Approximately Minwise Independence with Twisted Tabulation

TL;DR

This paper demonstrates that twisted tabulation hashing achieves near-minwise independence with low computational complexity, improving understanding of its effectiveness for similarity estimation in various set sizes.

Contribution

It proves that twisted tabulation hashing provides /u^{1/c}-minwise independence, a significant improvement over previous results, with a simpler analysis.

Findings

Achieves /u^{1/c}-minwise independence with constant-time lookups.

Requires only O(1) lookups and small space, suitable for large and small sets.

Simplifies previous complex analysis methods for minwise hashing.

Abstract

A random hash function $h$ is $\varepsilon$-minwise if for any set $S$, $|S|=n$, and element $x\in S$, $\Pr[h(x)=\min h(S)]=(1\pm\varepsilon)/n$. Minwise hash functions with low bias $\varepsilon$ have widespread applications within similarity estimation. Hashing from a universe $[u]$, the twisted tabulation hashing of P\v{a}tra\c{s}cu 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]. P\v{a}tra\c{s}cu 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.