Deterministically Estimating Data Stream Frequencies

Abstract

We consider updates to an n-dimensional frequency vector of a data stream, that is, the vector f is updated coordinate-wise by means of insertions or deletions in any arbitrary order. A fundamental problem in this model is to recall the vector approximately, that is to return an estimate $\hat{f}$ of f such that

where ε is an accuracy parameter and p is the index of the ℓ _p norm used to calculate the norm . This problem, denoted by , is fundamental in data stream processing and is used to solve a number of other problems, such as heavy hitters, approximating range queries and quantiles, approximate histograms, etc..

Suppressing poly-logarithmic factors in n and , for p = 1 the problem is known to have ${\it \tilde{\Theta}}(1/\epsilon)$ randomized space complexity [2,4] and ${\it \tilde{\Theta}}(1/\epsilon^2)$ deterministic space complexity [6,7]. However, the deterministic space complexity of this problem for any value of p > 1 is not known. In this paper, we show that the deterministic space complexity of the problem is ${\it \tilde{ \Theta}}(n^{2-2/p}/\epsilon^2)$ for 1 < p < 2, and $\it \Theta(n)$ for p ≥ 2.