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

A family of permutations F of [n] = {1, 2, . . . , n} is (ε, 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 F , π(x) is the minimum element of π(X), deviates from 1/|X| by at most ε/|X|. This notion can be defined for the uniform case, when the elements of F are picked according to a uniform distribution, or for the more general, biased case, in which the elements of 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 ε and all large n, the size of F must satisfy |F| ≥ Ω ( k ε2 log(1/ε) log n ) , as well as |F| ≥ Ω ( k ε log(1/ε) log n ) . This improves the best known previous estimates even for the uniform case. ∗School of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact Science, Tel Aviv University, Tel Aviv 69978, Israel. Research supported in part by a USA Israeli BSF grant, by a grant from the Israel Science Foundation, and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. †Global Scientific Information and Computing Center, Tokyo Institute of Technology, 2–12–1 O-okayama, Meguroku, Tokyo 152–8550, Japan. Research supported in part by Japan Society of the Promotion of Science, Grant-in-Aid for Scientific Research on Priority Areas, 16092205, 2006. ‡Advanced Technology R&D Center, Mitsubishi Electric Corporation, 8–1–1, Tsukaguchi-Honmachi, Amagasaki, Hyogo 661–8661, Japan.