We give a simple algorithm to efficiently sample the rows of a matrix while preserving the p-norms of its product with vectors. Given an n * d matrix A, we find with high probability and in input sparsity time an A' consisting of about d log d rescaled rows of A such that |Ax|1 is close to |A'x|1 for all vectors x. We also show similar results for all Lp that give nearly optimal sample bounds in input sparsity time. Our results are based on sampling by "Lewis weights", which can be viewed as statistical leverage scores of a reweighted matrix. We also give an elementary proof of the guarantees of this sampling process for L1.
{"title":"Lp Row Sampling by Lewis Weights","authors":"Michael B. Cohen, Richard Peng","doi":"10.1145/2746539.2746567","DOIUrl":"https://doi.org/10.1145/2746539.2746567","url":null,"abstract":"We give a simple algorithm to efficiently sample the rows of a matrix while preserving the p-norms of its product with vectors. Given an n * d matrix A, we find with high probability and in input sparsity time an A' consisting of about d log d rescaled rows of A such that |Ax|1 is close to |A'x|1 for all vectors x. We also show similar results for all Lp that give nearly optimal sample bounds in input sparsity time. Our results are based on sampling by \"Lewis weights\", which can be viewed as statistical leverage scores of a reweighted matrix. We also give an elementary proof of the guarantees of this sampling process for L1.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73837868","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 a classical iterative algorithm for the problem of balancing matrices in the L∞ norm via a scaling transformation. This algorithm, which goes back to Osborne and Parlett & Reinsch in the 1960s, is implemented as a standard preconditioner in many numerical linear algebra packages. Surprisingly, despite its widespread use over several decades, no bounds were known on its rate of convergence. In this paper we prove that, for a large class of irreducible n x n (real or complex) input matrices~$A$, a natural variant of the algorithm converges in O(n3 log(nρ/ε)) elementary balancing operations, where ρ measures the initial imbalance of A and ε is the target imbalance of the output matrix. (The imbalance of A is maxi |log(aiout/aiin)|, where aiout,aiin are the maximum entries in magnitude in the ith row and column respectively.) This bound is tight up to the log n factor. A balancing operation scales the ith row and column so that their maximum entries are equal, and requires O(m/n) arithmetic operations on average, where m is the number of non-zero elements in A. Thus the running time of the iterative algorithm is ~O(n2m). This is the first time bound of any kind on any variant of the Osborne-Parlett-Reinsch algorithm. The class of matrices for which the above analysis holds are those which satisfy a condition we call Unique Balance, meaning that the limit of the iterative balancing process does not depend on the order in which balancing operations are performed. We also prove a combinatorial characterization of the Unique Balance property, which had earlier been conjectured by Chen.
{"title":"Analysis of a Classical Matrix Preconditioning Algorithm","authors":"L. Schulman, A. Sinclair","doi":"10.1145/2746539.2746556","DOIUrl":"https://doi.org/10.1145/2746539.2746556","url":null,"abstract":"We study a classical iterative algorithm for the problem of balancing matrices in the L∞ norm via a scaling transformation. This algorithm, which goes back to Osborne and Parlett & Reinsch in the 1960s, is implemented as a standard preconditioner in many numerical linear algebra packages. Surprisingly, despite its widespread use over several decades, no bounds were known on its rate of convergence. In this paper we prove that, for a large class of irreducible n x n (real or complex) input matrices~$A$, a natural variant of the algorithm converges in O(n3 log(nρ/ε)) elementary balancing operations, where ρ measures the initial imbalance of A and ε is the target imbalance of the output matrix. (The imbalance of A is maxi |log(aiout/aiin)|, where aiout,aiin are the maximum entries in magnitude in the ith row and column respectively.) This bound is tight up to the log n factor. A balancing operation scales the ith row and column so that their maximum entries are equal, and requires O(m/n) arithmetic operations on average, where m is the number of non-zero elements in A. Thus the running time of the iterative algorithm is ~O(n2m). This is the first time bound of any kind on any variant of the Osborne-Parlett-Reinsch algorithm. The class of matrices for which the above analysis holds are those which satisfy a condition we call Unique Balance, meaning that the limit of the iterative balancing process does not depend on the order in which balancing operations are performed. We also prove a combinatorial characterization of the Unique Balance property, which had earlier been conjectured by Chen.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80116157","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}
George Giakkoupis, Maryam Helmi, L. Higham, Philipp Woelfel
The test-and-set object is a fundamental synchronization primitive for shared memory systems. This paper addresses the number of registers (supporting atomic reads and writes) required to implement a one-shot test-and-set object in the standard asynchronous shared memory model with n processes. The best lower bound is log n - 1 [12,21] for obstruction-free and deadlock-free implementations, and recently a deterministic obstruction-free implementation using O(√ n) registers was presented [11]. This paper closes the gap between these existing upper and lower bounds by presenting a deterministic obstruction-free implementation of a one-shot test-and-set object from Θ(log n) registers of size Θ(log n) bits. Combining our obstruction-free algorithm with techniques from previous research [11,12], we also obtain a randomized wait-free test-and-set algorithm from Θ(log n) registers, with expected step-complexity Θ(log* n) against the oblivious adversary. The core tool in our algorithm is the implementation of a deterministic obstruction-free sifter object, using only 6 registers. If k processes access a sifter, then when they have terminated, at least one and at most ⌊(2k+1)/3⌋ processes return "win" and all others return "lose".
