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114
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 159 (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.
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 93 (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 ...
Nearoptimal lower bounds on the multiparty communication complexity of set disjointness
 In IEEE Conference on Computational Complexity
, 2003
"... We study the communication complexity of the set disjointness problem in the general multiparty model. For t players, each holding a subset of a universe of size n, we establish a nearoptimal lower bound of Ω(n/(t log t)) on the communication complexity of the problem of determining whether their ..."
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Cited by 89 (7 self)
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We study the communication complexity of the set disjointness problem in the general multiparty model. For t players, each holding a subset of a universe of size n, we establish a nearoptimal lower bound of Ω(n/(t log t)) on the communication complexity of the problem of determining whether their sets are disjoint. In the more restrictive oneway communication model, in which the players are required to speak in a predetermined order, we improve our bound to an optimal Ω(n/t). These results improve upon the earlier bounds of Ω(n/t 2) in the general model, and Ω(ε 2 n/t 1+ε) in the oneway model, due to BarYossef, Jayram, Kumar, and Sivakumar [5]. As in the case of earlier results, our bounds apply to the unique intersection promise problem. This communication problem is known to have connections with the space complexity of approximating frequency moments in the data stream model. Our results lead to an improved space complexity lower bound of Ω(n 1−2/k / log n) for approximating the k th frequency moment with a constant number of passes over the input, and a technical improvement to Ω(n 1−2/k) if only one pass over the input is permitted. Our proofs rely on the information theoretic direct sum decomposition paradigm of BarYossef et al [5]. Our improvements stem from novel analytical tech
On Randomized OneRound Communication Complexity
 Computational Complexity
, 1995
"... We present several results regarding randomized oneround communication complexity. Our results include a connection to the VCdimension, a study of the problem of computing the inner product of two real valued vectors, and a relation between \simultaneous" protocols and oneround protocols. Ke ..."
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Cited by 75 (0 self)
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We present several results regarding randomized oneround communication complexity. Our results include a connection to the VCdimension, a study of the problem of computing the inner product of two real valued vectors, and a relation between \simultaneous" protocols and oneround protocols. Key words. Communication Complexity; Oneround and simultaneous protocols; VCdimension; Subject classications. 68Q25. 1.
Distributed streams algorithms for sliding windows
 In Proc. ACM Symp. on Parallel Algorithms and Architectures (SPAA
, 2002
"... Massive data sets often arise as physically distributed, parallel data streams, and it is important to estimate various aggregates and statistics on the union of these streams. This paper presents algorithms for estimating aggregate functions over a “sliding window ” of the N most recent data items ..."
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Cited by 62 (11 self)
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Massive data sets often arise as physically distributed, parallel data streams, and it is important to estimate various aggregates and statistics on the union of these streams. This paper presents algorithms for estimating aggregate functions over a “sliding window ” of the N most recent data items in one or more streams. Our results include: 1. For a single stream, we present the first ɛapproximation scheme for the number of 1’s in a sliding window that is optimal in both worst case time and space. We also present the first ɛapproximation scheme for the sum of integers in [0..R] in a sliding window that is optimal in both worst case time and space (assuming R is at most polynomial in N). Both algorithms are deterministic and use only logarithmic memory words. 2. In contrast, we show that any deterministic algorithm that estimates, to within a small constant relative error, the number of 1’s (or the sum of integers) in a sliding window on the union of distributed streams requires Ω(N) space.
Exponential separations for oneway quantum communication complexity, with applications to cryptography
 IN PROCEEDINGS OF 39TH ACM STOC
, 2007
"... We give an exponential separation between oneway quantum and classical communication protocols for a partial Boolean function (a variant of the Boolean Hidden Matching Problem of BarYossef et al.) Earlier such an exponential separation was known only for a relational problem. The communication pr ..."
