Triplet and Quartet DistancesBetween Trees of Arbitrary Degree

Gerth Stølting BrodalAarhus University

Rolf FagerbergUniversity of Southern Denmark

Thomas Mailund, Christian N. S. Pedersen, Andreas SandAarhus University, Bioinformatics Research Center

ETH Zürich, Switzerland, 22 November 2012

(paper to be presented at SODA’13)

Evolutionary Tree

Bonobo Chimpanzee Human Neanderthal Gorilla Orangutan

Tim

e

Rooted

Unrooted Evolutionary Tree

Dominant modern approach to study evolution is from DNA analysis

Constructing Evolutionary Trees – Binary or Arbitrary Degrees ?

Sequence dataDistance matrix

1 2 3 ··· n123···

n

Neighbor JoiningSaitou, Nei 1987

[ O(n3) Saitou, Nei 1987 ]

Refined Buneman TreesMoulton, Steel 1999

[ O(n3) Brodal et al. 2003 ]

Buneman TreesBuneman 1971

[ O(n3) Berry, Bryan 1999 ]

123···

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)

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.

Evolutionary Tree Comparison

?

split1357|2468

1

4

32

5

67

T2

81

6

3

2

5

47

T1

8

Common Only T1 Only T2

1357|2468 35|124678 57|123468

13567|248 48|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 Combinatorialmathematics, VI, Lecture Notes in Mathematics, pages 119–126. Springer, 1979.

8

8

Robinson-Foulds Distance (unrooted trees)

?

T1

Common Only T1 Only T2

(none) 12567|3481257|3468157|2346857|123468

125678|3412578|3461578|2346578|1234678|123456

1

6

2

5

4

7T2

3

1

6

2

5

4

7

3

RF-dist(T1 , T2) = 4 + 5 = 9RF-dist(T1\{8} , T2\{8}) = 0

Robinson-Foulds very sensitive to outliers

D. F. Robinson and L. R. Foulds. Comparison of weighted labeled trees. In Combinatorialmathematics, VI, Lecture Notes in Mathematics, pages 119–126. Springer, 1979.

resolved : ij|kl

Quartet Distance (unrooted trees)

Consider all quartets, i.e. topologies of subsets of 4 leaves {i,j,k,l}

Quartet T1 T2

{1,2,3,4} 14|23 14|23{1,2,3,5} 13|25 15|23{1,2,4,5} 14|25 1245{1,3,4,5} 14|35 1345{2,3,4,5} 25|34 23|45

i

j

k

l

i

j

k

l

unresolved : ijkl(only non-binary trees)

Quartet-dist(T1 , T2) = - # common quartets = 5 - 1 = 4

1

3

2 5

24

3 1

5

T1 T2

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.

Triplet Distance (rooted trees)

Consider all triplets, i.e. topologies of subsets of 3 leaves {i,j,k}

Triplet T1 T2

{1,2,3} 2|13 2|13{1,2,4} 1|24 4|12{1,2,5} 1|25 5|12{1,3,4} 4|13 4|13{1,3,5} 5|13 5|13{1,4,5} 1|45 1|45{2,3,4} 3|24 4|23{2,3,5} 3|25 5|23{2,4,5} 5|24 2|45{3,4,5} 3|45 3|45

resolved : k|iji j

k i j k

unresolved : ijk(only non-binary trees)

Triplet-dist(T1 , T2) = - # common triplets = 10 - 5 = 5

1

2

53

T1

44

1

52

T2

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.

RootedTriplet distance

UnrootedQuartet distance

BinaryO(n2)

O(nlog n)CPQ 1996

[SODA 2013]

O(n3)O(n2)

O(nlog2 n)O(nlog n)

D 1985

BTKL 2000

BFP 2001

BFP 2003

Degrees d O(n2)

O(nlog n)BDF 2011

[SODA 2013]

O(d 9nlog n)

O(n2.688)O(dnlog n)

SPMBF 2007

NKMP 2011

[SODA 2013]

Computational Results

12

534 1

3

2 5

4

12

534 6

712

3 110

6 7 1311

58

9

Distance Computation T2

Resolved Unresolved

T1

ResolvedA : Agree

C

B : Disagree

Unresolved D E

i jk

i j k i j k

i j k

i jk

j ki

i kj

i kj

Triplet-dist(T1 , T2) = A – E = B + C + D

A + B + C + D + E = D + E and C + E unresolved in one tree

Sufficient to compute A and E or A and B

i jk

j ki

O(n) triplets & quartets

O(n·log n) triplets

O(n·log n) triplets & quartets

O(d·n·log n) quartets

T2

Resolved Unresolved

T1

ResolvedA : Agree

C

B : Disagree

Unresolved D E

i jk

i j k i j k

i j k

i jk

j ki

i kj

i kj

i jk

j ki

Parameterized Triplet & Quartet DistancesB + α·(C + D) , 0 α 1

BDF 2011 O(n2) for triplet, NKMP 2011 O(n2.688) for quartet[SODA 13] O(n·log n) and O(d·n·log n), respectively

