A Nearly Linear Size 4-Min-Wise Independent Permutation Family by Finite Geometries

Abstract

Informally, a family ${\cal F} \subseteq S_{n}$ 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 ${\cal F} \subseteq S_{n}$ 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 ${\cal F} \subseteq S_{n}$ of permutations is k-restricted min-wise (resp. k-rankwise) independent, then $|{\cal F}| = \Omega(n^{\lfloor{(k-1)/2}\rfloor})$ (resp. $|{\cal F}| = \Omega(n^{\lfloor{k/2}\rfloor})$). In this paper, we construct families ${\cal F} \subseteq S_{n}$ of permutations of which size are close to those lower bounds for k=3,4, i.e., we construct a family ${\cal F} \subseteq S_{n}$ of 3-restricted (resp. 4-restricted) min-wise independent permutations such that $|{\cal F}| = O(n\lg^{2}n)$ (resp. $|{\cal F}| = O(n\lg^{3}n)$) by applying the affine plane AG(2,q), and a family ${\cal F} \subseteq S_{n}$ of 4-rankwise independent permutations such that $|{\cal F}| = O(n^{3} \lg^{6}n)$ by applying the projective plane PG(2,q). Note that if a family ${\cal F} \subseteq S_{n}$ of permutations is 4-rankwise independent, then $|{\cal F}| = \Omega(n^{2})$. Since a family ${\cal F} \subseteq S_{n}$ of 4-rankwise independent permutations is 4-restricted min-wise independent, our family ${\cal F} \subseteq S_{n}$ 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.