The NOF Multiparty Communication Complexity of Composed Functions

Ada, Anil; Chattopadhyay, Arkadev; Fawzi, Omar; Nguyen, Phuong Minh

doi:10.1007/978-3-642-31594-7_2

Cited by 5 publications

(4 citation statements)

References 22 publications

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“…Since then, the breadth and profoundness of this relation is better understood, see e.g. [20,5,7,1,16]. This note offers another strong bridge, showing that the Hales-Jewett theorem, a pillar of Ramsey theory, and related questions, are naturally formulated in this model of communication complexity.…”

Section: Discussion and Open Problemsmentioning

confidence: 99%

“…Thus, one way to relax D(f ) is to view it as a coloring problem. Denote by α(f ) the largest size of a 1-monochromatic cylinder intersection with respect to f , and by χ(f ) the least number of monochromatic cylinder intersections that form a partition of f −1 (1). Obviously, D(f ) ≥ log χ(f ), and as we shall see, for special families of functions this bound is nearly tight, including the function P art n,k that we are interested in.…”

Section: Introductionmentioning

confidence: 95%

“…1} be a weak graph function and C ⊆ f −1 (1). The set C is a (1-monochromatic) cylinder intersection with respect to f if and only if it does not contain a star.…”

Section: Lemma 7 ([15])mentioning

confidence: 99%

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A note on multiparty communication complexity and the Hales–Jewett theorem

Shraibman

2018

Information Processing Letters

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For integers n and k, the density Hales-Jewett number c n,k is defined as the maximal size of a subset of [k] n that contains no combinatorial line. We show that for k ≥ 3 the density Hales-Jewett number c n,k is equal to the maximal size of a cylinder intersection in the problem P art n,k of testing whether k subsets of [n] form a partition. It follows that the communication complexity, in the Number On the Forehead (NOF) model, of P art n,k , is equal to the minimal size of a partition of [k] n into subsets that do not contain a combinatorial line. Thus, the bound in [7] on P art n,k using the Hales-Jewett theorem is in fact tight, and the density Hales-Jewett number can be thought of as a quantity in communication complexity. This gives a new angle to this well studied quantity.As a simple application we prove a lower bound on c n,k , similar to the lower bound in [19] which is roughly c n,k /k n ≥ exp(−O(log n) 1/⌈log 2 k⌉ ). This lower bound follows from a protocol for P art n,k . It is interesting to better understand the communication complexity of P art n,k as this will also lead to the better understanding of the Hales-Jewett number. The main purpose of this note is to motivate this study.

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Section: Discussion and Open Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

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A note on multiparty communication complexity and the Hales–Jewett theorem

Shraibman

2018

Information Processing Letters

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“…Denote by χ : Z × p → {0, 1} the indicator function for quadratic residuosity modulo p, i.e., χ(x) = 1 iff x is a square in Z p . The pseudo-random qualities of χ have often made it an object of study in communication complexity [BNS92,BGKL03,ACFN12]. Moreover, the self-reducibility properties of χ are famously useful in cryptography, starting with [GM84].…”

Section: Multi-party Examplesmentioning

confidence: 99%

Communication lower bounds via critical block sensitivity

Göös

Pitassi

2014

Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing

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Abstract. We use critical block sensitivity, a new complexity measure introduced by Huynh and Nordström (STOC 2012), to study the communication complexity of search problems. To begin, we give a simple new proof of the following central result of Huynh and Nordström: if S is a search problem with critical block sensitivity b, then every randomised two-party protocol solving a certain two-party lift of S requires Ω(b) bits of communication. Besides simplicity, our proof has the advantage of generalising to the multi-party setting. We combine these results with new critical block sensitivity lower bounds for Tseitin and Pebbling search problems to obtain the following applications.• Monotone circuit depth: We exhibit a monotone n-variable function in NP whose monotone circuits require depth Ω(n/ log n); previously, a bound of Ω( √ n) was known (Raz and Wigderson, JACM 1992). Moreover, we prove a Θ( √ n) monotone depth bound for a function in monotone P.• Proof complexity: We prove new rank lower bounds as well as obtain the first length-space lower bounds for semi-algebraic proof systems, including Lovász-Schrijver and Lasserre (SOS) systems. In particular, these results extend and simplify the works of Beame et al. (SICOMP 2007) and Huynh and Nordström.

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