A Robinson space is a dissimilarity space $(X,d)$ on $n$ points for which there exists a compatible order, {\it i.e.} a total order $<$ on $X$ such that $x<y<z$ implies that $d(x,y)\le d(x,z)$ and $d(y,z)\leq d(x,z)$. Recognizing if a dissimilarity space is Robinson has numerous applications in seriation and classification. A PQ-tree is a classical data structure introduced by Booth and Lueker to compactly represent a set of related permutations on a set $X$. In particular, the set of all compatible orders of a Robinson space are encoded by a PQ-tree. An mmodule is a subset $M$ of $X$ which is not distinguishable from the outside of $M$, {\it i.e.} the distances from any point of $X\setminus M$ to all points of $M$ are the same. Mmodules define the mmodule-tree of a dissimilarity space $(X,d)$. Given $p\in X$, a $p$-copoint is a maximal mmodule not containing $p$. The $p$-copoints form a partition of $X\setminus \{p\}$. There exist two algorithms recognizing Robinson spaces in optimal $O(n^2)$ time. One uses PQ-trees and one uses a copoint partition of $(X, d)$. In this paper, we establish correspondences between the PQ-trees and the mmodule-trees of Robinson spaces. More precisely, we show how to construct the mmodule-tree of a Robinson dissimilarity from its PQ-tree and how to construct the PQ-tree from the odule-tree. To establish this translation, additionally to the previous notions, we introduce the notions of $\delta$-graph $G_\delta$ of a Robinson space and of $\delta$-mmodules, the connected components of $G_\delta$. We also use the dendrogram of the subdominant ultrametric of $d$. All these results also lead to optimal $O(n^2)$ time algorithms for constructing the PQ-tree and the mmodule tree of Robinson spaces.

翻译：罗宾逊空间是一个在n个点上的相异度空间$(X,d)$，存在一个相容序，即$X$上的全序$<$使得当$x<y<z$时，有$d(x,y)\le d(x,z)$且$d(y,z)\leq d(x,z)$。判断相异度空间是否为罗宾逊空间在序列化与分类中具有广泛应用。PQ-树是Booth与Lueker提出的经典数据结构，用于紧凑表示集合$X$上的一组相关排列。特别地，罗宾逊空间的所有相容序均由一棵PQ-树编码。mmodule是$X$的子集$M$，无法从$M$外部区分，即从$X\setminus M$中任意点到$M$中所有点的距离均相等。mmodule定义了相异度空间$(X,d)$的mmodule树。给定$p\in X$，$p$-copoint是不包含$p$的极大mmodule。所有$p$-copoints构成$X\setminus \{p\}$的一个划分。目前存在两种最优$O(n^2)$时间复杂度的罗宾逊空间识别算法：一种使用PQ-树，另一种使用$(X,d)$的copoint划分。本文建立了罗宾逊空间中PQ-树与mmodule树的对应关系。具体而言，我们展示了如何从PQ-树构建罗宾逊相异度的mmodule树，以及如何从mmodule树构建PQ-树。为建立这种转换，除已有概念外，我们引入了罗宾逊空间的$\delta$-图$G_\delta$和$\delta$-mmodule（即$G_\delta$的连通分量）的概念，并利用了$d$的次优超度量对应的树状图。所有这些结果也给出了罗宾逊空间PQ-树与mmodule树的最优$O(n^2)$时间构造算法。