The Complexity of the uspr Distance

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1 The omplexity of the uspr istance Maria Luisa onet Universtidad Politecnica de atalunya arcelona, Spain Published in I/M T 2010 Joint work with Katherine St. John Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

2 When are two trees similar? Nicotiana ampanula Scaevola Stokesia imorphotheca Senecio erbera azania chinops elicia Tagetes hromolaena lennosperma oreopsis Vernonia acosmia ichorium chillea arthamnus laveria Piptocarpa Helianthus Tragopogon hrysanthemum upatorium Lactuca arnadesia asyphyllum Nicotiana ampanula Scaevola Stokesia imorphotheca Senecio azania erbera chinops elicia Tagetes hromolaena lennosperma oreopsis Vernonia acosmia ichorium chillea arthamnus laveria Piptocarpa Helianthus Tragopogon hrysanthemum upatorium Lactuca arnadesia asyphyllum Phylogenies for sunflowers. ob Jansen (UT ustin). Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

3 istances etween Trees Robinson-oulds distance: # of branches that occur in only one tree. alculate in O(n) time using ay s lgorithm (1985). xtends naturally to weighted trees. Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

4 TR istance Tree-isection-Reconnect (TR) Move: The TR distance between two trees is the minimal number of TR moves needed to transform the first tree into the second tree. Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

5 TR istance Tree-isection-Reconnect (TR): TR is NP-hard. (llen & Steel 01) Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

6 SPR istance Subtree-prune-regraft (SPR): The SPR distance between two trees is the minimal number of SPR moves needed to transform the first tree into the second tree. d T R (T 1, T 2 ) d SP R (T 1, T 2 ) Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

7 SPR istance Subtree-prune-regraft (SPR): SPR for rooted trees is NP-hard. (ordewich & Semple 05) pproximation algorithm for SPR on rooted trees (onet, St. John, menta, & Mahindru 05) (orderwich, Mcartin, Semple). Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

8 Reduction Rules: Subtree Rule a a pplying Subtree Rule preserves TR distance (llen & Steel 01), SPR distance (llen &Steel 01, orderwich & Semple 04) and Hybrid number (orderwich & Semple 07). Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

9 Reduction Rules: hain Rule pplying the chain Rule preserves TR distance (llen & Steel 01), rspr distance (orderwich & Semple 04) and Hybrid number (orderwich & Semple 07). Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

10 $100 Problem Mike Steel posed the following questions: oes shrinking common chains preserve SPR distance? alculating SPR distance is fixed parameter tractable? alculating TR (llen & Steel 01), rspr (orderwich & Semple 04), or Hybrid Number (orderwich & Semple 07) is fixed parameter tractable. Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

11 Lower ounds for Subchain Reduction Hickey et al. showed that: alculating SPR distance is NP-hard. d usp R (T1 n, T2 n ) 2 d usp R (T1 3, T2 3 ). We improve this to: d usp R (T1 n, T2 n ) 1 d usp R (T1 3, T2 3 ) Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

12 SPR distance is PT We show that uspr is fixed parameter tractable with respect to parameter k: That is, for distance k, it can be decided in p(n)f(k) time that two trees are distance k, where p is a polynomial in n, the number of leaves in the tree, and f(k) does not depend on n. Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

13 Sketch: TR distance is PT Input: Pair of trees (T 1, T 2 ) and distance parameter k. Obtain (T 1, T 2) applying subtree and clain reduction. d T R (T 1, T 2 ) = d T R (T 1, T 2) unknown for SPR T 1 c.d T R (T 1, T 2 ) (lemma in llen-steel 2001) if T 1 > c.k then d T R (T 1, T 2 ) > k and return no. else check all posible k TR-moves from T 1 to T 2 and return yes or no. This can be done in time O(( T 1 2 ) k ) = O((d k) 2k ) Total time is O(k 2k p(n)). Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

14 New Reduction Rule: l-hain Rule l ck l Lemma: Let T 1 and T 2 be X-trees with SPR distance k. pplying the c k-chain Rule preserves SPR distance. ck Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

15 Size of Reduced Trees Lemma: Let T 1 and T 2 be X-trees with SPR distance k. Let T 1 and T 2 be the trees obtained from T 1 and T 2 applying subtree and c k-chain Rule. Then, T 1 dk 2. Proof: if d SP R (T 1, T 2 ) k then d T R (T 1, T 2 ) k. Show that if d T R (T 1, T 2 ) k, then T 1 dk 2 as in llen-steel 2001 but with c.k-chains instead of 3-chains. Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

16 Sketch: SPR distance is PT Input: Pair of trees (T 1, T 2 ) and distance parameter k. Obtain (T 1, T 2) applying subtree and ck-clain reduction. If d SP R (T 1, T 2 ) k, T 1 d k 2 and d SP R (T 1, T 2 ) = d SP R (T 1, T 2) if T 1 > d k 2 then d SP R (T 1, T 2 ) > k and return no. else check all posible k SPR-moves from T 1 to T 2 and return yes or no. This can be done in time O(( T 1 2 ) k ) = O((d k 2 ) 2k ) Total time is O(k 4k p(n)). Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

17 Idea for Improving Lower ound d usp R (T n 1, T n 2 ) 1 d usp R (T 3 1, T 3 2 ) Idea inspired from Hickey et al: treat common chains as subtrees to get 2 bound. o carefully by cases on a minimal set of moves to improve the bound to 1. Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

18 Idea for Improving Lower ound Let T 1 and T 2 X-trees with common chain 1 l and SPR dist k. We show d SP R (T 3 1, T 3 2 ) k 1 by cases. ase sketch: the minimal number of moves breaks 2 or 3 pendant edges from the 3-chain T 1 l+2... l+2 m 0 T m, m,..., m 1 2 k+1 k +1 T m T l+2 2 l+2 l+2 l+1 l+2 l+1 Maria Luisa onet, Phylogenetics follow-up Workshop, INI, June

Walks in Phylogenetic Treespace

Walks in Phylogenetic Treespace lan Joseph aceres Samantha aley John ejesus Michael Hintze iquan Moore Katherine St. John bstract We prove that the spaces of unrooted phylogenetic trees are Hamiltonian