Exponential Separation of Information and Communication for Boolean Functions

Ganor, Anat; Kol, Gillat; Raz, Ran

doi:10.1145/2746539.2746572

Cited by 27 publications

(30 citation statements)

References 35 publications

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“…While the reduction incurs an exponential loss, this loss turns out to be tight [28,29]. We note that the resulting communication protocol always runs in 2 rounds.…”

Section: Information Complexity Vs Communication Complexity Our Nexmentioning

confidence: 94%

“…First and foremost, we would like to understand to what extent interactive computation can be compressed. The recent results of [28,29] show that it is not the case that for all f , IC (f, ε) = Ω (R(f, ε)). Moreover, more recently, it was also shown [30] that it is not the case that IC ext (f, ε) = Ω(R(f, ε))-at least for relations-and there is every reason to believe that it is not the case for functions either.…”

Section: Properties Of the Interactive Information Complexitymentioning

confidence: 99%

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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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“…While the reduction incurs an exponential loss, this loss turns out to be tight [28,29]. We note that the resulting communication protocol always runs in 2 rounds.…”

Section: Information Complexity Vs Communication Complexity Our Nexmentioning

confidence: 94%

Section: Properties Of the Interactive Information Complexitymentioning

confidence: 99%

Interactive Information Complexity

Braverman¹

2017

SIAM Rev.

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“…Jain, Lee and Vishnoi defined the public coin partition bound, and showed that its logarithm is quadratically related to the communication complexity [JLV14]. In addition, Ganor et al introduced the adaptive relative discrepancy [GKR14b]. We study the relation between public coin partition and adaptive relative discrepancy and show the following:…”

Section: Introductionmentioning

confidence: 94%

“…In a recent breakthrough, Ganor et al [GKR14a,GKR14b] gave an example of a function f and a distribution µ, for which there is an exponential separation between the distributional information and communication complexity. Does this settle the question of communication versus information?…”

Section: Introductionmentioning

confidence: 99%

Relative Discrepancy Does Not Separate Information and Communication Complexity

Fontes

Jain

Kerenidis

et al. 2016

ACM Trans. Comput. Theory

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Does the information complexity of a function equal its communication complexity? We examine whether any currently known techniques might be used to show a separation between the two notions. Recently, Ganor et al. provided such a separation in the distributional setting for a specific input distribution µ. We show that in the nondistributional setting, the relative discrepancy bound they defined is, in fact, smaller than the information complexity, and hence it cannot be used to separate information and communication complexity. In addition, in the distributional case, we provide an equivalent linear program formulation for relative discrepancy and relate it to variants of the partition bound, resolving also an open question regarding the relation of the partition bound and information complexity. Last, we prove the equivalence between the adaptive relative discrepancy and the public-coin partition bound, which implies that the logarithm of the adaptive relative discrepancy bound is quadratically tight with respect to communication.

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“…But the quantum version of the embedding argument requires new methods. In the classical setting, using classical information cost IC, as soon as we have Alice and Bob privately sample the remaining inputs, the Shearer-type embedding follows almost directly from a Shearer like inequality for information [GKR15]. In the quantum setting, we would similarly like to use a Shearer-type inequality for quantum information [ATYY17].…”

Section: Proof Overviewmentioning

confidence: 99%

Quantum Log-Approximate-Rank Conjecture is Also False

Anshu¹,

Boddu²,

Touchette³

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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In a recent breakthrough result, Chattopadhyay, Mande and Sherif showed an exponential separation between the log approximate rank and randomized communication complexity of a total function f , hence refuting the log approximate rank conjecture of Lee and Shraibman [2009]. We provide an alternate proof of their randomized communication complexity lower bound using the information complexity approach. Using the intuition developed there, we derive a polynomially-related quantum communication complexity lower bound using the quantum information complexity approach, thus providing an exponential separation between the log approximate rank and quantum communication complexity of f . Previously, the best known separation between these two measures was (almost) quadratic, due to Anshu, Ben-David, Garg, Jain, Kothari and Lee [CCC, 2017]. This settles one of the main question left open by Chattopadhyay, Mande and Sherif, and refutes the quantum log approximate rank conjecture of Lee and Shraibman [2009]. Along the way, we develop a Shearer-type protocol embedding for product input distributions that might be of independent interest.

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Exponential Separation of Information and Communication for Boolean Functions

Cited by 27 publications

References 35 publications

Interactive Information Complexity

Interactive Information Complexity

Relative Discrepancy Does Not Separate Information and Communication Complexity

Quantum Log-Approximate-Rank Conjecture is Also False

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