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: • 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 ∑n i=1 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 ∑ ai while keeping a tab on the “approximation cost” ∑ i(1/ui). Here we refine previous work [Andoni, Krauthgamer, and Onak, FOCS 2010] which shows that as long as ∑ ai = Ω(1), a good multiplicative approximation can be achieved using total precision of only O(n log n). Work done in part while the author was a postdoctoral researcher at Princeton University/CCI, supported by NSF CCF 0832797. Supported in part by The Israel Science Foundation (grant #452/08), and by a Minerva grant. Supported in part by a Simons Postdoctoral Fellowship and NSF grants 0732334 and 0728645.