Streaming Algorithms via 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 eas- ily 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 ,...,x n ), which is useful for the following applications: 1) Estimating the F k -moment of x, for k >; 2. 2) Estimating the ℓ p -norm of x, for p ϵ 1, 2, with small update time. 3) Estimating cascaded norms ℓp(ℓq) for all p,q >; 0. 4) ℓ 1 sampling, where the goal is to produce an element i with probability (approximately) |x i |/||x|| 1 . It extends to similarly defined ℓ p -sampling, for p ϵ 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 Σ i=1 n a i from weak estimates of each real a i ϵ 0,1. More precisely, the estimator first chooses a desired precision u i ϵ (0,1] for each i ϵ n, and then it receives an estimate of every a i within additive u i . Its goal is to provide a good approximation to Σa i while keeping a tab on the "approximation cost" Σ i (1/u i )- Here we refine previous work (Andoni, Krauthgamer, and Onak, FOCS 2010) which shows that as long as Σa i = Ω(1), a good multiplicative approximation can be achieved using total precision of only O(n log n).