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Volume 14 (2018) Article 22 pp. 1-17
On Multiparty Communication with Large versus Unbounded Error
Received: September 3, 2016
Revised: March 24, 2017
Published: December 28, 2018
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Keywords: lower bounds, separation of complexity classes, multiparty communication complexity, unbounded-error communication complexity, PP, UPP
ACM Classification: F.1.3, F.2.3
AMS Classification: 68Q17, 68Q15

Abstract: [Plain Text Version]

The unbounded-error communication complexity of a Boolean function F is the limit of the ϵ-error randomized complexity of F as ϵ1/2. Communication complexity with weakly unbounded error is defined similarly but with an additive penalty term that depends on 1/2ϵ. Explicit functions are known whose two-party communication complexity with unbounded error is logarithmic compared to their complexity with weakly unbounded error. Chattopadhyay and Mande (ECCC TR16-095, Theory of Computing 2018) recently generalized this exponential separation to the number-on-the-forehead multiparty model. We show how to derive such an exponential separation from known two-party work, achieving a quantitative improvement along the way. We present several proofs here, some as short as half a page.

In more detail, we construct a k-party communication problem F:({0,1}n)k{0,1} that has complexity O(logn) with unbounded error and Ω(n/4k) with weakly unbounded error, reproducing the bounds of Chattopadhyay and Mande. In addition, we prove a quadratically stronger separation of O(logn) versus Ω(n/4k) using a nonconstructive argument.

A preliminary version of this paper appeared in ECCC, Report TR16-138, 2016.