{"title":"Test-and-Set in Optimal Space","authors":"George Giakkoupis, Maryam Helmi, L. Higham, Philipp Woelfel","doi":"10.1145/2746539.2746627","DOIUrl":"https://doi.org/10.1145/2746539.2746627","url":null,"abstract":"The test-and-set object is a fundamental synchronization primitive for shared memory systems. This paper addresses the number of registers (supporting atomic reads and writes) required to implement a one-shot test-and-set object in the standard asynchronous shared memory model with n processes. The best lower bound is log n - 1 [12,21] for obstruction-free and deadlock-free implementations, and recently a deterministic obstruction-free implementation using O(√ n) registers was presented [11]. This paper closes the gap between these existing upper and lower bounds by presenting a deterministic obstruction-free implementation of a one-shot test-and-set object from Θ(log n) registers of size Θ(log n) bits. Combining our obstruction-free algorithm with techniques from previous research [11,12], we also obtain a randomized wait-free test-and-set algorithm from Θ(log n) registers, with expected step-complexity Θ(log* n) against the oblivious adversary. The core tool in our algorithm is the implementation of a deterministic obstruction-free sifter object, using only 6 registers. If k processes access a sifter, then when they have terminated, at least one and at most ⌊(2k+1)/3⌋ processes return \"win\" and all others return \"lose\".","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"120 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87946182","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}
Due to the lack of unconditional polynomial lower bounds, it is now in fashion to prove conditional lower bounds in order to advance our understanding of the class P. The vast majority of these lower bounds are based on one of three famous hypotheses: the 3-SUM conjecture, the APSP conjecture, and the Strong Exponential Time Hypothesis. Only circumstantial evidence is known in support of these hypotheses, and no formal relationship between them is known. In hopes of obtaining "less conditional" and therefore more reliable lower bounds, we consider the conjecture that at least one of the above three hypotheses is true. We design novel reductions from 3-SUM, APSP, and CNF-SAT, and derive interesting consequences of this very plausible conjecture, including: Tight n3-o(1) lower bounds for purely-combinatorial problems about the triangles in unweighted graphs. New n1-o(1) lower bounds for the amortized update and query times of dynamic algorithms for single-source reachability, strongly connected components, and Max-Flow. New n1.5-o(1) lower bound for computing a set of n st-maximum-flow values in a directed graph with n nodes and ~O(n) edges. There is a hierarchy of natural graph problems on n nodes with complexity nc for c ∈ (2,3). Only slightly non-trivial consequences of this conjecture were known prior to our work. Along the way we also obtain new conditional lower bounds for the Single-Source-Max-Flow problem.
