamical systems. Through numerical experiments, it is confirmed that our algorithms run sufficiently fast for large-scale DAEs and output DAEs such that physical meanings of coefficients are easy to interpret. Our algorithms can also be applied to nonlinear DAEs by regarding nonlinear terms as parameters.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73183410","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}
“Mirror, mirror, on the wall, who is the fairest of them all?” The Evil Queen What is a fair way to assign rooms to several housemates and divide the rent between them? This is not just a theoretical question: many people have used the Spliddit website to obtain envy-free solutions to rent division instances. But envy freeness, in and of itself, is insufficient to guarantee outcomes that people view as intuitive and acceptable. We therefore focus on solutions that optimize a criterion of social justice, subject to the envy-freeness constraint, in order to pinpoint the “fairest” solutions. We develop a general algorithmic framework that enables the computation of such solutions in polynomial time. We then study the relations between natural optimization objectives and identify the maximin solution, which maximizes the minimum utility subject to envy freeness, as the most attractive. We demonstrate, in theory and using experiments on real data from Spliddit, that the maximin solution gives rise to significant gains in terms of our optimization objectives. Finally, a user study with Spliddit users as subjects demonstrates that people find the maximin solution to be significantly fairer than arbitrary envy-free solutions; this user study is unprecedented in that it asks people about their real-world rent division instances. Based on these results, the maximin solution has been deployed on Spliddit since April 2015.
{"title":"Which Is the Fairest (Rent Division) of Them All?","authors":"Y. Gal, Moshe Mash, A. Procaccia, Yair Zick","doi":"10.1145/3131361","DOIUrl":"https://doi.org/10.1145/3131361","url":null,"abstract":"“Mirror, mirror, on the wall, who is the fairest of them all?” The Evil Queen What is a fair way to assign rooms to several housemates and divide the rent between them? This is not just a theoretical question: many people have used the Spliddit website to obtain envy-free solutions to rent division instances. But envy freeness, in and of itself, is insufficient to guarantee outcomes that people view as intuitive and acceptable. We therefore focus on solutions that optimize a criterion of social justice, subject to the envy-freeness constraint, in order to pinpoint the “fairest” solutions. We develop a general algorithmic framework that enables the computation of such solutions in polynomial time. We then study the relations between natural optimization objectives and identify the maximin solution, which maximizes the minimum utility subject to envy freeness, as the most attractive. We demonstrate, in theory and using experiments on real data from Spliddit, that the maximin solution gives rise to significant gains in terms of our optimization objectives. Finally, a user study with Spliddit users as subjects demonstrates that people find the maximin solution to be significantly fairer than arbitrary envy-free solutions; this user study is unprecedented in that it asks people about their real-world rent division instances. Based on these results, the maximin solution has been deployed on Spliddit since April 2015.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77725284","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}
X. Goaoc, P. Paták, Zuzana Patáková, M. Tancer, Uli Wagner
We prove that for every d ≥ 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d ≥ 2 and k ≥ 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable is NP-hard. For d ≥ 3, both problems remain NP-hard when restricted to contractible pure d-dimensional complexes. Another simple corollary of our result is that it is NP-hard to decide whether a given poset is CL-shellable.
{"title":"Shellability is NP-complete","authors":"X. Goaoc, P. Paták, Zuzana Patáková, M. Tancer, Uli Wagner","doi":"10.1145/3314024","DOIUrl":"https://doi.org/10.1145/3314024","url":null,"abstract":"We prove that for every d ≥ 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d ≥ 2 and k ≥ 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable is NP-hard. For d ≥ 3, both problems remain NP-hard when restricted to contractible pure d-dimensional complexes. Another simple corollary of our result is that it is NP-hard to decide whether a given poset is CL-shellable.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84763884","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}
The Invited Article section of this issue consists of two papers. The paper “Communication Steps for Parallel Query Processing,” by Paul Beame, Paraschos Koutris, and Dan Suciu, was invited from the 32nd and 33rd Annual ACM Symposium on Principles of Distributed Computing (PODC’13–14). The paper “The matching polytope has exponential extension complexity,” by Thomas Rothvoss, won the best paper award at the 46th ACM Symposium on Theory of Computing (STOC’14). We want to thank the PODC’13 and PDOC’14 and STOC’14 Program Committees for their help in selecting these invited papers. We thank editor Phokion Kolaitis for handling the first of the two papers.
