Three-way symbolic tree-maps and ultrametrics

TL;DR

This paper introduces a new class of three-way symbolic tree-maps and ultrametrics derived from leaf-labelled trees with interior vertex labels, providing characterizations and algorithms for their identification, with applications in phylogenetics.

Contribution

It generalizes existing two-way ultrametrics to three-way symbolic ultrametrics and tree-maps, offering new characterizations and decision algorithms for these structures.

Findings

Characterization of three-way symbolic tree-maps and ultrametrics via $k$-point conditions.

Equivalence of unrooted case characterization to Gurvich's hypergraph class.

Development of a triplet-based algorithm using the BUILD method for recognition.

Abstract

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 B\"ocker 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.