Recovering tree-child networks from shortest inter-taxa distance information

TL;DR

This paper demonstrates that weighted tree-child phylogenetic networks can be reconstructed from a quadratic number of shortest inter-taxa distances, enabling efficient algorithms for their recovery.

Contribution

It shows that ignoring redundant edges allows reconstruction of weighted tree-child networks from quadratic inter-taxa distances, with cubic-time algorithms.

Findings

Networks are determined by shortest distances between taxa.

Quadratic number of distances suffices for reconstruction.

Constructive proofs lead to cubic-time algorithms.

Abstract

Phylogenetic networks are a type of leaf-labelled, acyclic, directed graph used by biologists to represent the evolutionary history of species whose past includes reticulation events. A phylogenetic network is tree-child if each non-leaf vertex is the parent of a tree vertex or a leaf. Up to a certain equivalence, it has been recently shown that, under two different types of weightings, edge-weighted tree-child networks are determined by their collection of distances between each pair of taxa. However, the size of these collections can be exponential in the size of the taxa set. In this paper, we show that, if we ignore redundant edges, the same results are obtained with only a quadratic number of inter-taxa distances by using the shortest distance between each pair of taxa. The proofs are constructive and give cubic-time algorithms in the size of the taxa sets for building such weighted networks.