{"title":"Matching Triangles and Basing Hardness on an Extremely Popular Conjecture","authors":"Amir Abboud, V. V. Williams, Huacheng Yu","doi":"10.1145/2746539.2746594","DOIUrl":"https://doi.org/10.1145/2746539.2746594","url":null,"abstract":"Due to the lack of unconditional polynomial lower bounds, it is now in fashion to prove conditional lower bounds in order to advance our understanding of the class P. The vast majority of these lower bounds are based on one of three famous hypotheses: the 3-SUM conjecture, the APSP conjecture, and the Strong Exponential Time Hypothesis. Only circumstantial evidence is known in support of these hypotheses, and no formal relationship between them is known. In hopes of obtaining \"less conditional\" and therefore more reliable lower bounds, we consider the conjecture that at least one of the above three hypotheses is true. We design novel reductions from 3-SUM, APSP, and CNF-SAT, and derive interesting consequences of this very plausible conjecture, including: Tight n3-o(1) lower bounds for purely-combinatorial problems about the triangles in unweighted graphs. New n1-o(1) lower bounds for the amortized update and query times of dynamic algorithms for single-source reachability, strongly connected components, and Max-Flow. New n1.5-o(1) lower bound for computing a set of n st-maximum-flow values in a directed graph with n nodes and ~O(n) edges. There is a hierarchy of natural graph problems on n nodes with complexity nc for c ∈ (2,3). Only slightly non-trivial consequences of this conjecture were known prior to our work. Along the way we also obtain new conditional lower bounds for the Single-Source-Max-Flow problem.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82731559","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. Henzinger, Sebastian Krinninger, Danupon Nanongkai, Thatchaphol Saranurak
Consider the following Online Boolean Matrix-Vector Multiplication problem: We are given an n x n matrix M and will receive n column-vectors of size n, denoted by v1, ..., vn, one by one. After seeing each vector vi, we have to output the product Mvi before we can see the next vector. A naive algorithm can solve this problem using O(n3) time in total, and its running time can be slightly improved to O(n3/log2 n) [Williams SODA'07]. We show that a conjecture that there is no truly subcubic (O(n3-ε)) time algorithm for this problem can be used to exhibit the underlying polynomial time hardness shared by many dynamic problems. For a number of problems, such as subgraph connectivity, Pagh's problem, d-failure connectivity, decremental single-source shortest paths, and decremental transitive closure, this conjecture implies tight hardness results. Thus, proving or disproving this conjecture will be very interesting as it will either imply several tight unconditional lower bounds or break through a common barrier that blocks progress with these problems. This conjecture might also be considered as strong evidence against any further improvement for these problems since refuting it will imply a major breakthrough for combinatorial Boolean matrix multiplication and other long-standing problems if the term "combinatorial algorithms" is interpreted as "Strassen-like algorithms" [Ballard et al. SPAA'11]. The conjecture also leads to hardness results for problems that were previously based on diverse problems and conjectures -- such as 3SUM, combinatorial Boolean matrix multiplication, triangle detection, and multiphase -- thus providing a uniform way to prove polynomial hardness results for dynamic algorithms; some of the new proofs are also simpler or even become trivial. The conjecture also leads to stronger and new, non-trivial, hardness results, e.g., for the fully-dynamic densest subgraph and diameter problems.
考虑下面的在线布尔矩阵向量乘法问题:我们给定一个n × n矩阵M，并将得到n个大小为n的列向量，记为v1，…， vn，一个接一个。在看到每个向量vi之后，我们必须在看到下一个向量之前输出乘积Mvi。一种朴素算法可以用总共O(n3)时间来解决这个问题，其运行时间可以略微提高到O(n3/log2 n) [Williams SODA'07]。我们证明了这个问题不存在真正的次三次(O(n3-ε))时间算法的猜想可以用来展示许多动态问题所共有的多项式时间硬度。对于许多问题，如子图连通性、Pagh问题、d-failure连通性、递减单源最短路径和递减传递闭包，这个猜想意味着紧硬度结果。因此，证明或反驳这个猜想将是非常有趣的，因为它要么暗示了几个严格的无条件下界，要么突破了阻碍这些问题进展的常见障碍。这一猜想也可能被认为是反对进一步改进这些问题的有力证据，因为反驳它将意味着组合布尔矩阵乘法和其他长期存在的问题的重大突破，如果术语“组合算法”被解释为“Strassen-like算法”[Ballard et al]。SPAA 11]。该猜想还导致以前基于不同问题和猜想的问题的硬度结果-例如3SUM，组合布尔矩阵乘法，三角形检测和多相-从而为证明动态算法的多项式硬度结果提供了统一的方法;一些新的证明也更简单，甚至变得微不足道。该猜想还导致更强的和新的，非平凡的，硬度的结果，例如，对于全动态密度子图和直径问题。
