Approximately Minwise Independence with Twisted Tabulation

Abstract

A random hash function $h$ is $\epsilon$-minwise if for any set $S$, $|S|=n$, and element $x\in S$, $\mathrm{Pr}[h(x)=\min h(S)]=(1\pm \epsilon )/n$. Minwise hash functions with low bias $\epsilon$ have widespread applications within similarity estimation. Hashing from a universe $[u]$, the twisted tabulation hashing of Pǎtraşcu and Thorup [SODA'13] makes $c=O(1)$ lookups in tables of size $u^{1/c}$. Twisted tabulation was invented to get good concentration for hashing based sampling. Here we show that twisted tabulation yields $\tilde{O}(1/u^{1/c})$-minwise hashing. In the classic independence paradigm of Wegman and Carter [FOCS'79] $\tilde{O}(1/u^{1/c})$-minwise hashing requires $\Omega (\log u)$-independence [Indyk SODA'99]. Pǎtraşcu and Thorup [STOC'11] had shown that simple tabulation, using same space and lookups yields $\tilde{O}(1/n^{1/c})$-minwise independence, which is good for large sets, but useless for small sets. Our analysis uses some of the same methods, but is much cleaner bypassing a complicated induction argument.

To appear in Proceedings of SWAT 2014