Streaming Algorithms via Precision Sampling

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 $O(n\log n)$.