Direct Products in Communication Complexity

• Published in: Annual Symposium on Foundations of Computer Science pp. 746 - 755
• Main Authors: Braverman, Mark, Rao, Anup, Weinstein, Omri, Yehudayoff, Amir
• Format: Conference Proceeding
• Language: English
• Published: IEEE 01.10.2013
• Subjects: Communication Complexity; Complexity theory; Computational modeling; Computer science; Direct Products; Educational institutions; Information Complexity; Mutual information; Protocols; Upper bound
• ISSN: 0272-5428
• DOI: 10.1109/FOCS.2013.85

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)).