Unbiased Insights: Optimal Streaming Algorithms for $\ell_p$ Sampling, the Forget Model, and Beyond
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914447137701888 |
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| author | Lin, Honghao Nguyen, Hoai-An Swartworth, William Woodruff, David P. |
| author_facet | Lin, Honghao Nguyen, Hoai-An Swartworth, William Woodruff, David P. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07067 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unbiased Insights: Optimal Streaming Algorithms for $\ell_p$ Sampling, the Forget Model, and Beyond Lin, Honghao Nguyen, Hoai-An Swartworth, William Woodruff, David P. Data Structures and Algorithms 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. |
| title | Unbiased Insights: Optimal Streaming Algorithms for $\ell_p$ Sampling, the Forget Model, and Beyond |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2508.07067 |