The One-Way Communication Complexity of Hamming Distance


Consider the following version of the Hamming distance problem for {1,-1}-vectors of length n: the promise is that the distance is either at least (n/2)+sqrt{n} or at most (n/2)-sqrt{n}, and the goal is to find out which of these two cases occurs. Woodruff (Proc. ACM-SIAM Symposium on Discrete Algorithms, 2004) gave a linear lower bound for the randomized one-way communication complexity of this problem. In this note we give a simple proof of this result. Our proof uses a simple reduction from the indexing problem and avoids the VC-dimension arguments used in the previous paper. As shown by Woodruff (loc. cit.), this implies an
Omega(1/epsilon^2)-space lower bound for approximating frequency moments within a factor 1+epsilon in the data stream model.