On the sample size of k-restricted min-wise independent permutations and other k-wise distributions

• Authors: Toshiya Itoh;Yoshinori Takei;Jun Tarui
• Venue: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
• Year: 2003

An explicit study of min-wise independent permutation families, together with their variants --- k-restricted, approximate, etc. --- was initiated by Broder, et al[4]. In this paper, we give a lower bound for the size of k-restricted min-wise independent permutation family. A family F of permutations on [0,n-1]=(0,1,...,n-1) is said to be k-restricted min-wise independent if for any subset X ⊆ [0,n-1] with |X| ≤ k and any x ∈ X, Pr[min(π(X))=π(x)] = 1/|X|, when π is randomly chosen from F according to a probability distribution D on the family F. For the minimum size of a family of k-restricted min-wise independent permutations, upper bounds of O(nk) for any fixed k have been shown for uniform and biased probability distributions on F. We show that if a family F of permutations on [0,n-1] is k-restricted min-wise independent, then |F| ≥ m(n-1,k-1), where m(n,d) = ∑i=0d/2(ni) if d is even; m(n,d)= ∑i=0(d-1)/2(ni) + (n-1(d-1)/2) otherwise. The lower bound for the size of F still holds when we allow an arbitrary probability distribution on F. Our proof technique is based on linear algebra methods, and can be regarded as a generalization of the result by Alon, Babai, and Itai[1], i.e., if random variables X1,X2,...,Xn: Ω → (0,1) are k-wise independent and Pr[Xi=1] = pi is neither 0 nor 1, then |Ω| ≥ m(n,k). By applying our proof technique, we also derive lower bounds for the sample size of the related notions, e.g., k-wise symmetrically independent distributions, k-rankwise independent permutation families, etc.