# AMS Without 4-Wise Independence on Product Domains

Abstract : In their seminal work, Alon, Matias, and Szegedy introduced several sketching techniques, including showing that 4-wise independence is sufficient to obtain good approximations of the second frequency moment. In this work, we show that their sketching technique can be extended to product domains $[n]^k$ by using the product of 4-wise independent functions on $[n]$. Our work extends that of Indyk and McGregor, who showed the result for $k = 2$. Their primary motivation was the problem of identifying correlations in data streams. In their model, a stream of pairs $(i,j) \in [n]^2$ arrive, giving a joint distribution $(X,Y)$, and they find approximation algorithms for how close the joint distribution is to the product of the marginal distributions under various metrics, which naturally corresponds to how close $X$ and $Y$ are to being independent. By using our technique, we obtain a new result for the problem of approximating the $\ell_2$ distance between the joint distribution and the product of the marginal distributions for $k$-ary vectors, instead of just pairs, in a single pass. Our analysis gives a randomized algorithm that is a $(1 \pm \epsilon)$ approximation (with probability $1-\delta$) that requires space logarithmic in $n$ and $m$ and proportional to $3^k$.
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Conference papers
Jean-Yves Marion and Thomas Schwentick. 27th International Symposium on Theoretical Aspects of Computer Science - STACS 2010, Mar 2010, Nancy, France. pp.119-130, 2010, Proceedings of the 27th Annual Symposium on the Theoretical Aspects of Computer Science

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Vladimir Braverman, Kai-Min Chung, Zhenming Liu, Michael Mitzenmacher, Rafail Ostrovsky. AMS Without 4-Wise Independence on Product Domains. Jean-Yves Marion and Thomas Schwentick. 27th International Symposium on Theoretical Aspects of Computer Science - STACS 2010, Mar 2010, Nancy, France. pp.119-130, 2010, Proceedings of the 27th Annual Symposium on the Theoretical Aspects of Computer Science. <inria-00455776>

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