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2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (2011)
Palm Springs, California USA
Oct. 22, 2011 to Oct. 25, 2011
ISSN: 0272-5428
ISBN: 978-0-7695-4571-4
pp: 363-372
ABSTRACT
A technique introduced by Indyk and Woodruff (STOC 2005) has inspired several recent advances in data-stream algorithms. We show that a number of these results follow easily from the application of a single probabilistic method called Precision Sampling. Using this method, we obtain simple data-stream algorithms that maintain a randomized sketch of an input vector $x=(x_1,x_2,\ldots,x_n)$, which is useful for the following applications:* Estimating the $F_k$-moment of $x$, for $k>2$.* Estimating the $\ell_p$-norm of $x$, for $p\in[1,2]$, with small update time.* Estimating cascaded norms $\ell_p(\ell_q)$ for all $p,q>0$.* $\ell_1$ sampling, where the goal is to produce an element $i$ with probability (approximately) $|x_i|/\|x\|_1$. It extends to similarly defined $\ell_p$-sampling, for $p\in [1,2]$. For all these applications the algorithm is essentially the same: scale the vector $x$ entry-wise by a well-chosen random vector, and run a heavy-hitter estimation algorithm on the resulting vector. Our sketch is a linear function of $x$, thereby allowing general updates to the vector $x$. Precision Sampling itself addresses the problem of estimating a sum $\sum_{i=1}^n a_i$ from weak estimates of each real $a_i\in[0,1]$. More precisely, the estimator first chooses a desired precision$u_i\in(0,1]$ for each $i\in[n]$, and then it receives an estimate of every $a_i$ within additive $u_i$. Its goal is to provide a good approximation to $\sum a_i$ while keeping a tab on the ``approximation cost'' $\sum_i (1/u_i)$. Here we refine previous work (Andoni, Krauthgamer, and Onak, FOCS 2010)which shows that as long as $\sum a_i=\Omega(1)$, a good multiplicative approximation can be achieved using total precision of only $O(n\log n)$.
INDEX TERMS
streaming, sampling, moments, cascaded norms
CITATION

R. Krauthgamer, K. Onak and A. Andoni, "Streaming Algorithms via Precision Sampling," 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science(FOCS), Palm Springs, California USA, 2011, pp. 363-372.
doi:10.1109/FOCS.2011.82
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