An algebraic approach to complexity of data stream computations

TL;DR

This paper investigates the space complexity of data stream algorithms for vector estimation, establishing new lower bounds for deterministic algorithms and analyzing the algebraic structure underlying the problem.

Contribution

It introduces a novel algebraic framework to analyze the complexity of data stream computations, providing tight bounds for deterministic algorithms.

Findings

Deterministic algorithms require space at least proportional to (\u03b5^{-2} \, ext{log} \, orm{f}_1) bits.

Established a lower bound matching known randomized upper bounds up to logarithmic factors.

Provided insights into the algebraic structure influencing data stream computation complexity.

Abstract

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, $\forall i ~(\abs{\hat{f}_i - f_i} \le \epsilon \norm{f}_1)$. 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.