Streaming Algorithms from Precision Sampling

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 $x=(x1,...xn)$, which is useful for the following applications. 1) Estimating the $Fk$-moment of $x$, for $k>2$. 2) Estimating the $\ellp$-norm of $x$, for $p\in[1,2]$, with small update time. 3) Estimating cascaded norms $\ellp(\ellq)$ for all $p,q>0$. 4) $\ell1$ sampling, where the goal is to produce an element $i$ with probability (approximately) $|xi|/|x|1$. It extends to similarly defined $\ellp$-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 $\sum{i=1}ⁿ ai$ from weak estimates of each real $ai\in[0,1]$. More precisely, the estimator first chooses a desired precision $ui\in(0,1]$ for each $i\in[n]$, and then it receives an estimate of every $ai$ within additive $ui$. Its goal is to provide a good approximation to $\sum ai$ while keeping a tab on the "approximation cost" $\sumi (1/ui)$. Here we refine previous work [Andoni, Krauthgamer, and Onak, FOCS 2010] which shows that as long as $\sum ai=\Omega(1)$, a good multiplicative approximation can be achieved using total precision of only $O(n\log n)$.