The NOF Multiparty Communication Complexity of Composed Functions

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

doi:10.1007/s00037-013-0078-4

Cited by 16 publications

(47 citation statements)

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“…See also the survey [45]. Both methods have since been studied in the two-party setting [35,43,48] and also the multiparty setting [33,3,11,50,51,38].…”

Section: Main Structural Result: Junta Theoremmentioning

confidence: 98%

Rectangles Are Nonnegative Juntas

Göös

Lovett

Meka

et al. 2015

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

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We develop a new method to prove communication lower bounds for composed functions of the form f • g n where f is any boolean function on n inputs and g is a sufficiently "hard" two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of f • g n can be simulated by a nonnegative combination of juntas. This is the strongest yet formalization for the intuition that each low-communication randomized protocol can only "query" few inputs of f as encoded by the gadget g. Consequently, we characterize the communication complexity of f • g n in all known one-sided zero-communication models by a corresponding query complexity measure of f . These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy.As applications, we resolve several open problems from prior work: We show that SBP cc (a class characterized by corruption) is not closed under intersection. An immediate corollary is that MA cc = SBP cc . These results answer questions of Klauck (CCC 2003) and Böhler et al. (JCSS 2006). We also show that approximate nonnegative rank of partial boolean matrices does not admit efficient error reduction. This answers a question of Kol et al. (ICALP 2014) for partial matrices.

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“…See also the survey [45]. Both methods have since been studied in the two-party setting [35,43,48] and also the multiparty setting [33,3,11,50,51,38].…”

Section: Main Structural Result: Junta Theoremmentioning

confidence: 98%

Rectangles Are Nonnegative Juntas

Göös

Lovett

Meka

et al. 2015

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

View full text Add to dashboard Cite

show abstract

“…See also the survey [She08]. Both methods have since been studied in the two-party setting [LZ10,RS10,She11b] and also the multiparty setting [LS09b,AC08,Cha08,She12b,She14,RY15].…”

Section: Main Structural Result: Junta Theoremmentioning

confidence: 99%

Rectangles Are Nonnegative Juntas

Göös¹,

Lovett²,

Meka³

et al. 2016

SIAM J. Comput.

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We develop a new method to prove communication lower bounds for composed functions of the form f • g n where f is any boolean function on n inputs and g is a sufficiently "hard" two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of f • g n can be simulated by a nonnegative combination of juntas. This is a new formalization for the intuition that each low-communication randomized protocol can only "query" few inputs of f as encoded by the gadget g. Consequently, we characterize the communication complexity of f • g n in all known one-sided (i.e., not closed under complement) zero-communication models by a corresponding query complexity measure of f. These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy. As applications, we resolve several open problems from prior work: We show that SBP cc (a class characterized by corruption) is not closed under intersection. An immediate corollary is that MA cc = SBP cc. These results answer questions of Klauck (CCC 2003) and Böhler et al. (JCSS 2006). We also show that approximate nonnegative rank of partial boolean matrices does not admit efficient error reduction. This answers a question of Kol et al. (ICALP 2014) for partial matrices. In subsequent work, our structure theorem has been applied to resolve the communication complexity of the Clique vs. Independent Set problem.

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“…In this section, we prove essentially an exp − √ N/4 k upper bound on the discrepancy of the GHR N k function where the first player gets N input bits. This gives us an exp −n Ω (1) upper bound on the discrepancy if k ≤ ε log(N) for any constant 0 < ε < 1/4. Goldmann et al [17] showed that when k = 2, if there is a low cost one-way protocol for GHR N 2 , then it must have low advantage.…”

Section: A Discrepancy Upper Bound For the Multiparty Ghr Functionmentioning

confidence: 97%

“…for some constant ε > 0 to give us a discrepancy upper bound of exp −n Ω (1) . For notational convenience, we may switch between the notations Ex and Ex∼µ X from now on.…”

Section: Fact 33 (Law Of Total Expectation)mentioning

confidence: 99%

“…The two features that make this model much more subtle than its classical two-party counterpart, are the mutual overlap of information and the fact that as k grows, each player misses less information. Indeed, starting with the surprising result of Grolmusz [18], several sets of authors (see, for example, [3,1,15]) constructed counter-intuitive protocols especially when k is greater than log n. This makes proving multi-party lower bounds on the cost of protocols quite challenging. However, researchers have been well motivated to take on this challenge due to many well-known applications of such lower bounds in diverse areas like circuit complexity, proof complexity, and pseudo-random generators.…”

Section: Introductionmentioning

confidence: 99%

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Untitled

Chattopadhyay¹,

Mande²

2018

Theory of Comput.

Self Cite

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We construct a simple function that has small unbounded-error communication complexity in the k-party number-on-forehead (NOF) model but every probabilistic protocol that solves it with subexponential advantage over random guessing has cost essentially Ω(√ n/4 k) bits. This separates these classes up to k ≤ δ log n players for any constant δ < 1/4, and gives the largest known separation by an explicit function in this regime of k. Our analysis is elementary and self-contained, inspired by the methods of Goldmann, Håstad, and Razborov (Computational Complexity, 1992). After initial publication of our work as a preprint (ECCC, 2016), Sherstov pointed out to us that an alternative proof of an Ω((n/4 k) 1/7) separation is implicit in his prior work (SICOMP, 2016). Furthermore, based on his prior work (SICOMP, 2013 and SICOMP, 2016), Sherstov gave an alternative proof of our constructive Ω(√ n/4 k) separation and also produced a stronger non-constructive Ω(n/4 k) separation. These results are explained in Sherstov's preprint (ECCC, 2016) and in article 14/22 in Theory of Computing.

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The NOF Multiparty Communication Complexity of Composed Functions

Abstract: Abstract. We study the k-party 'number on the forehead' communication

Cited by 16 publications

References 50 publications

Rectangles Are Nonnegative Juntas

Rectangles Are Nonnegative Juntas

Rectangles Are Nonnegative Juntas

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