Modules and PQ-trees in Robinson spaces

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.