Unbiased Insights: Optimal Streaming Algorithms for $\ell_p$ Sampling, the Forget Model, and Beyond

TL;DR

This paper introduces space-efficient streaming algorithms for $_p$ sampling and frequency moment estimation, including in models with deletions and continuous sampling, solving open problems and extending to entropy and arbitrary functions.

Contribution

It provides nearly optimal space algorithms for $_p$ sampling and $F_p$ estimation in advanced streaming models, including the forget and suffix-prefix deletion models, with broad applicability.

Findings

Achieves nearly space-optimal $_p$ samplers for $p eq 2$

Develops unbiased estimators for $F_p$ in models with deletions

Extends techniques to entropy estimation and arbitrary functions

Abstract

We study $\ell_p$ sampling and frequency moment estimation in a single-pass insertion-only data stream. For $p \in (0,2)$, we present a nearly space-optimal approximate $\ell_p$ sampler that uses $\widetilde{O}(\log n \log(1/\delta))$ bits of space and for $p = 2$, we present a sampler with space complexity $\widetilde{O}(\log^2 n \log(1/\delta))$. This space complexity is optimal for $p \in (0, 2)$ and improves upon prior work by a $\log n$ factor. We further extend our construction to a continuous $\ell_p$ sampler, which outputs a valid sample index at every point during the stream. Leveraging these samplers, we design nearly unbiased estimators for $F_p$ in data streams that include forget operations, which reset individual element frequencies and introduce significant non-linear challenges. As a result, we obtain near-optimal algorithms for estimating $F_p$ for all $p$ in this model, originally proposed by Pavan, Chakraborty, Vinodchandran, and Meel [PODS'24], resolving all three open problems they posed. Furthermore, we generalize this model to what we call the suffix-prefix deletion model, and extend our techniques to estimate entropy as a corollary of our moment estimation algorithms. Finally, we show how to handle arbitrary coordinate-wise functions during the stream, for any $g \in \mathbb{G}$, where $\mathbb{G}$ includes all (linear or non-linear) contraction functions.