The split decomposition of a k-dissimilarity map

A k-dissimilarity map on a finite set X is a function D:(Xk)->R assigning a real value to each subset of X with cardinality k, k>=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 [H.-J. Bandelt, A.W.M. Dress, A canonical decomposition theory for metrics on a finite set, 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 [L. Pachter, D.E. Speyer, Reconstructing trees from subtree weights, Appl. Math. Lett. 17 (2004) 615-621] for recovering k-dissimilarity maps from trees.