Journal of Mathematical Biology

, Volume 77, Issue 3, pp 571–594 | Cite as

Recovering normal networks from shortest inter-taxa distance information

  • Magnus Bordewich
  • Katharina T. Huber
  • Vincent Moulton
  • Charles SempleEmail author


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 have no “shortcuts”, that is, the networks are normal, 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.


Distance matrix Tree–child network Normal network 

Mathematics Subject Classification

05C85 68R10 



We thank the anonymous referees for their helpful comments. Katharina Huber and Vincent Moulton also thank the Biomathematics Research Centre at the University of Canterbury for its hospitality.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Magnus Bordewich
    • 1
  • Katharina T. Huber
    • 2
  • Vincent Moulton
    • 2
  • Charles Semple
    • 3
    Email author
  1. 1.Department of Computer ScienceDurham UniversityDurhamUK
  2. 2.School of Computing SciencesUniversity of East AngliaNorwichUK
  3. 3.School of Mathematics and StatisticsUniversity of CanterburyChristchurchNew Zealand

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