Direct Products in Communication Complexity

Braverman, Mark; Rao, Anup; Weinstein, Omri; Yehudayoff, Amir

doi:10.1109/focs.2013.85

Cited by 65 publications

(52 citation statements)

References 28 publications

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“…Nevertheless, the simple result above is not comparable with the known direct product theorems in [9], [11] and can be stronger in some regimes 11 .…”

Section: Amortized Regime: Second-order Asymptoticsmentioning

confidence: 55%

“…In the theoretical computer science literature, such converse results have been termed direct product theorems and have been considered in the context of the (distributional) communication complexity problem (for computing a given function) [9], [11], [26]. Our lower bound in Theorem 1, too, yields a direct product theorem for the communication complexity problem.…”

Section: Amortized Regime: Second-order Asymptoticsmentioning

confidence: 93%

“…and by B the set B = {(τ X , τ Y , τ, x, y) :X(x, y, τ, τ ) = x,Ŷ (x, y, τ, τ ) = y}, which is the same as E c error for E error in (9). Then, by (7) and (9)…”

Section: B From Partial Knowledge To Omnisciencementioning

confidence: 99%

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Information Complexity Density and Simulation of Protocols

Tyagi

Venkatakrishnan

Viswanath

et al. 2017

IEEE Trans. Inform. Theory

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Two parties observing correlated random variables seek to run an interactive communication protocol.How many bits must they exchange to simulate the protocol, namely to produce a view with a joint distribution within a fixed statistical distance of the joint distribution of the input and the transcript of the original protocol? We present an information spectrum approach for this problem whereby the information complexity of the protocol is replaced by its information complexity density. Our singleshot bounds relate the communication complexity of simulating a protocol to tail bounds for information complexity density. As a consequence, we obtain a strong converse and characterize the second-order asymptotic term in communication complexity for independent and identically distributed observation sequences. Furthermore, we obtain a general formula for the rate of communication complexity which applies to any sequence of observations and protocols. Connections with results from theoretical computer science and implications for the function computation problem are discussed.

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“…Nevertheless, the simple result above is not comparable with the known direct product theorems in [9], [11] and can be stronger in some regimes 11 .…”

Section: Amortized Regime: Second-order Asymptoticsmentioning

confidence: 55%

Section: Amortized Regime: Second-order Asymptoticsmentioning

confidence: 93%

See 1 more Smart Citation

Information Complexity Density and Simulation of Protocols

Tyagi

Venkatakrishnan

Viswanath

et al. 2017

IEEE Trans. Inform. Theory

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“…Direct product theorems are generally harder to prove than direct sum theorems. In the context of communication complexity, they have a long history, including recent developments [66,54,68,44,37,15,58,14]. In particular, in [15] a generic direct product result that roughly matches the direct sum results of [4] was given.…”

Section: Direct Product Theoremsmentioning

confidence: 99%

“…In the context of communication complexity, they have a long history, including recent developments [66,54,68,44,37,15,58,14]. In particular, in [15] a generic direct product result that roughly matches the direct sum results of [4] was given. Building on that result, most recently, in [18] a direct product theorem was finally given showing that o(n · IC (f, 1/3)) communication will lead to an exponentially small success probability in computing f n .…”

Section: Direct Product Theoremsmentioning

confidence: 99%

Interactive Information Complexity

Braverman¹

2017

SIAM Rev.

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Abstract. The primary goal of this paper is to define and study the interactive information complexity of functions. Let f (x, y) be a function, and suppose Alice is given x and Bob is given y. Informally, the interactive information complexity IC(f ) of f is the least amount of information Alice and Bob need to reveal to each other to compute f . Previously, information complexity has been defined with respect to a prior distribution on the input pairs (x, y). Our first goal is to give a definition that is independent of the prior distribution. We show that several possible definitions are essentially equivalent. We establish some basic properties of the interactive information complexity IC(f ). In particular, we show that IC(f ) is equal to the amortized (randomized) communication complexity of f . We also show a direct sum theorem for IC(f ) and give the first general connection between information complexity and (nonamortized) communication complexity. This connection implies that a nontrivial exchange of information is required when solving problems that have nontrivial communication complexity. We explore the information complexity of two specific problems: equality and disjointness. We show that only a constant amount of information needs to be exchanged when solving equality with no errors, while solving disjointness with a constant error probability requires the parties to reveal a linear amount of information to each other. This paper originally appeared in [SIAM J. Comput., 44 (2015), pp. 1698-1739]. The present version has been slightly updated with some recent developments.

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