Annual Symposium on Foundations of Computer Science

A Tight Bound for Set Disjointness in the Message-Passing Model

Mark Braverman, Faith Ellen, Rotem Oshman, Toniann Pitassi, Vinod Vaikuntanathan

Published Oct 1, 2013• 58 citations

Abstract

In a multiparty message-passing model of communication, there are k players. Each player has a private input, and they communicate by sending messages to one another over private channels. While this model has been used extensively in distributed computing and in secure multiparty computation, lower bounds on communication complexity in this model and related models have been somewhat scarce. In recent work [25], [29], [30], strong lower bounds of the form Ω(n·k) were obtained for several functions in the message-passing model; however, a lower bound on the classical set disjointness problem remained elusive. In this paper, we prove a tight lower bound of Ω(n · k) for the set disjointness problem in the message passing model. Our bound is obtained by developing information complexity tools for the message-passing model and proving an information complexity lower bound for set disjointness.

DOI: 10.1109/focs.2013.77

Related papers

Multiparty Communication Complexity of Disjointness

Arkadev Chattopadhyay, Anil Ada

We obtain a lower bound of n^Omega(1) on the k-party randomized communication complexity of the Disjointness function in the `Number on the Forehead' model of multiparty communication when k is a constant. For k=o(loglog n), the bounds remain super-polylogarithmic i.e. (log n)^omega(1). The previous best lower bound for three players until recently was Omega(log n). Our bound separates the communication complexity classes NP^{CC}_k and BPP^{CC}_k for k=o(loglog n). Furthermore, by the results of Beame, Pitassi and Segerlind \cite{BPS07}, our bound implies proof size lower bounds for tree-like, degree k-1 threshold systems and superpolynomial size lower bounds for Lovasz-Schrijver proofs. Sherstov \cite{She07b} recently developed a novel technique to obtain lower bounds on two-party communication using the approximate polynomial degree of boolean functions. We obtain our results by extending his technique to the multi-party setting using ideas from Chattopadhyay \cite{Cha07}. A similar bound for Disjointness has been recently and independently obtained by Lee and Shraibman.

arXiv:0801.3624Jan 23, 200863 citations

Multiparty Communication Complexity of AC 0

Paul W. Beame, Dang-Trinh Huynh-Ngoc

We prove n Ω(1) lower bounds on the multiparty communication complexity of AC 0 functions in the number-on-forehead (NOF) model for up to Θ(log n) players. These are the first lower bounds for any AC 0 function for ω(log log n) players. In particular we show that there are families of depth 3 read-once AC 0 formulas having k-player randomized multiparty NOF communication complexity n Ω(1) /2 O(k). We show similar lower bounds for depth 4 read-once AC 0 formulas that have nondeterministic communication complexity O(log 2 n), yielding exponential separations between k-party nondeterministic and randomized communication complexity for AC 0 functions. As a consequence of the latter bound, we obtain an n Ω(1/k) /2 O(k) lower bound on the k-party NOF communication complexity of set disjointness. This is non-trivial for up to Θ (√ log n) players which is significantly larger than the up to Θ(log log n) players allowed in the best previous lower bounds for multiparty set disjointness given by Lee and Shraibman [LS08] and Chattopadhyay and Ada [CA08] (though our complexity bounds themselves are not as strong as those in [LS08, CA08] for o(log log n) players). We derive these results by extending the k-party generalization in [CA08, LS08] of the pattern matrix method of Sherstov [She07, She08]. Using this technique, we derive a new sufficient criterion for strong communication complexity lower bounds based on functions having many diverse subfunctions that do not have good low-degree polynomial approximations. This criterion guarantees that such functions have orthogonalizing distributions that are “max-smooth” as opposed to the “min-smooth” orthogonalizing distributions used by Razborov and Sherstov [RS08] to analyze the sign-rank of AC⁰.

