## Low Discrepancy Sets Yield Approximate Min-Wise Independent Permutation Families

### Authors: Michael Saks, Aravind Srinivasan, Shiyu Zhou and David Zuckerman

ABSTRACT

Motivated by a problem of filtering near-duplicate Web documents, Broder, Charikar, Frieze \& Mitzenmacher defined the following notion of {\em $\epsilon$-approximate min-wise independent permutation families}. A multiset ${\cal F}$ of permutations of $\{0,1,\ldots,n-1\}$ is such a family if for all $K \subseteq \{0,1,\ldots,n-1\}$ and any $x \in K$, a permutation $\pi$ chosen uniformly at random from ${\cal F}$ satisfies $$|\ \Pr[\min\{\pi(K)\}=\pi(x)]-{1\over |K|}\ |\le{\eps\over |K|}.$$ We show connections of such families with {\em low discrepancy sets for geometric rectangles}, and give explicit constructions of such families ${\cal F}$ of size $n^{O(\sqrt{\log n})}$ for $\eps=1/n^{\Theta(1)}$, improving upon the previously best-known bound of Indyk. We also present polynomial-size constructions when the min-wise condition is required only for $|K| \le 2^{O(\log^{2/3} n)}$, with $\epsilon \geq 2^{-O(\log^{2/3} n)}$.

Paper Available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1999/99-34.ps.gz