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Triplet and Quartet Distances Between Trees of Arbitrary Degree Gerth Stølting Brodal Aarhus University Rolf Fagerberg University of Southern Denmark Thomas Mailund, Christian N. S. Pedersen, Andreas Sand Aarhus University, Bioinformatics Research Center ETH Zürich, Switzerland, 22 November 2012 (paper to be presented at SODA13)

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Evolutionary Tree BonoboChimpanzeeHumanNeanderthalGorillaOrangutan Time Rooted

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Unrooted Evolutionary Tree Dominant modern approach to study evolution is from DNA analysis

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Constructing Evolutionary Trees – Binary or Arbitrary Degrees ? Sequence data Distance matrix 123···n 1 2 3 n Neighbor Joining Saitou, Nei 1987 [ O(n 3 ) Saitou, Nei 1987 ] Refined Buneman Trees Moulton, Steel 1999 [ O(n 3 ) Brodal et al. 2003 ] Buneman Trees Buneman 1971 [ O(n 3 ) Berry, Bryan 1999 ] 1 2 3 ··· n.... Binary trees (despite no evidence in distance data) Arbitrary degrees (strong support for all edges ; few branches) Arbitrary degree (compromise ; good support for all edges)

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Data Analysis vs Expert Trees – Binary vs Arbitrary Degrees ? Linguistic expert classification (Aryon Rodrigues) Neighbor Joining on linguistic data Cultural Phylogenetics of the Tupi Language Family in Lowland South America. R. S. Walker, S. Wichmann, T. Mailund, C. J. Atkisson. PLoS One. 7(4), 2012.

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Evolutionary Tree Comparison ? split 1357|2468 1 4 3 2 5 6 7 T2T2 8 1 6 3 2 5 4 7 T1T1 8 CommonOnly T 1 Only T 2 1357|246835|12467857|123468 13567|24848|123567 Robinson-Foulds distance = # non-common splits = 2 + 1 = 3 [Day 1985] O(n) time algorithm using 2 x DFS + radix sort D. F. Robinson and L. R. Foulds. Comparison of weighted labeled trees. In Combinatorial mathematics, VI, Lecture Notes in Mathematics, pages 119–126. Springer, 1979.

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8 8 Robinson-Foulds Distance (unrooted trees) ? T1T1 CommonOnly T 1 Only T 2 (none)12567|348 1257|3468 157|23468 57|123468 125678|34 12578|346 1578|2346 578|12346 78|123456 1 6 2 5 4 7 T2T2 3 1 6 2 5 4 7 3 RF-dist(T 1, T 2 ) = 4 + 5 = 9 RF-dist(T 1 \{8}, T 2 \{8}) = 0 Robinson-Foulds very sensitive to outliers D. F. Robinson and L. R. Foulds. Comparison of weighted labeled trees. In Combinatorial mathematics, VI, Lecture Notes in Mathematics, pages 119–126. Springer, 1979.

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resolved : ij|kl Quartet Distance (unrooted trees) QuartetT1T1 T2T2 {1,2,3,4}14|23 {1,2,3,5}13|2515|23 {1,2,4,5}14|251245 {1,3,4,5}14|351345 {2,3,4,5}25|3423|45 i j k l i j k l unresolved : ijkl (only non-binary trees) 1 3 2 5 2 4 3 1 5 T1T1 T2T2 4 G. Estabrook, F. McMorris, and C. Meacham. Comparison of undirected phylogenetic trees based on subtrees of four evolutionary units. Systematic Zoology, 34:193-200, 1985.

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Triplet Distance (rooted trees) TripletT1T1 T2T2 {1,2,3}2|13 {1,2,4}1|244|12 {1,2,5}1|255|12 {1,3,4}4|13 {1,3,5}5|13 {1,4,5}1|45 {2,3,4}3|244|23 {2,3,5}3|255|23 {2,4,5}5|242|45 {3,4,5}3|45 resolved : k|ij ij k i j k unresolved : ijk (only non-binary trees) 1 2 5 3 T1T1 4 4 1 5 2 T2T2 3 D. E. Critchlow, D. K. Pearl, C. L. Qian: The triples distance for rooted bifurcating phylogenetic trees. Systematic Biology, 45(3):323-334, 1996.