{"title":"Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture","authors":"M. Henzinger, Sebastian Krinninger, Danupon Nanongkai, Thatchaphol Saranurak","doi":"10.1145/2746539.2746609","DOIUrl":"https://doi.org/10.1145/2746539.2746609","url":null,"abstract":"Consider the following Online Boolean Matrix-Vector Multiplication problem: We are given an n x n matrix M and will receive n column-vectors of size n, denoted by v1, ..., vn, one by one. After seeing each vector vi, we have to output the product Mvi before we can see the next vector. A naive algorithm can solve this problem using O(n3) time in total, and its running time can be slightly improved to O(n3/log2 n) [Williams SODA'07]. We show that a conjecture that there is no truly subcubic (O(n3-ε)) time algorithm for this problem can be used to exhibit the underlying polynomial time hardness shared by many dynamic problems. For a number of problems, such as subgraph connectivity, Pagh's problem, d-failure connectivity, decremental single-source shortest paths, and decremental transitive closure, this conjecture implies tight hardness results. Thus, proving or disproving this conjecture will be very interesting as it will either imply several tight unconditional lower bounds or break through a common barrier that blocks progress with these problems. This conjecture might also be considered as strong evidence against any further improvement for these problems since refuting it will imply a major breakthrough for combinatorial Boolean matrix multiplication and other long-standing problems if the term \"combinatorial algorithms\" is interpreted as \"Strassen-like algorithms\" [Ballard et al. SPAA'11]. The conjecture also leads to hardness results for problems that were previously based on diverse problems and conjectures -- such as 3SUM, combinatorial Boolean matrix multiplication, triangle detection, and multiphase -- thus providing a uniform way to prove polynomial hardness results for dynamic algorithms; some of the new proofs are also simpler or even become trivial. The conjecture also leads to stronger and new, non-trivial, hardness results, e.g., for the fully-dynamic densest subgraph and diameter problems.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76768309","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}
In the Submodular Welfare Maximization (SWM) problem, the input consists of a set of n items, each of which must be allocated to one of m agents. Each agent l has a valuation function vl, where vl(S) denotes the welfare obtained by this agent if she receives the set of items S. The functions vl are all submodular; as is standard, we assume that they are monotone and vl(∅) = 0. The goal is to partition the items into m disjoint subsets S1, S2, ... Sm in order to maximize the social welfare, defined as ∑l = 1m vl(Sl). A simple greedy algorithm gives a 1/2-approximation to SWM in the offline setting, and this was the best known until Vondrak's recent (1-1/e)-approximation algorithm [34]. In this paper, we consider the online version of SWM. Here, items arrive one at a time in an online manner; when an item arrives, the algorithm must make an irrevocable decision about which agent to assign it to before seeing any subsequent items. This problem is motivated by applications to Internet advertising, where user ad impressions must be allocated to advertisers whose value is a submodular function of the set of users / impressions they receive. There are two natural models that differ in the order in which items arrive. In the fully adversarial setting, an adversary can construct an arbitrary / worst-case instance, as well as pick the order in which items arrive in order to minimize the algorithm's performance. In this setting, the 1/2-competitive greedy algorithm is the best possible. To improve on this, one must weaken the adversary slightly: In the random order model, the adversary can construct a worst-case set of items and valuations, but does not control the order in which the items arrive; instead, they are assumed to arrive in a random order. The random order model has been well studied for online SWM and various special cases, but the best known competitive ratio (even for several special cases) is 1/2 + 1/n [9,10], barely better than the ratio for the adversarial order. Obtaining a competitive ratio of 1/2 + Ω(1) for the random order model has been an important open problem for several years. We solve this open problem by demonstrating that the greedy algorithm has a competitive ratio of at least 0.505 for online SWM in the random order model. This is the first result showing a competitive ratio bounded above 1/2 in the random order model, even for special cases such as the weighted matching or budgeted allocation problems (without the so-called 'large capacity' assumptions). For special cases of submodular functions including weighted matching, weighted coverage functions and a broader class of "second-order supermodular" functions, we provide a different analysis that gives a competitive ratio of 0.51. We analyze the greedy algorithm using a factor-revealing linear program, bounding how the assignment of one item can decrease potential welfare from assigning future items. We also formulate a natural conjecture which, if true, would improve the
