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123
The space complexity of approximating the frequency moments
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1996
"... The frequency moments of a sequence containing mi elements of type i, for 1 ≤ i ≤ n, are the numbers Fk = �n i=1 mki. We consider the space complexity of randomized algorithms that approximate the numbers Fk, when the elements of the sequence are given one by one and cannot be stored. Surprisingly, ..."
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Cited by 704 (12 self)
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The frequency moments of a sequence containing mi elements of type i, for 1 ≤ i ≤ n, are the numbers Fk = �n i=1 mki. We consider the space complexity of randomized algorithms that approximate the numbers Fk, when the elements of the sequence are given one by one and cannot be stored. Surprisingly, it turns out that the numbers F0, F1 and F2 can be approximated in logarithmic space, whereas the approximation of Fk for k ≥ 6 requires nΩ(1) space. Applications to data bases are mentioned as well.
A Parallel Repetition Theorem
 SIAM Journal on Computing
, 1998
"... We show that a parallel repetition of any twoprover oneround proof system (MIP(2, 1)) decreases the probability of error at an exponential rate. No constructive bound was previously known. The constant in the exponent (in our analysis) depends only on the original probability of error and on the t ..."
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Cited by 324 (11 self)
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We show that a parallel repetition of any twoprover oneround proof system (MIP(2, 1)) decreases the probability of error at an exponential rate. No constructive bound was previously known. The constant in the exponent (in our analysis) depends only on the original probability of error and on the total number of possible answers of the two provers. The dependency on the total number of possible answers is logarithmic, which was recently proved to be almost the best possible [U. Feige and O. Verbitsky, Proc. 11th Annual IEEE Conference on Computational Complexity, IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 7076].
An Information Statistics Approach to Data Stream and Communication Complexity
, 2003
"... We present a new method for proving strong lower bounds in communication complexity. ..."
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Cited by 153 (8 self)
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We present a new method for proving strong lower bounds in communication complexity.
Quantum lower bounds by quantum arguments
 In Proceedings of the ACM Symposium on Theory of Computing
, 2000
"... We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with one input and then modifies the input, we use a quantum adversary that runs the algorithm with a superposition of inputs. If the algorithm works correctly, its ..."
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Cited by 146 (15 self)
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We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with one input and then modifies the input, we use a quantum adversary that runs the algorithm with a superposition of inputs. If the algorithm works correctly, its state becomes entangled with the superposition over inputs. We bound the number of queries needed to achieve a sufficient entanglement and this implies a lower bound on the number of queries for the computation. Using this method, we prove two new Ω ( √ N) lower bounds on computing AND of ORs and inverting a permutation and also provide more uniform proofs for several known lower bounds which have been previously proven via variety of different techniques. 1
Quantum vs. classical communication and computation
 Proc. 30th Ann. ACM Symp. on Theory of Computing (STOC ’98
, 1998
"... We present a simple and general simulation technique that transforms any blackbox quantum algorithm (à la Grover’s database search algorithm) to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism. This allows us to obtain new positive and ne ..."
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Cited by 126 (15 self)
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We present a simple and general simulation technique that transforms any blackbox quantum algorithm (à la Grover’s database search algorithm) to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism. This allows us to obtain new positive and negative results. The positive results are novel quantum communication protocols that are built from nontrivial quantum algorithms via this simulation. These protocols, combined with (old and new) classical lower bounds, are shown to provide the first asymptotic separation results between the quantum and classical (probabilistic) twoparty communication complexity models. In particular, we obtain a quadratic separation for the boundederror model, and an exponential separation for the zeroerror model. The negative results transform known quantum communication lower bounds to computational lower bounds in the blackbox model. In particular, we show that the quadratic speedup achieved by Grover for the OR function is impossible for the PARITY function or the MAJORITY function in the boundederror model, nor is it possible for the OR function itself in the exact case. This dichotomy naturally suggests a study of boundeddepth predicates (i.e. those in the polynomial hierarchy) between OR and MAJORITY. We present blackbox algorithms that achieve near quadratic speed up for all such predicates.
Synopsis Data Structures for Massive Data Sets
"... Abstract. Massive data sets with terabytes of data are becoming commonplace. There is an increasing demand for algorithms and data structures that provide fast response times to queries on such data sets. In this paper, we describe a context for algorithmic work relevant to massive data sets and a f ..."
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Cited by 108 (13 self)
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Abstract. Massive data sets with terabytes of data are becoming commonplace. There is an increasing demand for algorithms and data structures that provide fast response times to queries on such data sets. In this paper, we describe a context for algorithmic work relevant to massive data sets and a framework for evaluating such work. We consider the use of "synopsis" data structures, which use very little space and provide fast (typically approximated) answers to queries. The design and analysis of effective synopsis data structures o er many algorithmic challenges. We discuss a number of concrete examples of synopsis data structures, and describe fast algorithms for keeping them uptodate in the presence of online updates to the data sets.
Quantum Communication Complexity of Symmetric Predicates
 Izvestiya of the Russian Academy of Science, Mathematics
, 2002
"... We completely (that is, up to a logarithmic factor) characterize the boundederror quantum communication complexity of every predicate f(x; y) (x; y [n]) depending only on jx\yj. Namely, for a predicate D on f0; 1; : : : ; ng let ` 0 (D) = max f` j 1 ` n=2 ^ D(`) 6 D(` 1)g and ` 1 (D) = ..."
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Cited by 87 (1 self)
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We completely (that is, up to a logarithmic factor) characterize the boundederror quantum communication complexity of every predicate f(x; y) (x; y [n]) depending only on jx\yj. Namely, for a predicate D on f0; 1; : : : ; ng let ` 0 (D) = max f` j 1 ` n=2 ^ D(`) 6 D(` 1)g and ` 1 (D) = max fn ` j n=2 ` < n ^ D(`) 6 D(` + 1)g. Then the boundederror quantum communication complexity of f D (x; y) = D(jx \ yj) is equal (again, up to a logarithmic factor) to ` 1 (D). In particular, the complexity of the set disjointness predicate is n). This result holds both in the model with prior entanglement and without it.
Monotone Circuits for Matching Require Linear Depth
"... We prove that monotone circuits computing the perfect matching function on nvertex graphs require\Omega\Gamma n) depth. This implies an exponential gap between the depth of monotone and nonmonotone circuits. ..."
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Cited by 77 (8 self)
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We prove that monotone circuits computing the perfect matching function on nvertex graphs require\Omega\Gamma n) depth. This implies an exponential gap between the depth of monotone and nonmonotone circuits.
Exponential Separation of Quantum and Classical Communication Complexity
, 1999
"... Communication complexity has become a central complexity model. In that model, we count the amount of communication bits needed between two parties in order to solve certain computational problems. We show that for certain communication complexity problems quantum communication protocols are expo ..."
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Cited by 77 (2 self)
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Communication complexity has become a central complexity model. In that model, we count the amount of communication bits needed between two parties in order to solve certain computational problems. We show that for certain communication complexity problems quantum communication protocols are exponentially faster than classical ones. More explicitly, we give an example for a communication complexity relation (or promise problem) P such that: 1. The quantum communication complexity of P is O(log m). 2. The classical probabilistic communication complexity of P is \Omega\Gamma m 1=4 = log m). (where m is the length of the inputs). This gives an exponential gap between quantum communication complexity and classical probabilistic communication complexity. Only a quadratic gap was previously known. Our problem P is of geometrical nature, and is a finite precision variation of the following problem: Player I gets as input a unit vector x 2 R n and two orthogonal subspaces M 0 ...