An algebraic approach to complexity of data stream computations

We consider a basic problem in the general data streaming model, namely, to estimate a vector $f \in \Z^n$ that is arbitrarily updated (i.e., incremented or decremented) coordinate-wise. The estimate $\hat{f} \in \Z^n$ must satisfy $\norm{\hat{f}-f}{\infty}\le \epsilon\norm{f}1 $, that is, $\forall i ~(\abs{\hat{f}i - fi} \le \epsilon \norm{f}1)$. 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 $n, \log \epsilon^{-1}, \norm{f}{\infty}$ and $\log \delta^{-1}$, where, $\delta$ is the error probability (for randomized algorithm).} We show that any deterministic algorithm for this problem requires space $\Omega(\epsilon^{-2} (\log \norm{f}_1))$ bits.