Automedian sets of permutations: direct sum and shuffle

Abstract

Given a set A⊆Sn of m permutations of {1,2,…,n} and a distance function d, the median problem consists in finding the set M(A) of all the permutations that are the “closest” of this set A. In this article we study the automedian case of the problem, i.e. when A=M(A), under the Kendall-τ distance. We show that automedian sets of permutations are closed under the direct sum operation and also, when some balancing properties are imposed on these sets, under the shuffle operation. These results allow us to derive a parallel algorithm that computes the medians of any separable set of permutations in O(k!+mn), where k is the length of its longest inseparable component.