Improved Separations between Nondeterministic and Randomized Multiparty Communication

Abstract

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.

Tensor Rank and Strong Quantum Nondeterminism in Multiparty Communication

“…We do so by applying Theorem 1 to a total function explicitly constructed for this task. This result could be considered as the quantum analog of a separation previously proved in [9]- [11] between classical nondetermistic and randomized NOF communication. † An exact quantum protocol accepts a correct input and rejects an incorrect input with probability 1.…”

Multiparty Communication Complexity and Threshold Circuit Size of AC^0

“…This function is expressible by a small 2-level ∨ of ∧s. As described in [11] the generalized discrepancy/correlation arguments work for any selector function that uses the inputs for players 1 to k − 1 to select which bits from player 0's input to pass on to f , but we need our more general formulation for some examples we consider in Appendix A.2.…”

“…We give a brief overview of the remainder of the argument in [9,11], which extends ideas of [23,25] from 2-party to k-party communication complexity.…”

“…Lee and Schraibman [16] and Chattopadhyay and Ada [9] applied the full method in [25] to pattern tensors to yield the first lower bounds for the general NOF multiparty communication complexity of set disjointness for k > 2 players, improving on a long line of research on the problem [3,27,6,29,14,7] and obtaining a lower bound of Ω(n 1 k+1 )/2 2 O(k) . This yields a separation between randomized and nondeterministic k-party models for k = o(log log n), which David, Pitassi, and Viola [11] improved to Ω(log n) players for other functions based on pseudorandom generators. They asked whether there was a separation for Ω(log n) players for AC 0 functions since their functions are only in AC 0 for k = O(log log n), a problem which our results resolve.…”

“…We also describe a general form of the method of [25,9,11] based on selector functions and orthogonalizing distributions for functions of large -approximate degree and briefly discuss its limitations.…”