{"title":"Online Submodular Welfare Maximization: Greedy Beats 1/2 in Random Order","authors":"Nitish Korula, V. Mirrokni, Morteza Zadimoghaddam","doi":"10.1145/2746539.2746626","DOIUrl":"https://doi.org/10.1145/2746539.2746626","url":null,"abstract":"In the Submodular Welfare Maximization (SWM) problem, the input consists of a set of n items, each of which must be allocated to one of m agents. Each agent l has a valuation function vl, where vl(S) denotes the welfare obtained by this agent if she receives the set of items S. The functions vl are all submodular; as is standard, we assume that they are monotone and vl(∅) = 0. The goal is to partition the items into m disjoint subsets S1, S2, ... Sm in order to maximize the social welfare, defined as ∑l = 1m vl(Sl). A simple greedy algorithm gives a 1/2-approximation to SWM in the offline setting, and this was the best known until Vondrak's recent (1-1/e)-approximation algorithm [34]. In this paper, we consider the online version of SWM. Here, items arrive one at a time in an online manner; when an item arrives, the algorithm must make an irrevocable decision about which agent to assign it to before seeing any subsequent items. This problem is motivated by applications to Internet advertising, where user ad impressions must be allocated to advertisers whose value is a submodular function of the set of users / impressions they receive. There are two natural models that differ in the order in which items arrive. In the fully adversarial setting, an adversary can construct an arbitrary / worst-case instance, as well as pick the order in which items arrive in order to minimize the algorithm's performance. In this setting, the 1/2-competitive greedy algorithm is the best possible. To improve on this, one must weaken the adversary slightly: In the random order model, the adversary can construct a worst-case set of items and valuations, but does not control the order in which the items arrive; instead, they are assumed to arrive in a random order. The random order model has been well studied for online SWM and various special cases, but the best known competitive ratio (even for several special cases) is 1/2 + 1/n [9,10], barely better than the ratio for the adversarial order. Obtaining a competitive ratio of 1/2 + Ω(1) for the random order model has been an important open problem for several years. We solve this open problem by demonstrating that the greedy algorithm has a competitive ratio of at least 0.505 for online SWM in the random order model. This is the first result showing a competitive ratio bounded above 1/2 in the random order model, even for special cases such as the weighted matching or budgeted allocation problems (without the so-called 'large capacity' assumptions). For special cases of submodular functions including weighted matching, weighted coverage functions and a broader class of \"second-order supermodular\" functions, we provide a different analysis that gives a competitive ratio of 0.51. We analyze the greedy algorithm using a factor-revealing linear program, bounding how the assignment of one item can decrease potential welfare from assigning future items. We also formulate a natural conjecture which, if true, would improve the ","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"134 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74847207","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 consider the problem of designing a simple, oblivious scheme to generate (almost) random permutations. We use the concept of switching networks and show that almost every switching network of logarithmic depth can be used to almost randomly permute any set of (1-ε) n elements with any ε > 0 (that is, gives an almost (1-ε) n$-wise independent permutation). Furthermore, we show that the result still holds for every switching network of logarithmic depth that has some special expansion properties, leading to an explicit construction of such networks. Our result can be also extended to an explicit construction of a switching network of depth O(log2n) and with O(n log n) switches that almost randomly permutes any set of n elements. We also discuss basic applications of these results in cryptography. Our results are obtained using a non-trivial coupling approach to study mixing times of Markov chains which allows us to reduce the problem to some random walk-like problem on expanders.
