Affine and Projective Tree Metric Theorems

TL;DR

This paper unifies tree and circular split system metrics using PC-trees and PQ-trees, extending concepts from clustering theory and providing a broader framework for understanding these metrics.

Contribution

It introduces a unifying combinatorial framework based on PC-trees and PQ-trees that generalizes tree metric theorems and related concepts.

Findings

Generalizes tree metric theorems using PC-trees and PQ-trees

Establishes links between PC-trees, PQ-trees, and Gromov products

Extends clustering concepts to the projective case

Abstract

The tree metric theorem provides a combinatorial four point condition that characterizes dissimilarity maps derived from pairwise compatible split systems. A similar (but weaker) four point condition characterizes dissimilarity maps derived from circular split systems (Kalmanson metrics). The tree metric theorem was first discovered in the context of phylogenetics and forms the basis of many tree reconstruction algorithms, whereas Kalmanson metrics were first considered by computer scientists, and are notable in that they are a non-trivial class of metrics for which the traveling salesman problem is tractable. We present a unifying framework for these theorems based on combinatorial structures that are used for graph planarity testing. These are (projective) PC-trees, and their affine analogs, PQ-trees. In the projective case, we generalize a number of concepts from clustering theory, including hierarchies, pyramids, ultrametrics and Robinsonian matrices, and the theorems that relate them. As with tree metrics and ultrametrics, the link between PC-trees and PQ-trees is established via the Gromov product.