On (ε,k)-min-wise independent permutations

A family of permutations of $\cal{F}$ [n] = l1,2,…,nr is (e,k)-min-wise independent if for every nonempty subset X of at most k elements of [n], and for any x ∈ X, the probability that in a random element π of $\cal{F}$, π(x) is the minimum element of π(X), deviates from 1/mXm by at most e/mXm. This notion can be defined for the uniform case, when the elements of $\cal{F}$ are picked according to a uniform distribution, or for the more general, biased case, in which the elements of $\cal{F}$ are chosen according to a given distribution D. It is known that this notion is a useful tool for indexing replicated documents on the web. We show that even in the more general, biased case, for all admissible k and e and all large n, the size of $\cal{F}$ must satisfy $$|{\cal{F}}| \ge \Omega \left({k \over \varepsilon^2\log(1/\varepsilon)} \log n\right),$$ as well as $$|{\cal{F}}| \ge \Omega \left({k^2 \over \varepsilon\log(1/\varepsilon)} \log n\right).$$ This improves the best known previous estimates even for the uniform case. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007