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- W4380993914 abstract "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 $Xsetminus M$ to all points of $M$ are the same. Mmodules define the mmodule-tree of a dissimilarity space $(X,d)$. Given $pin X$, a $p$-copoint is a maximal mmodule not containing $p$. The $p$-copoints form a partition of $Xsetminus {p}$. 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." @default.
- W4380993914 created "2023-06-17" @default.
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- W4380993914 date "2023-06-14" @default.
- W4380993914 modified "2023-09-26" @default.
- W4380993914 title "Modules and PQ-trees in Robinson spaces" @default.
- W4380993914 doi "https://doi.org/10.48550/arxiv.2306.08800" @default.
- W4380993914 hasPublicationYear "2023" @default.
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