Communications in Algebra

Min-Wise Independent Families with Respect to any Linear Order

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DOI:: 10.1080/00927870701404812

Peter J. Cameron^a^* & Pablo Spiga^a

pages 3026-3033

Available online: 25 Sep 2007

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Abstract

A set of permutations 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 , 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.