Volume 8 (2012)
Article 8 pp. 197-208

The Communication Complexity of Gap Hamming Distance

Received: August 8, 2011

Published: May 17, 2012

Published: May 17, 2012

**Comments and updates on this paper:**

"Comparing three proofs of the randomized communication complexity of Hamming distance" by Thomas Vidick, February 26, 2013

**Keywords:**communication complexity, gap Hamming distance, Talagrand's concentration inequality

**Categories:**short, complexity theory, lower bounds, communication complexity, Hamming distance, Talagrand's concentration inequality, commented on

**ACM Classification:**F.1.3

**AMS Classification:**68Q17, 68Q25

**Abstract:**
[Plain Text Version]

In the *gap Hamming distance* problem, two parties
must determine whether their respective strings
$x,y\in\{0,1\}^n$ are at Hamming distance less than
$n/2-\sqrt n$ or greater than $n/2+\sqrt n.$ In a
recent tour de force, Chakrabarti and Regev (2010)
proved the long-conjectured $\Omega(n)$ bound on the
randomized communication complexity of this problem.
In follow-up work, Vidick (2010)
discovered a simpler proof. We contribute a new proof,
which is simpler yet and a page-and-a-half long.