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,x2,…,xn), which is useful for the following applications:
- Estimating the Fk-moment of x, for k>2.
- Estimating the ℓp-norm of x, for p∈[1,2], with small update time.
- Estimating cascaded norms ℓp(ℓq) for all p,q>0.
- ℓ1 sampling, where the goal is to produce an element i with probability (approximately) |xi|/\|x\|1. It extends to similarly defined ℓp-sampling, for p∈ [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 sumi=1n ai from weak estimates of each real ai∈[0,1]. More precisely, the estimator first chooses a desired precision ui∈(0,1] for each i∈[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=Ω(1), a good multiplicative approximation can be achieved using total precision of only O(nlog n).