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).