Separating Deterministic from Nondeterministic NOF Multiparty Communication Complexity

Beame, Paul; David, Matei; Pitassi, Toniann; Woelfel, Philipp

doi:10.1007/978-3-540-73420-8_14

Cited by 18 publications

(47 citation statements)

References 8 publications

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“…We also note that Theorem 1.1 in particular implies an exponential separation between nondeterministic and deterministic protocols (hence, NP cc k ⊂ P cc k for k = δ log n players). Similar separations follow from [BDPW07], but only for non-explicit functions.…”

Section: Introductionsupporting

confidence: 59%

“…One of the most fundamental questions in number-on-forehead communication complexity, and the main question addressed in this paper, is to separate these classes. In [BDPW07], Beame et al give an exponential separation between randomized and deterministic protocols for k ≤ n O(1) players (in particular, RP cc k = P cc k for k ≤ n O(1) ). The breakthrough work by Sherstov [She09,She08a] sparked a flurry of exciting results in communication complexity [Cha07,LS08,CA08] which gave an exponential separation between nondeterministic and randomized protocols for k < log log n players (in particular, NP cc k ⊂ BPP cc k for k < log log n).…”

Section: Introductionmentioning

confidence: 99%

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Improved Separations between Nondeterministic and Randomized Multiparty Communication

David

Pitassi

Viola

Lecture Notes in Computer Science

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We exhibit an explicit function f : {0, 1} n → {0, 1} that can be computed by a nondeterministic number-on-forehead protocol communicating O(log n) bits, but that requires n Ω(1) bits of communication for randomized number-on-forehead protocols with k = δ · log n players, for any fixed δ < 1. Recent breakthrough results for the Set-Disjointness function (Lee Shraibman, CCC '08; Chattopadhyay Ada, ECCC '08) based on the work of (Sherstov, STOC '08) imply such a separation but only when the number of players is k < log log n.We also show that for any k = A log log n the above function f is computable by a small circuit whose depth is constant whenever A is a (possibly large) constant. Recent results again give such functions but only when the number of players is k < log log n.

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Section: Introductionsupporting

confidence: 59%

Section: Introductionmentioning

confidence: 99%

Improved Separations between Nondeterministic and Randomized Multiparty Communication

David

Pitassi

Viola

Lecture Notes in Computer Science

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“…f0; 1g is a k-party number-on-the-forehead communication problem, is considered to be efficiently solvable by a given class of protocols if F n has communication complexity at most log c n; for a large enough constant c > 1 and all n > c: This convention allows one to define BPP k ; NP k ; coNP k ; and MA k as the classes of families with efficient randomized, nondeterministic, co-nondeterministic, and Merlin-Arthur protocols, respectively. In recent years, the relationships among these classes have been almost fully determined [9,35,19,22,11,10,23]. It particular, for k .log n/; it is known [10,23] that coNP k is not contained in BPP k ; NP k ; or even MA k : As a corollary to Theorem 1.4, we show that coNP k can be separated from all these classes by a particularly simple function, set disjointness.…”

Section: Introductionmentioning

confidence: 93%

“…In recent years, the relationships among these classes have been almost fully determined [9,35,19,22,11,10,23]. It particular, for k .log n/; it is known [10,23] that coNP k is not contained in BPP k ; NP k ; or even MA k : As a corollary to Theorem 1.4, we show that coNP k can be separated from all these classes by a particularly simple function, set disjointness.…”

Section: Introductionmentioning

confidence: 93%

The multiparty communication complexity of set disjointness

Sherstov

2012

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

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We study the set disjointness problem in the number-on-theforehead model of multiparty communication.(i) We prove that k-party set disjointness has communication complexity˝.n=4 k / 1=4 in the randomized and nondeterministic models and˝.n=4 k / 1=8 in the Merlin-Arthur model. These lower bounds are close to tight. Previous lower bounds (2007)(2008) for k 3 parties were weaker than˝.n=2 k 3 / 1=.kC1/ in all three models.(ii) We prove that solving`instances of set disjointness requires ˝.n=4 k / 1=4 bits of communication, even to achieve correctness probability exponentially close to 1=2: This gives the first directproduct result for multiparty set disjointness, solving an open problem due to Beame, Pitassi, Segerlind, and Wigderson (2005).(iii) We construct a read-once f^; _g-circuit of depth 3 with exponentially small discrepancy for up to k 1 2 log n parties. This result is optimal with respect to depth and solves an open problem due to Beame and Huynh-Ngoc (FOCS '09), who gave a depth-6 construction. Applications to circuit complexity are given.

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“…Non-explicit separations for a large number of parties are given in [5]. Recent progress on explicit separations [35,34,19,27,16,23,8] was sparked by the work by Sherstov [35,34], see his survey [33].…”

Section: Theorem 12 (Nondeterminism Vs Determinism)mentioning

confidence: 99%

One-way multiparty communication lower bound for pointer jumping with applications

Viola

Wigderson

2009

Combinatorica

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In this paper we study the one-way multiparty communication model, in which every party speaks exactly once in its turn. For every k, we prove a tight lower bound of Ω`n 1/(k−1)ó n the probabilistic communication complexity of pointer jumping in a k-layered tree, where the pointers of the i-th layer reside on the forehead of the i-th party to speak. The lower bound remains nontrivial even for k = (log n) 1/2− parties, for any constant > 0. Previous to our work a lower bound was known only for k = 3 (Wigderson, see [7]), and in restricted models for k > 3 [22,24,18,4,13]. Our results have the following consequences to other models and problems, extending previous work in several directions.The one-way model is strong enough to capture general (not one-way) multiparty protocols with a bounded number of rounds. Thus we generalize two problem areas previously studied in the 2-party model (cf. [30, 21, 29]). The first is a rounds hierarchy: we give an exponential separation between the power of r and 2r rounds in general probabilistic k-party protocols, for any k and r. The second is the relative power of determinism and nondeterminism: we prove an exponential separation between nondeterministic and deterministic communication complexity for general k-party protocols with r rounds, for any k, r.The pointer jumping function is weak enough to be a special case of the well-studied disjointness function. Thus we obtain a lower bound of Ω`n 1/(k−1)´o n the probabilistic complexity of k-set disjointness in the one-way model, which was known only for k = 3 parties. Our result also extends a similar lower bound for the weaker simultaneous model, in which parties simultaneously send one message to a referee [12].Finally, we infer an exponential separation between the power of any two different orders in which parties send messages in the one-way model, for every k. Previous results [29,7] separated orders based on who speaks first. EMANUELE VIOLA, AVI WIGDERSONOur lower bound technique, which handles functions of high discrepancy over cylinder intersections, provides a "party-elimination" induction, based on a restricted form of a direct-product result, specific to the pointer jumping function.

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Separating Deterministic from Nondeterministic NOF Multiparty Communication Complexity

Cited by 18 publications

References 8 publications

Improved Separations between Nondeterministic and Randomized Multiparty Communication

Improved Separations between Nondeterministic and Randomized Multiparty Communication

The multiparty communication complexity of set disjointness

One-way multiparty communication lower bound for pointer jumping with applications

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