AI 中文总结
本文明确了k-最小哈希所需有界独立性的紧程度，证明Θ(k+log1/δ)阶独立性是充分必要的，改进了上界并给出匹配下界，还分析了F₂上随机仿射哈希函数的误差特性。
AI 中文摘要
最小哈希及其k-最小扩展是采样、草图和相似度估计的基础工具，构造这类哈希族的标准方法是有界独立性。对于普通最小哈希，所需的独立性程度已完全明确：Θ(log1/δ)阶独立性是充分且必要的。然而对于k-最小哈希，此前最佳结果仅表明O(k log log1/δ + log1/δ)阶独立性足够，却无匹配的下界。本文给出k-最小哈希所需有界独立性的紧刻画，证明Θ(k + log1/δ)阶独立性是充分且必要的，这改进了此前的上界并提供了匹配的下界。因此，有界独立哈希族的标准构造的种子长度为O((k + log1/δ)log(N/δ))；特别地，对于任意多项式小误差δ和任意k=Ω(log N)，该构造达到最优种子长度O(k log N)。本文还研究了有限域F₂上的随机仿射哈希函数，表明尽管它们是两两独立的，即使对于普通最小哈希，也可能产生Ω(log n)的乘性误差。
英文摘要
Min-wise hashing and its $k$-min-wise extension are fundamental tools in sampling, sketching, similarity estimation, etc. A standard approach to constructing such families is bounded independence. For ordinary min-wise hashing, the required degree of independence is fully understood: $Θ(\log 1/δ)$-wise independence is both sufficient and necessary. For $k$-min-wise hashing, however, the best previous result only showed that $O(k\log\log1/δ+\log1/δ)$-wise independence suffices, with no matching lower bound. We give a tight characterization of the amount of bounded independence required for $k$-min-wise hashing, proving that $Θ(k+\log 1/δ)$-wise independence is both sufficient and necessary. This improves the previous upper bound and provides a matching lower bound. Consequently, the standard construction of bounded-independent hash families has seed length $O\bigl((k+\log 1/δ)\cdot\log(N/δ)\bigr)$. In particular, for polynomially small $δ$ and any $Ω(\log N) \le k \le N^{1 - c}$, it achieves the optimal seed length $O(k\log N)$. We further investigate two standard low-independence hash families. For random affine functions over $\mathbb{F}_2$, which form a pairwise independent family, we show that the multiplicative error is $Ω(\log N)$ even for ordinary min-wise hashing. For simple tabulation hashing, which is $3$-wise independent and performs well for ordinary min-wise hashing, we show that it incurs a multiplicative error $Ω(N)$ for $k$-min-wise hashing whenever $k\ge 4$.
CommentsThis paper merges and extends two concurrent and independent preprints: arXiv:2607.27157v1 and arXiv:2607.10255v2
