A nearly linear size 4-min-wise independent permutation family by finite geometries

Abstract

Informally, a family ℱ ⊆ Sn of permutations is k-restricted min-wise independent if for any X ⊆ [0, n-1] with |X| ≤ k, each x ∈ X is mapped to the minimum among π(X) equally likely, and a family ℱ ⊆ Sn of permutations is k-rankwise independent if for any X ⊆ [0, n - 1] with |X| ≤ k, all elements in X are mapped in any possible order equally likely. It has been shown that if a family ℱ ⊆ Sn of permutations is k-restricted min-wise (resp. k-rankwise) independent, then |ℱ| = Ω(n⌊(k-1)/2⌋) (resp. |ℱ| = Ω(n⌊k/2⌋)). In this paper, we construct families ℱ ⊆ Sn of permutations of which size are close to those lower bounds for k = 3, 4, i.e., we construct a family ℱ ⊆ Sn of 3-restricted (resp. 4-restricted) min-wise independent permutations such that |ℱ| = O(n lg2 n) (resp. |ℱ| = O(n lg3 n)) by applying the affine plane AG(2, q), and a family ℱ ⊆ Sn of 4-rankwise independent permutations such that |ℱ| = O(n3 lg6 n) by applying the projective plane PG(2, q). Note that if a family ℱ ⊆ Sn of permutations is 4-rankwise independent, then |ℱ| = Ω(n2). Since a family ℱ ⊆ Sn of 4-rankwise independent permutations is 4-restricted min-wise independent, our family ℱ ⊆ Sn of 4-restricted min-wise independent permutations is the witness that properly separates the notion of 4-rankwise independence and that of 4-restricted min-wise independence. © Springer-Verlag Berlin Heidelberg 2003.