T-Robinson空间：结构、识别及在真实数据中的应用
T-Robinson Spaces: Structure, Recognition, and Applications to Real Data
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中文总结 AI 辅助
该研究提出T-Robinson空间这一Robinson空间的树基推广，建立其结构刻画，开发复杂度为O(Kn²)的识别算法，引入结构定量度量，并探讨其在真实关系数据分析中的应用。
中文摘要 AI 辅助
我们研究T-Robinson空间，这是Robinson空间基于树的推广，其中兼容树的每条路径都诱导出一个Robinson子空间。该框架将经典的Robinson表示概念从线性序扩展到树结构，支持分层和分支数据的建模。我们通过证明其与若干图和超图理论性质等价，建立了T-Robinson空间的完整组合刻画。特别地，我们证明一个不相似空间是T-Robinson空间当且仅当它的所有层图都是具有公共兼容树的对偶弦图。结合Brucker建立的超树刻画，这得到了关联簇、球和2球超图为超树的等价刻画。基于这些结构结果，我们开发了复杂度为\textit{O(K n\textit{\textasciicircum{}}2)}的识别算法，其中\textit{K}表示不相似空间的最小生成树数量，当\textit{K}保持适中时，该算法优于现有的基于超树的方法。我们还引入了T-Robinson结构的定量度量，用于评估任意不相似空间接受类树表示的程度。最后，我们讨论其在真实世界数据集上的应用，说明T-Robinson空间如何为关系数据的分析与组织提供可解释的框架。
英文摘要
We study \emph{$T$-Robinson spaces}, a tree-based generalization of Robinson spaces in which every path of a compatible tree induces a Robinson subspace. This framework extends the classical notion of Robinsonian representations from linear orderings to tree structures, allowing the modeling of hierarchical and branching data. We establish a complete combinatorial characterization of $T$-Robinson spaces by proving their equivalence with several graph- and hypergraph-theoretic properties. In particular, we show that a dissimilarity space is $T$-Robinson if and only if all its level graphs are dually chordal with a common compatible tree. Combined with the characterization of hypertrees established by Brucker~\cite{brucker2005hypertrees}, this yields the equivalent characterization in terms of the associated cluster, ball, and 2-ball hypergraphs being hypertrees. Building upon these structural results, we develop a recognition algorithm with complexity \(O(K n^{2})\), where \(K\) denotes the number of minimum spanning trees of the dissimilarity space, improving upon existing hypertree-based approaches whenever \(K\) remains moderate. We further introduce a quantitative measure of $T$-Robinson structure that evaluates the extent to which an arbitrary dissimilarity space admits a tree-like representation. Finally, we discuss applications to real-world datasets, illustrating how $T$-Robinson spaces provide an interpretable framework for analyzing and organizing relational data.
