Discrete time trees:
theory and open problems

Alex Gavryushkin

Joint work with:
  • Chris Whidden, Fred Hutchinson Cancer Research Center, Seattle, WA, USA
  • Erick Matsen, FHCRC, U. of Washington, Seattle, WA, USA
  • Alexei Drummond, U. of Auckland, NZ

November 15, 2016

(Discrete) Time tree

Sampled ancestor tree

NNI graph

Why is this important?

  • Tree search algorithms
  • Model testing/selection and other simulation studies

Why is this hard?

Trees are many!

Discrete time tree space

Sampled ancestor tree space

Graph = Metric

Trees at distance 2

Trees at distance 4

Main idea

(my failed proof)

History of the NNI graph

  • Over 25 year of work!
  • Over 7 erroneous papers published!

What's wrong with the NNI graph?

  1. The Split Theorem

  2. The merge and sort trick

Merge and sort trick

Merge and sort trick

RNNI is free from all these!

  • Split theorem. Tick.
  • Merge and sort doesn't work. Tick
  • Efficient polynomial algorithm?

What is an approximate (?) algorithm

$\frac{1}{2} \log_3 \frac{(n-1)!n!}{6^{n-1}} \leq \mathrm{Diam} \leq n^2 - 3n - \frac{5}{8}$

What we've done?

  • Introduced the RNNI graph on ranked trees (to the best of our knowledge)

  • Established basic geometric properties of the graph

  • Designed an efficient approximate algorithm for computing shortest paths

  • Proved that all the fancy NNI methods, e.g. Sleator-Tarjan-Thurston and merge-and-sort argument, don't work

  • Failed to prove that RNNI is NP-hard

What has to be done?

  • Is RNNI polynomial? Complexity?
  • Split Theorem
  • Are these two related?
  • ...

What about branch lengths?

Hence, time trees are (more or less) fine

What about sampled ancestor trees?

Looks like a problem

[Stadler, JTB 2010]  must be cheating then :)

But then we can cheat a bit too!

... and introduce imaginary nodes


  • Sleator, Tarjan, and Thurston. Short Encodings of Evolving Structures. (1992)
  • Dasgupta, He, Jiang, Li, Tromp, and Zhang. On Computing the Nearest Neighbor Interchange Distance. (1999)
  • Stadler. Sampling-through-time in birth–death trees. (2010)
  • Gavryushkin and Drummond. The space of ultrametric phylogenetic trees. (2015)
  • Gavryushkin, Whidden, and Matsen. The combinatorics of discrete time trees: theory and open problems. (bioRxiv, 2016)

Thanks for your attention!