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Rooted Triplet distance Unrooted Quartet distance Binary O(n 2 ) O(n log n) CPQ 1996 [SODA 2013] O(n 3 ) O(n 2 ) O(n log 2 n) O(n log n) D 1985 BTKL 2000 BFP 2001 BFP 2003 Degrees d O(n 2 ) O(n log n) BDF 2011 [SODA 2013] O(d 9 n log n) O(n 2.688 ) O(d n log n) SPMBF 2007 NKMP 2011 [SODA 2013] Computational Results 1 2 5 3 4 1 3 2 5 4 1 2 5 3 46 7 12 3 1 10 6 7 13 11 5 8 9

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Distance Computation T 2 ResolvedUnresolved T1T1 Resolved A : Agree C B : Disagree UnresolvedDE ij k i j ki j ki j k ij k jk i ik j ik j ij k jk i O( n ) triplets & quartets O( n ·log n ) triplets O( n ·log n ) triplets & quartets O( d · n ·log n ) quartets

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T 2 ResolvedUnresolved T1T1 Resolved A : Agree C B : Disagree UnresolvedDE ij k i j ki j ki j k ij k jk i ik j ik j ij k jk i Parameterized Triplet & Quartet Distances B + α·(C + D), 0 α 1 BDF 2011 O(n 2 ) for triplet, NKMP 2011 O(n 2.688 ) for quartet [SODA 13] O(n·log n) and O(d·n·log n), respectively

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Counting Unresolved Triplets in One Tree n 1 n 2 n 3 ··· n d v Computable in O(n) time using DFS + dynamic programming n 1 n 2 n 3 ··· n d v Triplet anchored at v Quartet anchored at v Quartets (root tree arbitrary)

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Counting Agreeing Triplets (Basic Idea) v 1i j d T1T1 j T2T2 c i w 0

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Efficient Computation Limit recolorings in T 1 (and T 2 ) to O(n·log n) v 1 1 1 0 v 1 2 d 0 v 1 0 v 0 v 1 0 v 1 0... Count T 2 contribution (precondition) Recolor Recurse Recolor & recurse T1T1 Reduce recoloring cost in T 2 to O(n·log 2 n) 1 2 3 4 5 6 7 89 1 2 4 7 9 85 63 Reduce recoloring cost in T 2 from O(n·log 2 n) to O(n·log n) T 2 arbitrary height degree H(T 2 ) binary height O(log n) Contract T 2 and reconstruct H(T 2 ) during recursion

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Counting Agreeing Triplets (II) v 1i j d T1T1 0 node in H(T 2 ) = component composition in T 2 + Contribution to agreeing triplets at node in H(T 2 ) i ij i i j j i i C2C2 C1C1

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From O(n·log 2 n) to O(n·log n) Update O(1) counters for all colors through node Colored path lengths v 1i j d T1T1 0 nvnv nini w Compressed version of T 2 of size O(n v ) Total cost for updating counters a (1) a (2) l=a (0) a (3) a (4) a (5) T1T1

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Counting Quartets... Bottleneck in computing disagreeing resolved-resolved quartets v 1i j d T1T1 0 i j i j G1G1 G2G2 T2T2 double-sum factor d time Root T 1 and T 2 arbitrary Keep up to 15+38d different counters per node in H(T 2 )...

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Distance Computation T 2 ResolvedUnresolved T1T1 Resolved A : Agree C B : Disagree UnresolvedDE ij k i j ki j ki j k ij k jk i ik j ik j ij k jk i O( n ) triplets & quartets O( n ·log n ) triplets O( n ·log n ) triplets & quartets O( d · n ·log n ) quartets

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Rooted Triplet distance Unrooted Quartet distance Binary O(n 2 ) O(n log n) CPQ 1996 [SODA 2013] O(n 3 ) O(n 2 ) O(n log 2 n) O(n log n) D 1985 BTKL 2000 BFP 2001 BFP 2003 Degrees d O(n 2 ) O(n log n) BDF 2011 [SODA 2013] O(d 9 n log n) O(n 2.688 ) O(d n log n) SPMBF 2007 NKMP 2011 [SODA 2013] Summary d = maximal degree of any node in T 1 and T 2 1 2 5 3 4 1 3 2 5 4 1 2 5 3 46 7 12 3 1 10 6 7 13 11 5 8 9 O(n·log n) ? o(n·log n) ?

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