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 $\widetilde{O}(\log n \log(1/δ))$ bits of space and for $p = 2$, we present a sampler with space complexity $\widetilde{O}(\log^2 n \log(1/δ))$. 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：我们研究单遍仅插入数据流中的 $\ell_p$ 采样与频率矩估计。对于 $p \in (0,2)$，我们提出一种近乎空间最优的近似 $\ell_p$ 采样器，其空间复杂度为 $\widetilde{O}(\log n \log(1/δ))$ 比特；对于 $p = 2$，我们提出一种空间复杂度为 $\widetilde{O}(\log^2 n \log(1/δ))$ 的采样器。该空间复杂度在 $p \in (0, 2)$ 范围内达到最优，较之前工作改进了 $\log n$ 因子。我们进一步将构造推广至连续 $\ell_p$ 采样器，可在流中任意时刻输出有效样本索引。利用这些采样器，我们设计了针对包含遗忘操作的数据流中 $F_p$ 的近乎无偏估计器——遗忘操作会重置单个元素频率，并引入显著的非线性挑战。由此，我们为 Pavan、Chakraborty、Vinodchandran 与 Meel [PODS'24] 最初提出的模型获得了所有 $p$ 值下 $F_p$ 估计的近乎最优算法，解决了他们提出的全部三个开放问题。此外，我们将该模型推广至所谓的前缀-后缀删除模型，并将技术手段扩展至熵估计，作为矩估计算法的推论。最后，我们展示了如何处理流中任意坐标函数（对于任意 $g \in \mathbb{G}$，其中 $\mathbb{G}$ 包含所有线性或非线性收缩函数）。