The Split Decomposition of a k-Dissimilarity Map

Details The Split Decomposition of a k-Dissimilarity Map The Split Decomposition of a k-Dissimilarity Map

2010-08-10

arxiv.org/abs/1008.1703v1

Mathematics

Combinatorics

21 pages, 4 figures

Scientific paper

A k-dissimilarity map on a finite set X is a function D: \binom X k -> R assigning a real value to each subset of X with cardinality k. Such functions arise naturally in the area of phylogenetics, where they are commonly used to reconstruct evolutionary trees or networks. 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 decomposition is the well-known split decomposition of a distance due to Bandelt and Dress [Adv. Math. 92 (1992), 47-105]. 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.

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