An optimal algorithm for the distinct elements problem
Proceedings of the twenty-ninth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
On the exact space complexity of sketching and streaming small norms
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Better gap-hamming lower bounds via better round elimination
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
An optimal lower bound on the communication complexity of gap-hamming-distance
Proceedings of the forty-third annual ACM symposium on Theory of computing
Tight bounds for distributed functional monitoring
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Arthur-Merlin streaming complexity
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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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 $n/2 + \sqrt n$) or small (i.e., at most $n/2 - \sqrt n$); they do not care if it is neither large nor small. This $\Theta(\sqrt n)$ 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 $\epsilon$-approximately counting the number of distinct elements in a data stream requires $\Omega(1/\epsilon^2)$ 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 $n - \Theta(\sqrt{n}\log n)$ 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.