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.

翻译：我们研究单次遍历仅插入数据流中的$\ell_p$采样与频率矩估计问题。对于$p \in (0,2)$，我们提出一种近似空间最优的$\ell_p$采样器，其空间复杂度为$\widetilde{O}(\log n \log(1/\delta))$；对于$p = 2$，我们提出的采样器空间复杂度为$\widetilde{O}(\log^2 n \log(1/\delta))$。该空间复杂度在$p \in (0, 2)$范围内达到最优，并较前人工作提升了一个$\log n$因子。我们进一步将构造扩展至连续$\ell_p$采样器，该采样器可在数据流处理过程中的任意时刻输出有效的样本索引。基于这些采样器，我们针对包含遗忘操作（该操作会重置单个元素频率并引入显著的非线性挑战）的数据流，设计了近乎无偏的$F_p$估计器。由此，我们在Pavan、Chakraborty、Vinodchandran和Meel [PODS'24]最初提出的模型中，获得了针对所有$p$值估计$F_p$的近似最优算法，从而解决了他们提出的全部三个开放性问题。此外，我们将该模型推广至后缀-前缀删除模型，并扩展我们的技术以熵估计作为矩估计算法的推论。最后，我们展示了如何在数据流中处理任意坐标函数，其中$g \in \mathbb{G}$，而$\mathbb{G}$包含所有（线性或非线性）收缩函数。