{"title":"Invited Articles Foreword","authors":"É. Tardos","doi":"10.1145/3151720","DOIUrl":"https://doi.org/10.1145/3151720","url":null,"abstract":"The Invited Article section of this issue consists of two papers. The paper “Communication Steps for Parallel Query Processing,” by Paul Beame, Paraschos Koutris, and Dan Suciu, was invited from the 32nd and 33rd Annual ACM Symposium on Principles of Distributed Computing (PODC’13–14). The paper “The matching polytope has exponential extension complexity,” by Thomas Rothvoss, won the best paper award at the 46th ACM Symposium on Theory of Computing (STOC’14). We want to thank the PODC’13 and PDOC’14 and STOC’14 Program Committees for their help in selecting these invited papers. We thank editor Phokion Kolaitis for handling the first of the two papers.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87939603","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}
We show that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings x and y is equal, up to logarithmic precision, to the length of the longest shared secret key that two parties—one having x and the complexity profile of the pair and the other one having y and the complexity profile of the pair—can establish via a probabilistic protocol with interaction on a public channel. For ℓ > 2, the longest shared secret that can be established from a tuple of strings (x1, …, xℓ) by ℓ parties—each one having one component of the tuple and the complexity profile of the tuple—is equal, up to logarithmic precision, to the complexity of the tuple minus the minimum communication necessary for distributing the tuple to all parties. We establish the communication complexity of secret key agreement protocols that produce a secret key of maximal length for protocols with public randomness. We also show that if the communication complexity drops below the established threshold, then only very short secret keys can be obtained.
{"title":"An Operational Characterization of Mutual Information in Algorithmic Information Theory","authors":"Andrei E. Romashchenko, Marius Zimand","doi":"10.1145/3356867","DOIUrl":"https://doi.org/10.1145/3356867","url":null,"abstract":"We show that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings x and y is equal, up to logarithmic precision, to the length of the longest shared secret key that two parties—one having x and the complexity profile of the pair and the other one having y and the complexity profile of the pair—can establish via a probabilistic protocol with interaction on a public channel. For ℓ > 2, the longest shared secret that can be established from a tuple of strings (x1, …, xℓ) by ℓ parties—each one having one component of the tuple and the complexity profile of the tuple—is equal, up to logarithmic precision, to the complexity of the tuple minus the minimum communication necessary for distributing the tuple to all parties. We establish the communication complexity of secret key agreement protocols that produce a secret key of maximal length for protocols with public randomness. We also show that if the communication complexity drops below the established threshold, then only very short secret keys can be obtained.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89793857","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}
We study the problem of computing conjunctive queries over large databases on parallel architectures without shared storage. Using the structure of such a query q and the skew in the data, we study tradeoffs between the number of processors, the number of rounds of communication, and the per-processor load—the number of bits each processor can send or can receive in a single round—that are required to compute q. Since each processor must store its received bits, the load is at most the number of bits of storage per processor. When the data are free of skew, we obtain essentially tight upper and lower bounds for one round algorithms, and we show how the bounds degrade when there is skew in the data. In the case of skewed data, we show how to improve the algorithms when approximate degrees of the (necessarily small number of) heavy-hitter elements are available, obtaining essentially optimal algorithms for queries such as skewed simple joins and skewed triangle join queries. For queries that we identify as treelike, we also prove nearly matching upper and lower bounds for multi-round algorithms for a natural class of skew-free databases. One consequence of these latter lower bounds is that for any ϵ > 0, using p processors to compute the connected components of a graph, or to output the path, if any, between a specified pair of vertices of a graph with m edges and per-processor load that is O(m/p1−ϵ) requires Ω(logp) rounds of communication. Our upper bounds are given by simple structured algorithms using MapReduce. Our one-round lower bounds are proved in a very general model, which we call the Massively Parallel Communication (MPC) model, that allows processors to communicate arbitrary bits. Our multi-round lower bounds apply in a restricted version of the MPC model in which processors in subsequent rounds after the first communication round are only allowed to send tuples.
{"title":"Communication Steps for Parallel Query Processing","authors":"P. Beame, Paraschos Koutris, Dan Suciu","doi":"10.1145/3125644","DOIUrl":"https://doi.org/10.1145/3125644","url":null,"abstract":"We study the problem of computing conjunctive queries over large databases on parallel architectures without shared storage. Using the structure of such a query q and the skew in the data, we study tradeoffs between the number of processors, the number of rounds of communication, and the per-processor load—the number of bits each processor can send or can receive in a single round—that are required to compute q. Since each processor must store its received bits, the load is at most the number of bits of storage per processor. When the data are free of skew, we obtain essentially tight upper and lower bounds for one round algorithms, and we show how the bounds degrade when there is skew in the data. In the case of skewed data, we show how to improve the algorithms when approximate degrees of the (necessarily small number of) heavy-hitter elements are available, obtaining essentially optimal algorithms for queries such as skewed simple joins and skewed triangle join queries. For queries that we identify as treelike, we also prove nearly matching upper and lower bounds for multi-round algorithms for a natural class of skew-free databases. One consequence of these latter lower bounds is that for any ϵ > 0, using p processors to compute the connected components of a graph, or to output the path, if any, between a specified pair of vertices of a graph with m edges and per-processor load that is O(m/p1−ϵ) requires Ω(logp) rounds of communication. Our upper bounds are given by simple structured algorithms using MapReduce. Our one-round lower bounds are proved in a very general model, which we call the Massively Parallel Communication (MPC) model, that allows processors to communicate arbitrary bits. Our multi-round lower bounds apply in a restricted version of the MPC model in which processors in subsequent rounds after the first communication round are only allowed to send tuples.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78017564","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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