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, i.e. a total order < on X such that $x<y<z$ implies that $d(x,y)\le d(x,z)$ and $d(y,z)\le d(x,z)$. Recognizing whether a dissimilarity space is Robinson has numerous applications in seriation and classification. The set of all compatible orders of a Robinson space can be succintly represented by a PQ-tree, a classical data structure introduced by Booth and Lueker. An mmodule is a subset M of X which is not distinguishable from the outside of M, i.e. the distances from any point of $X\setminus M$ to all points of M are the same. The hierarchical structure of mmodules can also be represented by a tree: the mmodule-tree of a dissimilarity space $(X,d)$.

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 vice versa. To establish this translation, we introduce the notions of δ-graph $G_{\bar{\delta }}$ of a Robinson space and of δ-mmodules, the connected components of $G_{\bar{\delta }}$. It also involves the dendrogram of the subdominant ultrametric of d. All these results also lead to optimal $O\left(n^2\right)$ time algorithms for constructing the PQ-tree and the mmodule tree of Robinson spaces.