三向符号树映射与超度量
Three-way symbolic tree-maps and ultrametrics
摘要：
三向不相似性是（双向）不相似性的概括，可用于表示任何三个对象之间缺乏同质性或相似性。这种图谱在聚类分析中有应用，并已被用于心理学和遗传学等领域，其中可能出现三向数据表。这种相异的特殊例子是三向树度量和超度量，它们产生于叶子标记树，其边由正实数标记。在这里，我们考虑三路映射产生的叶标记的树，而不是内部顶点标记的任意一组值。对于无根树有根树，我们称这种映射为三向符号树映射;对于有根树有根树，我们称它们为三向符号超度量，因为它们可以被认为是Böcker和Dress的（双向）符号超度量的推广。我们表明，与两个和三路树度量和超度量，三路符号树映射和超度量可以通过某些$k$ -点条件的特点。在无根的情况下，我们的特征在数学上等价于Gurvich提出的一类边标号超图的特征。我们还表明，它可以决定是否是一个任意的三路符号映射是一个树映射或符号超度量使用基于三元组的方法，依赖于所谓的BUILD算法，决定当一组3叶树或三元组可以显示一棵树。我们设想，我们的研究结果将是有用的，在开发新的方法和算法来理解3路数据，特别是在该地区的遗传学。
英文摘要：
Three-way dissimilarities are a generalization of (two-way) dissimilarities which can be used to indicate the lack of homogeneity or resemblance between any three objects. Such maps have applications in cluster analysis, and have been used in areas such as psychology and phylogenetics, where three-way data tables can arise. Special examples of such dissimilarities are three-way tree-metrics and ultrametrics, which arise from leaf-labelled trees with edges labelled by positive real numbers. Here we consider three-way maps which arise from leaf-labelled trees where instead the interior vertices are labelled by an arbitrary set of values. For unrooted trees we call such maps three-way symbolic tree-maps; for rooted trees we call them three-way symbolic ultrametrics since they can be considered as a generalization of the (two-way) symbolic ultrametrics of Böcker and Dress. We show that, as with two- and three-way tree-metrics and ultrametrics, three-way symbolic tree-maps and ultrametrics can be characterized via certain $k$-point conditions. In the unrooted case, our characterization is mathematically equivalent to one presented by Gurvich for a certain class of edge-labelled hypergraphs. We also show that it can be decided whether or not an arbitrary three-way symbolic map is a tree-map or a symbolic ultrametric using a triplet-based approach that relies on the so-called BUILD algorithm for deciding when a set of 3-leaved trees or triplets can be displayed by a single tree. We envisage that our results will be useful in developing new approaches and algorithms for understanding 3-way data, especially within the area of phylogenetics.
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