An algebraic approach to complexity of data stream computations

Abstract

We consider a basic problem in the general data streaming model, namely, to estimate a vector $f\in {\mathbb{Z}}^n$ that is arbitrarily updated (i.e., incremented or decremented) coordinate-wise. The estimate $\hat{f}\in {\mathbb{Z}}^n$ must satisfy $\norm{\hat{f}-f}_{\infty}\le ε\norm{f}_1 $, that is, $\forall i ~(\abs{\hat{f}_i - f_i} \le ε\norm{f}_1)$. It is known to have $\tilde{O}({\epsilon }^{-1})$ randomized space upper bound \cite{cm:jalgo}, $\Omega ({\epsilon }^{-1}\log (\epsilon n))$ space lower bound \cite{bkmt:sirocco03} and deterministic space upper bound of $\tilde{\Omega }({\epsilon }^{-2})$ bits.\footnote{The $\tilde{O}$ and $\tilde{\Omega }$ notations suppress poly-logarithmic factors in $n, \log ε^{-1}, \norm{f}_{\infty}$ and $\log {\delta }^{-1}$, where, $\delta$ is the error probability (for randomized algorithm).} We show that any deterministic algorithm for this problem requires space $Ω(ε^{-2} (\log \norm{f}_1))$ bits.

Revised version