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 \( \epsilon \geq 2 ^{-O(\log^{2/3} n)}\).
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© 1999 Springer-Verlag Berlin Heidelberg
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Saks, M., Srinivasan, A., Zhou, S., Zuckerman, D. (1999). Low Discrepancy Sets Yield Approximate Min-Wise Independent Permutation Families. In: Hochbaum, D.S., Jansen, K., Rolim, J.D.P., Sinclair, A. (eds) Randomization, Approximation, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 1999 1999. Lecture Notes in Computer Science, vol 1671. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-48413-4_2
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DOI: https://doi.org/10.1007/978-3-540-48413-4_2
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