{"title":"Random Permutations using Switching Networks","authors":"A. Czumaj","doi":"10.1145/2746539.2746629","DOIUrl":"https://doi.org/10.1145/2746539.2746629","url":null,"abstract":"We consider the problem of designing a simple, oblivious scheme to generate (almost) random permutations. We use the concept of switching networks and show that almost every switching network of logarithmic depth can be used to almost randomly permute any set of (1-ε) n elements with any ε > 0 (that is, gives an almost (1-ε) n$-wise independent permutation). Furthermore, we show that the result still holds for every switching network of logarithmic depth that has some special expansion properties, leading to an explicit construction of such networks. Our result can be also extended to an explicit construction of a switching network of depth O(log2n) and with O(n log n) switches that almost randomly permutes any set of n elements. We also discuss basic applications of these results in cryptography. Our results are obtained using a non-trivial coupling approach to study mixing times of Markov chains which allows us to reduce the problem to some random walk-like problem on expanders.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88580477","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}
Mika Göös, Shachar Lovett, R. Meka, Thomas Watson, David Zuckerman
We develop a new method to prove communication lower bounds for composed functions of the form f o gn where f is any boolean function on n inputs and g is a sufficiently "hard" two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of f o gn can be simulated by a nonnegative combination of juntas. This is the strongest yet formalization for the intuition that each low-communication randomized protocol can only "query" few inputs of f as encoded by the gadget g. Consequently, we characterize the communication complexity of f o gn in all known one-sided zero-communication models by a corresponding query complexity measure of f. These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy. As applications, we resolve several open problems from prior work: We show that SBPcc (a class characterized by corruption) is not closed under intersection. An immediate corollary is that MAcc ≠ SBPcc. These results answer questions of Klauck (CCC 2003) and Bohler et al. (JCSS 2006). We also show that approximate nonnegative rank of partial boolean matrices does not admit efficient error reduction. This answers a question of Kol et al. (ICALP) for partial matrices.
{"title":"Rectangles Are Nonnegative Juntas","authors":"Mika Göös, Shachar Lovett, R. Meka, Thomas Watson, David Zuckerman","doi":"10.1145/2746539.2746596","DOIUrl":"https://doi.org/10.1145/2746539.2746596","url":null,"abstract":"We develop a new method to prove communication lower bounds for composed functions of the form f o gn where f is any boolean function on n inputs and g is a sufficiently \"hard\" two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of f o gn can be simulated by a nonnegative combination of juntas. This is the strongest yet formalization for the intuition that each low-communication randomized protocol can only \"query\" few inputs of f as encoded by the gadget g. Consequently, we characterize the communication complexity of f o gn in all known one-sided zero-communication models by a corresponding query complexity measure of f. These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy. As applications, we resolve several open problems from prior work: We show that SBPcc (a class characterized by corruption) is not closed under intersection. An immediate corollary is that MAcc ≠ SBPcc. These results answer questions of Klauck (CCC 2003) and Bohler et al. (JCSS 2006). We also show that approximate nonnegative rank of partial boolean matrices does not admit efficient error reduction. This answers a question of Kol et al. (ICALP) for partial matrices.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87300295","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 Random-Facet pivoting rule of Kalai and of Matousek, Sharir and Welzl is an elegant randomized pivoting rule for the simplex algorithm, the classical combinatorial algorithm for solving linear programs (LPs). The expected number of pivoting steps performed by the simplex algorithm when using this rule, on any linear program involving n inequalities in d variables, is 2O(√{(n-d),log({d}/{√{n-d}}},), where log n=max{1,log n}. A dual version of the algorithm performs an expected number of at most 2O(√{d,log({(n-d)}/√d},) dual pivoting steps. This dual version is currently the fastest known combinatorial algorithm for solving general linear programs. Kalai also obtained a primal pivoting rule which performs an expected number of at most 2O(√d,log n) pivoting steps. We present an improved version of Kalai's pivoting rule for which the expected number of primal pivoting steps is at most min{2O(√(n-d),log(d/(n-d),)},2O(√{d,log((n-d)/d}},)}. This seemingly modest improvement is interesting for at least two reasons. First, the improved bound for the number of primal pivoting steps is better than the previous bounds for both the primal and dual pivoting steps. There is no longer any need to consider a dual version of the algorithm. Second, in the important case in which n=O(d), i.e., the number of linear inequalities is linear in the number of variables, the expected running time becomes 2O(√d) rather than 2O(√d log d). Our results, which extend previous results of Gartner, apply not only to LP problems, but also to LP-type problems, supplying in particular slightly improved algorithms for solving 2-player turn-based stochastic games and related problems.
