On the communication complexity of greater-than

Ramamoorthy, Sivaramakrishnan Natarajan; Sinha, Makrand

doi:10.1109/allerton.2015.7447037

Cited by 3 publications

(1 citation statement)

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“…In another closely related work, Natarajan Ramamoorthy and the second author [22] used one of the technical lemmas (Lemma 4.1) clarified in this paper to give a simpler proof showing that the randomized communication complexity of the Greater-Than function on n bits is Ω(log n). The same lower bound was previously proved by Viola [28] and also by Braverman and Weinstein [7] using different techniques.…”

Section: Closely Related Workmentioning

confidence: 99%

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Rao

Sinha

2018

Theory of Comput.

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We give an example of a Boolean function whose information complexity is exponentially smaller than its communication complexity. Such a separation was first demonstrated by Ganor, Kol and Raz (J. ACM 2016). We give a simpler proof of the same result. In the course of this simplification, we make several new contributions: we introduce a new communication lower-bound technique, the notion of a fooling distribution, which allows us to separate information and communication complexity, and we also give a more direct proof of the information complexity upper bound. We also prove a generalization of Shearer's Lemma that may be of independent interest. A version of Shearer's original lemma bounds the expected mutual information of a subset of random variables with another random variable, when the subset is chosen independently of all the random variables that are involved. Our generalization allows some dependence between the random subset and the random variables involved, and still gives us similar bounds with an appropriate error term.

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Section: Closely Related Workmentioning

confidence: 99%

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Rao

Sinha

2018

Theory of Comput.

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Dimension-free bounds and structural results in communication complexity

Hambardzumyan

Hatami

2022

Isr. J. Math.

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The goal of this thesis is to continue to build the bridge between communication complexity and analysis. More specifically, the purpose is to initiate a systematic study of dimension-free relations between basic communication complexity and query complexity measures and various matrix norms. In other words, our goal is to establish qualitative equivalences between complexity measures, namely to bound a measure solely as a function of another measure. This is in contrast to the more common framework in communication complexity where quantitative equivalences are the main focus of study and poly-logarithmic dependencies on the number of input bits are tolerated.Dimension-free bounds are closely related to structural results, where one seeks to describe the structure of Boolean matrices and functions that have low complexity. We restate and propose several conjectures in this nature such as: Does every matrix with small randomized communication complexity contain a large all-zero or all-one submatrix [CLV19]? Does every Boolean function with small approximate Fourier algebra norm have large affine subspace on which the function is constant?We consider such questions for several communication and query complexity measures as well as various matrix and operator norms. In several cases, we achieve satisfying answers, while for some cases we show that such bounds do not exist.We establish that, in addition to applications in complexity theory, these problems arise naturally in operator theory and Harmonic analysis. We show that these problems are central to characterization of the idempotents of the algebra of Schur multipliers, and could lead to new extensions of Cohen's celebrated idempotent theorem regarding the Fourier algebra. i ContributionWork included The novel parts of this thesis are Chapters 4, 5, 6 and 7. These are based on the following joint work. Section 4.6 is not included in any manuscript.

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