Informational complexity and the direct sum problem for simultaneous message complexity

Chakrabarti, Amit; Shi, Yuanming; Wirth, Anthony; Yao, Andrew Chi-Chih

doi:10.1109/sfcs.2001.959901

Cited by 214 publications

(183 citation statements)

References 7 publications

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“…We proved that R 1/6 (DISJ N ·n ) < (α/2) · (N · n) for all sufficiently large N . This contradicts our assumption (22) and completes the proof of the theorem.…”

Section: The Protocolmentioning

confidence: 64%

“…Chakrabarti et al [22] were the first to define the external information cost explicitly for the purpose of studying communication complexity. In [22], which defined what is known today as external information complexity, information theory is used to prove a direct sum theorem for problems with one simultaneous round of communication.…”

mentioning

confidence: 99%

“…In [22], which defined what is known today as external information complexity, information theory is used to prove a direct sum theorem for problems with one simultaneous round of communication. Direct sum theorems (and the related direct product theorems) are theorems giving lower bounds for the complexity of n copies of a certain problem in terms of the complexity of one copy.…”

mentioning

confidence: 99%

“…Direct sum theorems (and the related direct product theorems) are theorems giving lower bounds for the complexity of n copies of a certain problem in terms of the complexity of one copy. In the context of communication complexity, direct sum theorems in various contexts have been the focus of much work [26,22,66,39,31,4,50,33]. More recently, information-theoretic quantities have been studied in the context of differential privacy as well [57].…”

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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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“…We proved that R 1/6 (DISJ N ·n ) < (α/2) · (N · n) for all sufficiently large N . This contradicts our assumption (22) and completes the proof of the theorem.…”

Section: The Protocolmentioning

confidence: 64%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Interactive Information Complexity

Braverman¹

2017

SIAM Rev.

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“…Communication complexity lower bounds via information theory The informational complexity paradigm, introduced by [CSWY01], and used in [SS02, BYJKS02, CKS03, BYJKS04, JKS03], provides a way to prove lower bounds on communication complexity via information theory. We are given a k-party function f and we want to show that any δ-error randomized NOF protocol Π for f requires high communication.…”

Section: H(x Y ) ≤ H(x)+h(y ) With Equality If and Only If X And Y Amentioning

confidence: 99%

Hellinger volume and number-on-the-forehead communication complexity

Lee

Leonardos

Saks

et al. 2016

Journal of Computer and System Sciences

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Information-theoretic methods have proven to be a very powerful tool in communication complexity, in particular giving an elegant proof of the linear lower bound for the two-party disjointness function, and tight lower bounds on disjointness in the multi-party number-in-the-hand (NIH) model. In this paper, we study the applicability of information theoretic methods to the multiparty number-on-the-forehead model (NOF), where determining the complexity of disjointness remains an important open problem.There are two basic parts to the NIH disjointness lower bound: a direct sum theorem and a lower bound on the one-bit AND function using a beautiful connection between Hellinger distance and protocols revealed by Bar-Yossef, Jayram, Kumar & Sivakumar [BYJKS04]. Inspired by this connection, we introduce the notion of Hellinger volume. We show that it lower bounds the information cost of multi-party NOF protocols and provide a small toolbox that allows one to manipulate several Hellinger volume terms and lower bound a Hellinger volume when the distributions involved satisfy certain conditions. In doing so, we prove a new upper bound on the difference between the arithmetic mean and the geometric mean in terms of relative entropy. We then apply these new tools to obtain a lower bound on the informational complexity of the AND k function in the NOF setting. Finally, we discuss the difficulties of proving a direct sum theorem for information cost in the NOF model.

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NOF-Multiparty Information Complexity Bounds for Pointer Jumping

Gronemeier

2006

Lecture Notes in Computer Science

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Informational complexity and the direct sum problem for simultaneous message complexity

Cited by 214 publications

References 7 publications

Interactive Information Complexity

Interactive Information Complexity

Hellinger volume and number-on-the-forehead communication complexity

NOF-Multiparty Information Complexity Bounds for Pointer Jumping

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