Electronic colloquium on computational complexityJan 1, 200810 citations

New Strong Direct Product Results in Communication Complexity

Rahul Jain

We show two new direct product results in two different models of communication complexity. Our first result is in the one-way public-coin model. Let f ⊆ X × Y × Z be a relation and ϵ > 0 be a constant. Let R 1,pub ϵ ( f ) represent the communication complexity of f , with worst-case error ϵ in this model. We show that if for computing f k ( k independent copies of f ) in this model, o ( k ċ R 1, pub 1/3 ( f )) communication is used, then the success is exponentially small in k . We show a new tight characterization of communication complexity in this model which strengthens the tight characterization shown in Jain et al. [2008]. We use this new characterization to show our direct product result and this characterization may also be of independent interest. Our second direct product result is in the model of two-way public-coin communication complexity. We show a direct product result for all relations in this model in terms of a new complexity measure that we define. Our new measure is a generalization to nonproduct distributions, of the two-way product subdistribution bound of Jain et al. [2008]. Our direct product result therefore generalizes to nonproduct distributions, their direct product result in terms of the two-way product subdistribution bound. As an application of our new direct product result, we reproduce (via completely different arguments) strong direct product result for the set-disjointness problem which was previously shown by Klauck [2010]. We show this by proving that our new complexity measure gives a tight lower bound of Ω( n ) for the set-disjointness problem on n -bit inputs (this strengthens the linear lower bound on the rectangle/corruption bound for set-disjointness shown by Razborov [1992]). In addition, we show that many previously known direct product results in this model are uniformly implied and often strengthened by our result.

Journal of the ACMJun 30, 201522 citations

New strong direct product results in communication complexity.

Rahul Jain

We show two new direct product results in two different models of communication complexity. Our first result is in the one-way public-coin model. Let f ⊆ X × Y × Z be a relation and e > 0 be a constant. Let R1,pube(f) represent the communication complexity of f, with worst-case error e in this model. We show that if for computing fk (k independent copies of f) in this model, o(k c R1, pub1/3(f)) communication is used, then the success is exponentially small in k. We show a new tight characterization of communication complexity in this model which strengthens the tight characterization shown in Jain et al. [2008]. We use this new characterization to show our direct product result and this characterization may also be of independent interest.Our second direct product result is in the model of two-way public-coin communication complexity. We show a direct product result for all relations in this model in terms of a new complexity measure that we define. Our new measure is a generalization to nonproduct distributions, of the two-way product subdistribution bound of Jain et al. [2008]. Our direct product result therefore generalizes to nonproduct distributions, their direct product result in terms of the two-way product subdistribution bound. As an application of our new direct product result, we reproduce (via completely different arguments) strong direct product result for the set-disjointness problem which was previously shown by Klauck [2010]. We show this by proving that our new complexity measure gives a tight lower bound of Ω(n) for the set-disjointness problem on n-bit inputs (this strengthens the linear lower bound on the rectangle/corruption bound for set-disjointness shown by Razborov [1992]). In addition, we show that many previously known direct product results in this model are uniformly implied and often strengthened by our result.

Electronic colloquium on computational complexityJan 1, 201116 citations

Communication Complexity of Set-Disjointness for All Probabilities.

Mika Göös, Thomas W. Watson

We study set-disjointness in a generalized model of randomized two-party communication where the probability of acceptance must be at least α(n) on yes-inputs and at most β(n) on no-inputs, for some functions α(n) > β(n). Our main result is a complete characterization of the private-coin communication complexity of set-disjointness for all functions α and β, and a near-complete characterization for public-coin protocols. In particular, we obtain a simple proof of a theorem of Braverman and Moitra (STOC 2013), who studied the case where α = 1/2+ (n) and β = 1/2− (n). The following contributions play a crucial role in our characterization and are interesting in their own right. (1) We introduce two communication analogues of the classical complexity class that captures small bounded-error computations: we define a “restricted” class SBP (which lies between MA and AM) and an “unrestricted” class USBP. The distinction between them is analogous to the distinction between the well-known communication classes PP and UPP. (2) We show that the SBP communication complexity is precisely captured by the classical corruption lower bound method. This sharpens a theorem of Klauck (CCC 2003). (3) We use information complexity arguments to prove a linear lower bound on the USBP complexity of set-disjointness.

Electronic colloquium on computational complexityJan 1, 20140 citations

Simplified Lower Bounds on the Multiparty Communication Complexity of Disjointness

Anup Rao, Amir Yehudayoff

We show that the deterministic number-on-forehead communication complexity of set disjointness for k parties on a universe of size n is Omega(n/4^k). This gives the first lower bound that is linear in n, nearly matching Grolmusz's upper bound of O(log^2(n) + k^2n/2^k). We also simplify the proof of Sherstov's Omega(sqrt(n)/(k2^k)) lower bound for the randomized communication complexity of set disjointness.

Leibniz international proceedings in informaticsJan 1, 201519 citations