Tracking the $\ell_2$ Norm with Constant Update Time

TL;DR

This paper presents a streaming algorithm for the 2 tracking problem that achieves constant update time independent of the accuracy parameter, while maintaining near-optimal space complexity, improving upon previous methods.

Contribution

The authors introduce the first 2 tracking algorithm with update time independent of psilons, using CountSketch, and achieve nearly optimal space complexity.

Findings

Achieves update time of O(log 1/elta) independent of psilons

Uses CountSketch for efficient 2 norm estimation

Maintains near-optimal space complexity of O(psilons^{-2}log(1/elta)) words

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 $O(\epsilon^{-2}\log(1/\delta))$ words and $O(\epsilon^{-2}\log(1/\delta))$ update time to obtain an $\epsilon$-accurate estimate with probability at least $1-\delta$. We give the first algorithm that achieves update time of $O(\log 1/\delta)$ which is independent of the accuracy parameter $\epsilon$, together with the nearly optimal space using $O(\epsilon^{-2}\log(1/\delta))$ words. Our algorithm is obtained using the \textsf{CountSketch} of [Charilkar-Chen-Farach-Colton, ICALP 2002].