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39
A brief introduction to Fourier analysis on the Boolean cube
 Theory of Computing Library– Graduate Surveys
, 2008
"... Abstract: We give a brief introduction to the basic notions of Fourier analysis on the ..."
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Abstract: We give a brief introduction to the basic notions of Fourier analysis on the
A strong direct product theorem for disjointness
 In 42nd ACM Symposium on Theory of Computing (STOC
, 2010
"... A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then the overall success probability will be exponentially small in k. We establish such a theorem for the randomized communication co ..."
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Cited by 28 (1 self)
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A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then the overall success probability will be exponentially small in k. We establish such a theorem for the randomized communication complexity of the Disjointness problem, i.e., with communication const · kn the success probability of solving k instances can only be exponentially small in k. We show that this bound even holds in an AM communication protocol with limited ambiguity. The main result implies a new lower bound for Disjointness in a restricted 3player NOF protocol, and optimal communicationspace tradeoffs for Boolean matrix product. Our main result follows from a solution to the dual of a linear programming problem, whose feasibility comes from a socalled Intersection Sampling Lemma that generalizes a result by Razborov [Raz92]. We also discuss a new lower bound technique for randomized communication complexity called the generalized rectangle bound that we use in our proof. 1
Some applications of hypercontractive inequalities in quantum information
, 2012
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Oneway multiparty communication lower bound for pointer jumping with applications
, 2007
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STRONG DIRECT PRODUCT THEOREMS FOR QUANTUM COMMUNICATION AND QUERY COMPLEXITY
"... A strong direct product theorem (SDPT) states that solving n instances of a problem requires ˝.n / times the resources for a single instance, even to achieve success probability 2 ˝.n / : We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound for a singl ..."
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A strong direct product theorem (SDPT) states that solving n instances of a problem requires ˝.n / times the resources for a single instance, even to achieve success probability 2 ˝.n / : We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound for a single instance is proved by the generalized discrepancy method, the strongest technique in that model. We prove that quantum query complexity obeys an SDPT whenever the query lower bound for a single instance is proved by the polynomial method, one of the two main techniques in that model. In both models, we prove the corresponding XOR lemmas and threshold direct product theorems.
Quantum Proofs for Classical Theorems
, 2009
"... Alongside the development of quantum algorithms and quantum complexity theory in recent years, quantum techniques have also proved instrumental in obtaining results in classical (nonquantum) areas. In this paper we survey these results and the quantum toolbox they use. ..."
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Cited by 15 (4 self)
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Alongside the development of quantum algorithms and quantum complexity theory in recent years, quantum techniques have also proved instrumental in obtaining results in classical (nonquantum) areas. In this paper we survey these results and the quantum toolbox they use.
Constructive proofs of concentration bounds
 In Proceedings of the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and 14th International Workshop on Randomization and Computation (APPROXRANDOM ’10
, 2010
"... We give a simple combinatorial proof of the ChernoffHoeffding concentration bound [Che52, Hoe63], which says that the sum of independent {0, 1}valued random variables is highly concentrated around the expected value. Unlike the standard proofs, our proof does not use the method of higher moments, ..."
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We give a simple combinatorial proof of the ChernoffHoeffding concentration bound [Che52, Hoe63], which says that the sum of independent {0, 1}valued random variables is highly concentrated around the expected value. Unlike the standard proofs, our proof does not use the method of higher moments, but rather uses a simple and intuitive counting argument. In addition, our proof is constructive in the following sense: if the sum of the given random variables is not concentrated around the expectation, then we can efficiently find (with high probability) a subset of the random variables that are statistically dependent. As simple corollaries, we also get the concentration bounds for [0, 1]valued random variables and Azuma’s inequality for martingales [Azu67]. We interpret the ChernoffHoeffding bound as a statement about Direct Product Theorems. Informally, a Direct Product Theorem says that the complexity of solving all k instances of a hard problem increases exponentially with k; a Threshold Direct Product Theorem says that it is exponentially hard in k to solve even a significant fraction of the given k instances of a hard problem. We show the equivalence between optimal Direct Product Theorems and optimal Threshold Direct Product Theorems. As an application of this connection, we get the Chernoff bound for expander walks [Gil98] from the (simpler to prove) hitting property [AKS87], as well as an optimal (in a certain range of parameters) Threshold Direct Product Theorem for weakly verifiable puzzles from the optimal Direct Product Theorem [CHS05]. We also get a simple constructive proof of Unger’s result [Ung09] saying that XOR Lemmas imply Threshold Direct
Multiparty Communication Complexity and Threshold Circuit Size of AC⁰
, 2008
"... We prove an nΩ(1) /2O(k) lower bound on the randomized kparty communication complexity of readonce depth 4 AC0 functions in the numberonforehead (NOF) model for up to Θ(log n) players. These are the first nontrivial lower bounds for general NOF multiparty communication complexity for any AC0 fu ..."
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We prove an nΩ(1) /2O(k) lower bound on the randomized kparty communication complexity of readonce depth 4 AC0 functions in the numberonforehead (NOF) model for up to Θ(log n) players. These are the first nontrivial lower bounds for general NOF multiparty communication complexity for any AC0 function for ω(log log n) players. For nonconstant k the bounds are larger than all previous lower bounds for any AC0 function even for simultaneous communication complexity. Our lower bounds imply the first superpolynomial lower bounds for the simulation of AC0 by general MAJ ◦ SYMM ◦ AND circuits, showing that the wellknown quasipolynomial simulations of AC0 by such circuits are qualitatively optimal, even for readonce formulas of small constant depth. We also exhibit a readonce depth 5 formula in NP cc k − BPPcc k
Direct product theorems for classical communication complexity . . .
, 2007
"... A basic question in complexity theory is whether the computational resources required for solving k independent instances of the same problem scale as k times the resources required for one instance. We investigate this question in various models of classical communication complexity. We introduce a ..."
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Cited by 10 (3 self)
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A basic question in complexity theory is whether the computational resources required for solving k independent instances of the same problem scale as k times the resources required for one instance. We investigate this question in various models of classical communication complexity. We introduce a new measure, the subdistribution bound, which is a relaxation of the wellstudied rectangle or corruption bound in communication complexity. We nonetheless show that for the communication complexity of Boolean functions with constant error, the subdistribution bound is the same as the latter measure, up to a constant factor. We prove that the oneway version of this bound tightly captures the oneway publiccoin randomized communication complexity of any relation, and the twoway version bounds the twoway publiccoin randomized communication complexity from below. More importantly, we show that the bound satisfies the strong direct product property under product distributions for both one and twoway protocols, and the weak direct product property under arbitrary distributions for twoway protocols. These results subsume and strengthen, in a unified manner, several recent results on the direct product question. The