Direct Products in Communication Complexity - Braverman, Mark; Rao, Anup; Weinstein, Omri; Yehudayoff, Amir (Annual Symposium on Foundations of Computer Science, 2013)
We give exponentially small upper bounds on the success probability for computing the direct product of any function over any distribution using a communication protocol. Let suc(μ, f, C) denote the maximum success probability of a 2-party communication protocol for computing the boolean function f(x, y) with C bits of communication, when the inputs (x, y) are drawn from the distribution μ. Let μ n be the product distribution on n inputs and f n denote the function that computes n copies of f on these inputs. We prove that if T log 3/2 T ≪ (C - 1)√n and suc(μ, f, C) <; 2/3, then suc(μ n , f n , T) ≤ exp(-Ω(n)). When μ is a product distribution, we prove a nearly optimal result: as long as T log 2 T ≪ Cn, we must have suc(μ n , f n , T) ≤ exp(-Ω(n)).