A Multi-Round Communication Lower Bound for Gap Hamming and Some Consequences

The gap-Hamming-distance problem arose in the context of proving space lower bounds for a number of key problems in the data stream model. In this problem, Alice and Bob have to decide whether the Hamming distance between their n-bit input strings is large (i.e., at least n/2 + radic(n)) or small (i.e., at most n/2 - radic(n)); they do not care if it is neither large nor small. This Theta(radic(n)) gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm. Thus far, for randomized communication, an Omega(n) lower bound on this problem was known only in the one-way setting. We prove an Omega(n) lower bound for randomized protocols that use any constant number of rounds. As a consequence we conclude, for instance, that epsiv-approximately counting the number of distinct elements in a data stream requires Omega(1/epsiv²) space, even with multiple (a constant number of) passes over the input stream. This extends earlier one-pass lower bounds, answering a long-standing open question. We obtain similar results for approximating the frequency moments and for approximating the empirical entropy of a data stream. In the process, we also obtain tight n - Theta(radic(n)log n) lower and upper bounds on the one-way deterministic communication complexity of the problem. Finally, we give a simple combinatorial proof of an Omega(n) lower bound on the one-way randomized communication complexity.