Constructions of $k$-Min-Wise Hash from Bounded Independence

Min-wise hashing and its $k$-min-wise extension are fundamental tools in sampling, sketching, and similarity estimation. A standard approach to constructing such families is bounded independence. For ordinary min-wise hashing, the required degree of independence is fully understood: $\mathrm{\Theta }(\log 1\mathrm{/}\delta )$-wise independence is both sufficient and necessary. For $k$-min-wise hashing, however, the best previous result only showed that $O(k\log \log 1\mathrm{/}\delta +\log 1\mathrm{/}\delta )$-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 $\mathrm{\Theta }(k+\log 1\mathrm{/}\delta )$-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 $O((k+\log 1\mathrm{/}\delta )\log (N\mathrm{/}\delta ))$. In particular, for any polynomially small error $\delta$ and any $k=\mathrm{\Omega }(\log N)$, it achieves the optimal seed length $O(k\log N)$. We also study random affine hash functions over ${\mathbb{F}}_2$ and show that, despite being pairwise independent, they may incur multiplicative error $\mathrm{\Omega }(\log n)$ even for ordinary min-wise hashing.