{"title":"An Improved Version of the Random-Facet Pivoting Rule for the Simplex Algorithm","authors":"Thomas Dueholm Hansen, Uri Zwick","doi":"10.1145/2746539.2746557","DOIUrl":"https://doi.org/10.1145/2746539.2746557","url":null,"abstract":"The Random-Facet pivoting rule of Kalai and of Matousek, Sharir and Welzl is an elegant randomized pivoting rule for the simplex algorithm, the classical combinatorial algorithm for solving linear programs (LPs). The expected number of pivoting steps performed by the simplex algorithm when using this rule, on any linear program involving n inequalities in d variables, is 2O(√{(n-d),log({d}/{√{n-d}}},), where log n=max{1,log n}. A dual version of the algorithm performs an expected number of at most 2O(√{d,log({(n-d)}/√d},) dual pivoting steps. This dual version is currently the fastest known combinatorial algorithm for solving general linear programs. Kalai also obtained a primal pivoting rule which performs an expected number of at most 2O(√d,log n) pivoting steps. We present an improved version of Kalai's pivoting rule for which the expected number of primal pivoting steps is at most min{2O(√(n-d),log(d/(n-d),)},2O(√{d,log((n-d)/d}},)}. This seemingly modest improvement is interesting for at least two reasons. First, the improved bound for the number of primal pivoting steps is better than the previous bounds for both the primal and dual pivoting steps. There is no longer any need to consider a dual version of the algorithm. Second, in the important case in which n=O(d), i.e., the number of linear inequalities is linear in the number of variables, the expected running time becomes 2O(√d) rather than 2O(√d log d). Our results, which extend previous results of Gartner, apply not only to LP problems, but also to LP-type problems, supplying in particular slightly improved algorithms for solving 2-player turn-based stochastic games and related problems.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86336083","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}
Alice receives a tuple (a1,...,at) of t elements from the group G = SL(2,q). Bob similarly receives a tuple of t elements (b1,...,bt). They are promised that the interleaved product prodi ≤ t ai bi equals to either g and h, for two fixed elements g,h ∈ G. Their task is to decide which is the case. We show that for every t ≥ 2 communication Ω(t log |G|) is required, even for randomized protocols achieving only an advantage ε = |G|-Ω(t) over random guessing. This bound is tight, improves on the previous lower bound of Ω(t), and answers a question of Miles and Viola (STOC 2013). An extension of our result to 8-party number-on-forehead protocols would suffice for their intended application to leakage-resilient circuits. Our communication bound is equivalent to the assertion that if (a1,...,at) and (b1,...,bt) are sampled uniformly from large subsets A and B of Gt then their interleaved product is nearly uniform over G = SL(2,q). This extends results by Gowers (Combinatorics, Probability & Computing, 2008) and by Babai, Nikolov, and Pyber (SODA 2008) corresponding to the independent case where A and B are product sets. We also obtain an alternative proof of their result that the product of three independent, high-entropy elements of G is nearly uniform. Unlike the previous proofs, ours does not rely on representation theory.
{"title":"The communication complexity of interleaved group products","authors":"T. Gowers, Emanuele Viola","doi":"10.1145/2746539.2746560","DOIUrl":"https://doi.org/10.1145/2746539.2746560","url":null,"abstract":"Alice receives a tuple (a1,...,at) of t elements from the group G = SL(2,q). Bob similarly receives a tuple of t elements (b1,...,bt). They are promised that the interleaved product prodi ≤ t ai bi equals to either g and h, for two fixed elements g,h ∈ G. Their task is to decide which is the case. We show that for every t ≥ 2 communication Ω(t log |G|) is required, even for randomized protocols achieving only an advantage ε = |G|-Ω(t) over random guessing. This bound is tight, improves on the previous lower bound of Ω(t), and answers a question of Miles and Viola (STOC 2013). An extension of our result to 8-party number-on-forehead protocols would suffice for their intended application to leakage-resilient circuits. Our communication bound is equivalent to the assertion that if (a1,...,at) and (b1,...,bt) are sampled uniformly from large subsets A and B of Gt then their interleaved product is nearly uniform over G = SL(2,q). This extends results by Gowers (Combinatorics, Probability & Computing, 2008) and by Babai, Nikolov, and Pyber (SODA 2008) corresponding to the independent case where A and B are product sets. We also obtain an alternative proof of their result that the product of three independent, high-entropy elements of G is nearly uniform. Unlike the previous proofs, ours does not rely on representation theory.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76527505","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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