Given a set $\mathcal{A}\subseteq {\mathbb{S}}_n$ of m permutations of $\{1,2,\dots ,n\}$ and a distance function d, the median problem consists in finding the set $\mathcal{M}(\mathcal{A})$ of all the permutations that are the “closest” of this set $\mathcal{A}$. In this article we study the automedian case of the problem, i.e. when $\mathcal{A}=\mathcal{M}(\mathcal{A})$, 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 $\mathcal{O}(k!+mn)$, where k is the length of its longest inseparable component.

给定一套 $\mathcal{一种}\subseteq {\mathbb{小号}}_{ñ}$的m个排列$\{\mathrm{1个}，2，\dots ，ñ\}$和距离函数d，中位数问题在于找到集合$\mathcal{中号}（\mathcal{一种}）$ 这组“最接近”的所有排列中 $\mathcal{一种}$。在本文中，我们研究问题的自动中值情况，即何时$\mathcal{一种}=\mathcal{中号}（\mathcal{一种}）$，在Kendall- τ距离下。我们证明了自动中位数排列在直接和运算下是封闭的，并且在对这些集合施加一些平衡属性时在随机运算下是封闭的。这些结果使我们能够导出一个并行算法，该算法可计算出任何可分离的排列集中的中位数$\mathcal{Ø}（ķ！+米ñ）$，其中k是其最长不可分成分的长度。