Low Discrepancy Sets Yield Approximate Min-Wise Independent Permutation Families

Abstract

Motivated by a problem of filtering near-duplicate Web documents, Broder, Charikar, Frieze & Mitzenmacher defined the following notion of ε-approximate min-wise independent permutation families [2]. A multiset $\mathcal{F}$ of permutations of {0,1, ... , n–1} is such a family if for all K ⊆ {0,1, ..., n–1} and any x ∈ K, a permutation π chosen uniformly at random form $\mathcal{F}$ statisfies

$| Pr[min\{\pi(K)\} = \pi(x)] - \frac{1}{|K|}| \leq \frac{\epsilon}{|K|}$.

We show connections of such families with low discrepancy sets for geometric rectangles, and give explicit constructions of such families $\mathcal{F}$ of size $n ^{O(\sqrt{\log n})}$ for ε = 1 / n ^θ(1), improving upon the previously best-known bound of Indyk [4]. We also present polynomial-size constructions when the min-wise condition is required only for $\vert K\vert \leq 2 ^{O(\log^{2/3} n)}$, with $\epsilon \geq 2 ^{-O(\log^{2/3} n)}$.