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Cited by 58 (16 self)
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We give an exponential separation between oneway quantum and classical communication protocols for a partial Boolean function (a variant of the Boolean Hidden Matching Problem of BarYossef et al.) Earlier such an exponential separation was known only for a relational problem. The communication problem corresponds to a strong extractor that fails against a small amount of quantum information about its random source. Our proof uses the Fourier coefficients inequality of Kahn, Kalai, and Linial. We also give a number of applications of this separation. In particular, we show that there are privacy amplification schemes that are secure against classical adversaries but not against quantum adversaries; and we give the first example of a keyexpansion scheme in the model of boundedstorage cryptography that is secure against classical memorybounded adversaries but not against quantum ones.
The Quantum Communication Complexity of Sampling
 SIAM J. Comput
, 1998
"... Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f : X 1} and a probability distribution over X Y , we define the sampling complexity of (f, as the minimum number of bits Alice and Bob must communica ..."
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Cited by 55 (3 self)
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Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f : X 1} and a probability distribution over X Y , we define the sampling complexity of (f, as the minimum number of bits Alice and Bob must communicate for Alice to pick x X and Bob to pick y Y as well as a value z such that the resulting distribution of (x, y, z) is close to the distribution (D, f(D)).
How to Compress Interactive Communication
, 2009
"... We describe new ways to simulate 2party communication protocols to get protocols with potentially smaller communication. We show that every communication protocol that communicates C bits and reveals I bits of information to the participating parties can be simulated by a new protocol involving at ..."
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Cited by 53 (8 self)
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We describe new ways to simulate 2party communication protocols to get protocols with potentially smaller communication. We show that every communication protocol that communicates C bits and reveals I bits of information to the participating parties can be simulated by a new protocol involving at most Õ(√CI) bits of communication. In the case that the parties have inputs that are independent of each other, we get much better results, showing how to carry out the simulation with Õ(I) bits of communication. These results lead to a direct sum theorem for randomized communication complexity. Ignoring polylogarithmic factors, we show that for worst case computation, computing n copies of a function requires √ n times the communication required for computing on copy of the function. For average case complexity, given any distribution µ on inputs, computing n copies of the function on n independent inputs sampled according to µ requires √ n times the communication for computing one copy. If µ is a product distribution, computing n copies on n independent inputs sampled according to µ requires n times the communication required for computing the function. We also study the complexity of computing the sum (or parity) of n evaluations of f,
Exponential separation of quantum and classical oneway communication complexity
 SIAM J. Comput
"... Abstract. We give the first exponential separation between quantum and boundederror randomized oneway communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ {0, 1} n and Bob gets a perfect matching M on the n coordinates. Bob’s goal is t ..."
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Cited by 48 (4 self)
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Abstract. We give the first exponential separation between quantum and boundederror randomized oneway communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ {0, 1} n and Bob gets a perfect matching M on the n coordinates. Bob’s goal is to output a tuple 〈i, j, b 〉 such that the edge (i, j) belongs to the matching M and b = xi ⊕ xj. We prove that the quantum oneway communication complexity of HMn is O(log n), yet any randomized oneway protocol with bounded error must use Ω ( √ n) bits of communication. No asymptotic gap for oneway communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM) and hence we provide the first exponential separation between quantum SM and randomized SM with public coins. For a Boolean decision version of HMn, we show that the quantum oneway communication complexity remains O(log n) and that the 0error randomized oneway communication complexity is Ω(n). We prove that any randomized linear oneway protocol with bounded error for this problem requires Ω ( 3 √ n log n) bits of communication. Key words. Communication complexity, quantum computation, separation, hidden matching AMS subject classifications. 68P30,68Q15,68Q17,81P68 1. Introduction. The
Two Applications of Information Complexity
, 2003
"... We show the following new lower bounds in two concrete complexity models: (1) In the twoparty communication complexity model, we show that the tribes function on n inputs [6] has twosided error randomized complexity # n), while its nondeterminstic complexity and conondeterministic complexity are ..."
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Cited by 40 (1 self)
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We show the following new lower bounds in two concrete complexity models: (1) In the twoparty communication complexity model, we show that the tribes function on n inputs [6] has twosided error randomized complexity # n), while its nondeterminstic complexity and conondeterministic complexity are both #( # n). This separation between randomized and nondeterministic complexity is the best possible and it settles an open problem in Kushilevitz and Nisan [17], which was also posed by Beame and Lawry [5].