The Multiparty Communication Complexity of Exact-T: Improved Bounds and New Problems

Beigel, Richard; Gasarch, William; Glenn, James R.

doi:10.1007/11821069_13

Cited by 9 publications

(19 citation statements)

References 20 publications

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“…(We believe that the latter is, in fact, super-polylogarithmic.) The best previously known deterministic lower bound for a function with efficient randomized NOF protocols is the Ω (log log n) lower bound given by Beigel, Gasarch, and Glenn [9] for the Exact-T function, which was originally investigated in [12] in the special case of k = 3 players. As a corollary, we also obtain the fact that the function families we define have Ω (log log n) private-coin randomized NOF communication complexity but only O (1) public-coin randomized NOF communication complexity.…”

Section: Introductionmentioning

confidence: 99%

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Beame¹,

David²,

Pitassi³

et al. 2010

Theory of Comput.

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We solve some fundamental problems in the number-on-forehead (NOF) kplayer communication model. We show that there exists a function which has at most logarithmic communication complexity for randomized protocols with one-sided falsepositives error probability of 1/3, but which has linear communication complexity for deterministic protocols and, in fact, even for the more powerful nondeterministic protocols. The result holds for every ε > 0 and every k ≤ 2 (1−ε)n players, where n is the number of bits on each player's forehead. As a consequence, we obtain the NOF communication class separation coRP ⊂ NP. This in particular implies that P = RP and NP = coNP. We also show that for every ε > 0 and every k ≤ n 1−ε , there exists a function which has constant randomized complexity for public coin protocols but at least logarithmic complexity for private coin protocols. No larger gap between private and public coin protocols is possible. Our lower bounds are existential; no explicit function is known to satisfy nontrivial lower bounds for k ≥ log n players. However, for every ε > 0 and every k ≤ (1 − ε) • log n players, the NP = coNP separation (and even the coNP ⊂ MA separation) was obtained independently by Gavinsky and Sherstov (2010) using an explicit construction. In this work, for k ≤ (1/9) • log n players, we exhibit an explicit function which has communication

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Section: Introductionmentioning

confidence: 99%

Untitled

Beame¹,

David²,

Pitassi³

et al. 2010

Theory of Comput.

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“…an Ω(log log log n) lower bounds for the communication complexity of any two-dimensional permutation [1,6,4,8]. For higher-dimensional permutations the best lower bound is Ω(log * n) [8], but it is outside the scope of techniques discussed here.…”

Section: Previous Results a Closer Lookmentioning

confidence: 99%

“…Namely, a graph function always has a very simple type of protocol in which one of the players is oblivious and the others send only one bit. This protocol was also previously used for a specific graph function by Chandra, Furst and Lipton in [5] and for a more general family of permutations in [4].…”

Section: Introductionmentioning

confidence: 99%

Nondeterministic communication complexity with help and graph functions

Shraibman

2019

Theoretical Computer Science

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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%

“…Proof The protocol uses a known reduction to the Exactly-n function, see e.g. [5,7]. Define Exactly n,k (x 1 , .…”

Section: Lemma 11mentioning

confidence: 99%

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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The Multiparty Communication Complexity of Exact-T: Improved Bounds and New Problems

Cited by 9 publications

References 20 publications

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Nondeterministic communication complexity with help and graph functions

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

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