Small Error Versus Unbounded Error Protocols in the NOF Model

We show that a simple function has small unbounded error communication complexity in the k-party number-on-forehead (NOF) model but every probabilistic protocol that solves it with sub-exponential advantage over random guessing has cost essentially Ω (√ n 4 ) bits. Such a separation was first shown for k = 2 independently by Buhrman et al. [8] and Sherstov [24]. A very recent work of Sherstov [27] can be combined with an earlier work of Beigel [7] to yield such a separation for up to k = Θ(log log n) players. To the best of our knowledge, our result provides the first such separation that works all the way up to k ≤ δ log n players, where δ < 1 is a constant. Additionally, our communication lower bounds for k-party probabilistic protocols for a function that has efficient unbounded error protocols are quantitatively stronger than previous bounds. In particular, for any constant k, our bound on communication for such a function is Ω ( n ) in contrast to the best known previous bound of Ω ( n ) . This has the following consequence for boolean Threshold circuits: let THR and MAJ denote respectively the classes of linear threshold functions that have unbounded weights and polynomially bounded weights. Further, let PARk (ANYk) denote the class of functions that are parities of k bits (any k-junta). For every 2 ≤ k ≤ δ log n, we show that there exists a function in linear size THR ◦ PARk that needs 2 Ω(1) size to be computed by every circuit in the class MAJ ◦ SYM ◦ ANYk−1, where SYM represents the class of all symmetric functions. Applying a result of Goldmann et al. [15] to the above, similar lower bounds on the size of circuits of the form MAJ ◦THR ◦ANYk−1 for computing the function follow. The main technical ingredient of our result is to show that a composed function of the form f ◦ PAR has exponentially small discrepancy while f has sign degree just 1. ∗[email protected] †[email protected] ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 95 (2016)