Counting Unresolved Triplets in One Tree

n1 n2 n3 ··· nd

v∑v

∑i < j < k

n i ·n j·n k

∑v ( ∑

i < j <k <l

n i· n j· n k · n l+(n−∑l n l ) ∑i < j < k n i ·n j·n k )

Computable in O(n) time using DFS + dynamic programming

n1 n2 n3 ··· nd

v

Triplet anchored at v

Quartet anchored at v

Quartets (root tree arbitrary)

Counting Agreeing Triplets(Basic Idea)

v

1 i j d

∑v T1

∑wT 2

∑c

∑1≤i ≤d

(n i❑c

2 ) (nw−nc−n i❑w+n i

❑c )

T1

j

T2

c

i i

∑1≤ i≤d

n i❑w

w0

Efficient ComputationLimit recolorings in T1 (and T2) to O(n·log n)

v

1 1 1

0v

1 2 d

0v

1

0v

0

v

1

0

v

1

0...

Count T2 contribution

(precondition)

Recolor Recolor Recurse

Recolor & recurse

T1

Reduce recoloring cost in T2 to O(n·log2 n)

1

2 3

4

56

7 891

24

79

8563

Reduce recoloring cost in T2 from O(n·log2 n) to O(n·log n)

T2

arbitrary

height

degree

H(T2)

binary

height

O(log n)

Contract T2 and reconstruct H(T2) during recursion

Counting Agreeing Triplets (II)

v

1 i j d

T10

node in H(T2) =component

composition in T2

∑1≤ i≤d

(n i❑C1

2 )(n∗❑C2−n i❑C2 )∑

1≤ i≤d(n∗❑C1−n i

❑C1 ) n( ii )❑ C2∑1≤ i≤d

n i❑C1 ·n i↑∗

❑ C2+ +

Contribution to agreeing triplets at node in H(T2)

i

i j

ii

j

j

ii

C2

C1

From O(n·log2 n) to O(n·log n)Update O(1) counters for all

colors through node

∑2≤ i≤d

log (¿ T2∨¿n i )= ∑2≤i ≤d

n i ∙ lognv

n iColored path lengths

v

1 i j d

T10

nv

ni

w H(T2)

Compressed version of T2 of size O(nv)

Total cost for updating counters

∑leaf  l∈ T1

∑ancestor   a( j)not   heavy   child

logna ( j +1)

na ( j ) =  n ∙ log na(1)

a(2)

l=a(0)

a(3)

a(4) a(5)

T1

Counting Quartets...

Bottleneck in computing disagreeing resolved-resolved quartets

v

1 i j d

T10

i j i j

∑1≤ i<d  

∑i < j ≤ d

n ( ij )G1 ·n ( ij )G2G1 G2

T2

double-sum factor d time

Root T1 and T2 arbitrary Keep up to 15+38d different counters per node in H(T2)...

Distance Computation T2

Resolved Unresolved

T1

ResolvedA : Agree

C

B : Disagree

Unresolved D E

i jk

i j k i j k

i j k

i jk

j ki

i kj

i kj

Triplet-dist(T1 , T2) = A – E = B + C + D

A + B + C + D + E = D + E and C + E unresolved in one tree

Sufficient to compute A and E or A and B

i jk

j ki

O(n) triplets & quartets

O(n·log n) triplets

O(n·log n) triplets & quartets

O(d·n·log n) quartets

RootedTriplet distance

UnrootedQuartet distance

BinaryO(n2)

O(nlog n)CPQ 1996

[SODA 2013]

O(n3)O(n2)

O(nlog2 n)O(nlog n)

D 1985

BTKL 2000

BFP 2001

BFP 2003

Degrees dO(n2)

O(nlog n)BDF 2011

[SODA 2013]

O(d 9nlog n)

O(n2.688)O(dnlog n)

SPMBF 2007

NKMP 2011

[SODA 2013]

Summary

d = maximal degree of any node in T1 and T2

12

534 1

3

2 5

4

12

534 6

712

3 110

6 7 1311

58

9

O(n·log n) ?

o(n·log n) ? The End