Tracking the $\ell_2$ Norm with Constant Update Time

Abstract

The \emph{$\ell_2$ tracking problem} is the task of obtaining a streaming algorithm that, given access to a stream of items $a_1,a_2,a_3,\ldots$ from a universe $[n]$, outputs at each time $t$ an estimate to the $\ell_2$ norm of the \textit{frequency vector} $f^{(t)}\in \mathbb{R}^n$ (where $f^{(t)}_i$ is the number of occurrences of item $i$ in the stream up to time $t$). The previous work [Braverman-Chestnut-Ivkin-Nelson-Wang-Woodruff, PODS 2017] gave an streaming algorithm with (the optimal) space using $O(ε^{-2}\log(1/δ))$ words and $O(ε^{-2}\log(1/δ))$ update time to obtain an $ε$-accurate estimate with probability at least $1-δ$. We give the first algorithm that achieves update time of $O(\log 1/δ)$ which is independent of the accuracy parameter $ε$, together with the nearly optimal space using $O(ε^{-2}\log(1/δ))$ words. Our algorithm is obtained using the \textsf{CountSketch} of [Charilkar-Chen-Farach-Colton, ICALP 2002].