Min-Wise Independent Families with Respect to any Linear Order

Abstract

A set of permutations 𝒮 on a finite linearly ordered set Ω is said to be k-min-wise independent, k-MWI for short, if Pr (min (π(X)) = π(x)) = 1/|X| for every X ⊆ Ω such that |X| ≤ k and for every x ∈ X. (Here π(x) and π(X) denote the image of the element x or subset X of Ω under the permutation π, and Pr refers to a probability distribution on 𝒮, which we take to be the uniform distribution.) We are concerned with sets of permutations which are k-MWI families for any linear order. Indeed, we characterize such families in a way that does not involve the underlying order. As an application of this result, and using the Classification of Finite Simple Groups, we deduce a complete classification of the k-MWI families that are groups, for k ≥ 3.

Key Words:

• Linear order
• Min-wise independent family
• Permutation group

Mathematics Subject Classification:

• 20B10
• 05E99

ACKNOWLEDGMENT

The second author is supported by a grant from the Istituto Nazionale di Alta Matematica, INdAM.

Notes

Communicated by A. Turull.