  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.
