The Split Decomposition of a k-Dissimilarity Map

TL;DR

This paper introduces a canonical way to decompose k-dissimilarity maps into simpler components using regular subdivisions of the kth hypersimplex, generalizing the split decomposition known for distances, with applications in phylogenetics.

Contribution

It extends the split decomposition method from distances to general k-dissimilarity maps using geometric subdivisions, providing a new characterization and proof techniques.

Findings

Decomposition of k-dissimilarity maps via hypersimplex subdivisions

Characterization of splits in the decompositions

New proof for reconstructing k-dissimilarity maps from trees

Abstract

A k-dissimilarity map on a finite set X is a function D : X \choose k \rightarrow R assigning a real value to each subset of X with cardinality k, k \geq 2. Such functions, also sometimes known as k-way dissimilarities, k-way distances, or k-semimetrics, are of interest in many areas of mathematics, computer science and classification theory, especially 2-dissimilarity maps (or distances) which are a generalisation of metrics. In this paper, we show how regular subdivisions of the kth hypersimplex can be used to obtain a canonical decomposition of a k-dissimilarity map into the sum of simpler k-dissimilarity maps arising from bipartitions or splits of X. In the special case k = 2, this is nothing other than the well-known split decomposition of a distance due to Bandelt and Dress [Adv. Math. 92 (1992), 47-105], a decomposition that is commonly to construct phylogenetic trees and networks. Furthermore, we characterise those sets of splits that may occur in the resulting decompositions of k-dissimilarity maps. As a corollary, we also give a new proof of a theorem of Pachter and Speyer [Appl. Math. Lett. 17 (2004), 615-621] for recovering k-dissimilarity maps from trees.