The Communication Complexity of Gap Hamming Distance

In the gap Hamming distance problem, two parties must determine whether their respective strings x; y2f0; 1g n are at Hamming distance less than n=2 p n or greater than n=2+ p n: In a recent tour de force, Chakrabarti and Regev (2010) proved the long- conjectured W(n) lower 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.

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