On restricted min‐wise independence of permutations

Abstract

AbstractA family of permutations ℱ ⊆ Sn with a probability distribution on it is called k‐restricted min‐wise independent if we have Pr[min π(X) = π(x)] = 1/|X| for every subset X ⊆ [n] with |X| ≤ k, every x ∈ X, and π ∈ ℱ chosen at random. We present a simple proof of a result of Norin: every such family has size at least Some features of our method might be of independent interest. The best available upper bound for the size of such family is 1 + ∑ (j − 1)(). We show that this bound is tight if the goal is to imitate not the uniform distribution on Sn, but a distribution given by assigning suitable priorities to the elements of [n] (the stationary distribution of the Tsetlin library, or self‐organizing lists). This is analogous to a result of Karloff and Mansour for k‐wise independent random variables. We also investigate the cases where the min‐wise independence condition is required only for sets X of size exactly k (where we have only an Ω(log log n + k) lower bound), or for sets of size k and k − 1 (where we already obtain a lower bound of n − k + 2). © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2003