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Geometry of Ranked Nearest Neighbour Interchange Space of Phylogenetic Trees

Lena Collienne, Kieran Elmes, Mareike Fischer, David Bryant, View ORCID ProfileAlex Gavryushkin
doi: https://doi.org/10.1101/2019.12.19.883603
Lena Collienne
1Department of Computer Science, University of Otago, New Zealand
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Kieran Elmes
1Department of Computer Science, University of Otago, New Zealand
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Mareike Fischer
2Institute of Mathematics and Computer Science, University of Greifswald, Germany
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David Bryant
3Department of Mathematics and Statistics, University of Otago, New Zealand
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Alex Gavryushkin
1Department of Computer Science, University of Otago, New Zealand
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  • ORCID record for Alex Gavryushkin
  • For correspondence: alex@biods.org
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Abstract

In this paper we study the graph of ranked phylogenetic trees where the adjacency relation is given by a local rearrangement of the tree structure. Our work is motivated by tree inference algorithms, such as maximum likelihood and Markov Chain Monte Carlo methods, where the geometry of the search space plays a central role for efficiency and practicality of optimisation and sampling. We hence focus on understanding the geometry of the space (graph) of ranked trees, the so-called ranked nearest neighbour interchange (RNNI) graph. We find the radius and diameter of the space exactly, improving the best previously known estimates. Since the RNNI graph is a generalisation of the classical nearest neighbour interchange (NNI) graph to ranked phylogenetic trees, we compare geometric and algorithmic properties of the two graphs. Surprisingly, we discover that both geometric and algorithmic properties of RNNI and NNI are quite different. For example, we establish convexity of certain natural subspaces in RNNI which are not convex is NNI. Our results suggest that the complexity of computing distances in the two graphs is different.

Footnotes

  • E-mail addresses: lena.collienne{at}postgrad.otago.ac.nz, kelmes{at}cs.otago.ac.nz, email{at}mareikefischer.de, david.bryant{at}otago.ac.nz.

  • A few improvements in text, plus new section on lattices.

Copyright 
The copyright holder for this preprint is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under a CC-BY 4.0 International license.
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Posted February 11, 2020.
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Geometry of Ranked Nearest Neighbour Interchange Space of Phylogenetic Trees
Lena Collienne, Kieran Elmes, Mareike Fischer, David Bryant, Alex Gavryushkin
bioRxiv 2019.12.19.883603; doi: https://doi.org/10.1101/2019.12.19.883603
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Geometry of Ranked Nearest Neighbour Interchange Space of Phylogenetic Trees
Lena Collienne, Kieran Elmes, Mareike Fischer, David Bryant, Alex Gavryushkin
bioRxiv 2019.12.19.883603; doi: https://doi.org/10.1101/2019.12.19.883603

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