Min-wise independent linear permutations

Abstract

A set of permutations F ⊆ Sn is min-wise independent if for any set X ⊆ [n] and any x ∈ X, when π is chosen at random in F we have ℙ (min{π(X)} = π(x)) = 1/[X] . This notion was introduced by Broder, Charikar, Frieze and Mitzenmacher and is motivated by an algorithm for filtering near-duplicate web documents. Linear permutations are an important class of permutations. Let p be a (large) prime and let Fp = {πa,b : 1 ≤ a ≤ p - 1, 0 ≤ b ≤ p - 1} where for x ∈ [p] = {0, 1, . . . , p - 1}, πa,b(x) = ax + b mod p. For X ⊆ [p] we let F(X) = maxx⊆X {ℙ a,b(min{π(X)} = (x))} where ℙa,b is over π chosen uniformly at random from Fp. We show that as k, p→1∞, EX[F(X)] = 1/k +O((log k)3/k 3/2) confirming that a simply chosen random linear permutation will suffice for an average set from the point of view of approximate min-wise independence.