Reconstructing One-Articulated Networks with Distance Matrices

Abstract

Given a distance matrix M that represents evolutionary distances between any two species, an edge-weighted phylogenetic network N is said to satisfy M if between any pair of species, there exists a path in N with length equal to the corresponding entry in M. In this paper, we consider a special class of networks called 1-articulated network which is a proper superset of galled trees. We show that if the distance matrix M is derived from an ultrametric 1-articulated network N (i.e., for any species X and Y, the entry M(X, Y) is equal to the shortest distance between X and Y in N), we can re-construct an network that satisfies M in $O(n^2)$ time, where n denotes the number of species; furthermore, the reconstructed network is guaranteed to be the simplest, in a sense that the number of hybrid nodes is minimized. In addition, one may easily index a 1-articulated network N with minimum number of hybrid nodes in O(n) space, such that on given any phylogenetic tree T, we can determine if T is contained in N (i.e., if a spanning subtree $T'$ of N is a subdivision of T) in O(n) time.