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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 xK, 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 \( \epsilon \geq 2 ^{-O(\log^{2/3} n)}\).

keywords and phrases

Min-wise independent permutations document filtering pseudorandom permutations explicit constructions 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Michael Saks
    • 1
  • Aravind Srinivasan
    • 2
  • Shiyu Zhou
    • 3
  • David Zuckerman
    • 4
  1. 1.Department of MathematicsRutgers University, Hill CenterPiscatawayUSA
  2. 2.Bell Laboratories, Lucent TechnologiesMurray HillUSA
  3. 3.Department of Computer & Information ScienceUniversity of Pennsylvania 
  4. 4.Computer Science DivisionUniversity of CaliforniaBerkeleyUSA

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