The One-Way Communication Complexity of Hamming Distance

Consider the following version of the Hamming distance problem for ±1 vectors of length n: the promise is that the distance is either at least 2 + √ n or at most 2 − √ 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 Ω(1/ε2)-space lower bound for approximating frequency moments within a factor 1+ ε in the data stream model. ACM Classification: F.2.2 AMS Classification: 68Q25