Constructing a level-2 phylogenetic network from a dense set of input triplets

1. Constructing a level-2 phylogenetic network from a dense set of input triplets Leo van Iersel1, Judith Keijsper1, Steven Kelk2, Leen Stougie12 (1) Technische Universiteit Eindhoven (TU/e) (2) Centrum voor Wiskunde en Informatica (CWI), Amsterdam Email: S.M.Kelk@cwi.nl Web: http://homepages.cwi.nl/~kelk

2. Triplet-based methods (1)

3. Triplet-based methods (2)

4. Triplet-based methods (2)

5. From trees to networks�

6. From trees to networks (2)

7. From trees to networks (2)

8. From trees to networks (2)

9. Level-k phylogenetic networks

10. Level-1 Networks

12. Algorithm, basic idea The basic idea behind Aho�s algorithm for trees is that we are able to determine, recursively, which species belong to which of the two subtrees hanging from some root vertex. For the level-1 and level-2 networks if there again exists such a clear dichotomy, we iterate on the two subsets.

15. Algorithm, high-level idea For level-2 networks the idea is similar:

17. Definition: simple level-2 networks

19. Level-2 network algorithm Assume some oracle gives us the partition of the leaves into sub-networks Treat each subnetwork as a leaf and construct a simple level-2 network The simple level-2 network algorithm Guess the right �recombination leaf� Remove it and remove the triplets that contain this leaf 1 recombination vertex left with below it a caterpillar

22. Level-2 network algorithm Assume some oracle gives us the partition of the leaves into sub-networks Treat each subnetwork as a leaf and construct a simple level-2 network The simple level-2 network algorithm Guess the right �recombination leaf� Remove it and remove the triplets that contain this leaf 1 recombination vertex left with below it a caterpillar Guess the right �caterpillar set�

23. Caterpillar set A caterpillar set with respect to a dense triplet set T is the set of leaves of a caterpillar subgraph of a network consistent with T

25. Level-2 network algorithm Assume some oracle gives us the partition of the leaves into sub-networks Treat each subnetwork as a leaf and construct a simple level-2 network The simple level-2 network algorithm Guess the right �recombination leaf� Remove it and remove the triplets that contain this leaf 1 recombination vertex left with below it a caterpillar Guess the right �caterpillar set� Remove it and remove the triplets that contain any element of this set Construct the unique tree for the remaining triplets [Jansson&Sung 2006]

30. Level-2 network algorithm Assume some oracle gives us the partition of the leaves into sub-networks Treat each subnetwork as a leaf and construct a simple level-2 network The simple level-2 network algorithm Guess the right �recombination leaf� Remove it and remove the triplets that contain this leaf 1 recombination vertex left with below it a caterpillar Guess the right �caterpillar set� Remove it and remove the triplets that contain any element of this set Construct the unique tree for the remaining triplets [Jansson&Sung 2006] Insert the caterpillar set and the recombination leaf in the tree in the correct way For each pair of guesses try all 4 backbone structures

31. Simple level-2 algorithm Theorem: The simple level-2 network algorithm works in O(|T|^3)

32. SN-sets to partition the set of leaves

34. Definition highest cut-edge In a phylogenetic network N, a cut-edge (x,y) is an edge whose removal disconnects the undirected graph. A cut-edge (x,y) is said to be a trivial cut edge iff y is a leaf. A cut-edge (x,y) is said to be highest iff there is no cut-edge (p,q) such that there is a directed path from q to x in N.

37. The algorithm Determine the maximal SN-sets Guess the right SN-set to be split Treat the max SN-sets and the two split sets as leaves {S1,S2,�,Sq} Adapt T to a new triplet set T�: SiSk|Sh ? T� if and only if there exist x?Si, y?Sk,z?Sh s.t. xy|z ? T Construct a simple level-2 network for T� Recursively find the sub-networks for the sets S1,S2,�,Sq

38. Conclusions & open problems