1. By Patrick Prosser Presented by Chris Unsworth at CP06
Species Trees & Constraint Programming: recent progress and new challenges
By Patrick Prosser
Presented by Chris Unsworth at CP06

2. Outline Tree of life (what’s that then?)
Previous work (conventional and CP model)
What’s new? (enhanced model, new problems)
Conclusions (what have I told you!?)
Future work (will this never end?)

3. Tree of life A central goal of systematics construct the tree of life
a tree that represents the relationship between all living things
The leaf nodes of the tree are species
The interior nodes are hypothesized species
extinct, where species diverged

5. Not to be confused with this

6. Not to be confused with this

7. Not to be
confused with
this either

8. Something
like
this

15. To date, biologists have cataloged about 1
To date, biologists have cataloged about 1.7 million species yet estimates
of the total number of species ranges from 4 to 100 million.
“Of the 1.7 million species identified only about 80,000 species have
been placed in the tree of life”
E. Pennisi “Modernizing the Tree of Life” Science 300:

16. Properties of a Species Tree
We have a set of leaf nodes, each labelled with a species
the interior nodes have no labels (maybe)
each interior node has 2 children and one parent (maybe/ideally)
a bifurcating tree (maybe/ideally)
Note: recently there has been a requirements that
interior nodes have divergence dates
leaf nodes correspond to other trees (such as a leaf “cats”)
trees might not bifurcate

17. Super Trees We are given two trees, T1 and T2
S1 and S2 are the sets of leaves for T1 and T2 respectively
remember, leaves are species!
S1 and S2 have a non-empty intersection
some species appear in both trees
We want to combine T1 and T2
respecting the relationships in T1 and T2
form a “super tree”

18. superTree
combine

19. Overlap is highlighted in the trees and
the superTree

20. A simple wee example
Overlap is leafs “a” and “f”

21. Most Recent Common Ancestors (mrca)
mrca(a,c) = mrca(b,c) a b c
mrca(a,b)
We have 3 species, a, b, and c
Species a and b are more closely related
to each other than they are to c
mrca(a,b)  mrca(a,c)
mrca(a,b)  mrca(b,c) mrca(a,c)  mrca(b,c)
The most recent common ancestor of a and b
is further from the root than the most recent
common ancestor of a and c (and b and c)
a is closer to b than c
NOTE: mrca(x,y) = mrca(y,x)

22. Most Recent Common Ancestors (mrca)
mrca(a,c) = mrca(b,c) a b c
mrca(a,b)
mrca(a,b)  mrca(a,c)
mrca(a,b)  mrca(b,c) mrca(a,c)  mrca(b,c)
Note: this
defines that
Think of mrca(x,y) having integer value “depth”

23. Ultrametric relationship
Given 3 leaf nodes labelled a, b, and c there are
only 4 possible situations a b c a c b b c a b c a
fan
triples

27. That’s all that there can be, for 3 leafs

28. Another view a b c a b c a c b b c a
A space made up of triangles a b c
Given any three vertices the triangle is either
isosceles or equilateral

29. Ultrametric relationship
Given 3 leaf nodes labelled a, b, and c there are
only 4 possible situations
We can represent this using primitive constraints
Where D[i,j] is a constrained integer variable representing
the depth in the tree of the most recent common ancestor
of the ith and jth species

30. Ultrametric constraint
Therefore the ultrametric constraint is as follows
Constraint acting between leaf nodes/species a, b, and c
Where D[x,y] is depth in tree of mrca(x,y)
D[x,y] can also be thought of as distance

31. How it goes (part 1)
Conventional technology (circa 1981)
Take 2 species trees T1 and T2
Use the “breakUp” algorithm (Ng & Wormald 1996) on T1 then T2
- This produces a set of triples and fans
Use the “oneTree” algorithm (Ng & Wormald 1996)
- Generates a superTree or fails
This is the “conventional” (non-CP) approach
Different versions of oneTree and breakUp from Semple and Steel
(I think) that treats fans differently (ignores them)
oneTree is essentially the algorithm of Aho, Sagiv,
Szymanski and Ullman in SIAM J.Compt 1981

32. breakUp generates constraints!D E F G
1. Find deepest interior node
2. Get its descendants (leaf nodes)
3. Get a cousin or uncle leaf node
4. Generate a triple or fan
5. Delete one of the leafs in 2
6. Take the other leaf in 2 and make its parent that leaf
7. Go to 1 unless we are at the root with degree 2

33. breakUp generates constraints!D E F G
A deepest interior node
Generate triple AB|C
This is the constraint D[A,C] = D[B,C] < D[A,B]

34. breakUp generates constraints!D E F G
A deepest interior node
Generate triple DE|C
This is the constraint D[D,C] = D[E,C] < D[D,E]

35. breakUp generates constraints!F G C E B
A deepest interior node
Generate fan BCE
This is the constraint D[B,C] = D[B,E] = D[C,E]

36. breakUp generates constraints!F G
A deepest interior node
Generate triple FG|E
This is the constraint D[E,F] = D[F,G] < D[F,G]

37. breakUp generates constraints!
Done
The triples and fans can be viewed as constraints that break
the ultrametric disjunctions

38. The 1st CP approach

39. How it goes (part 2)
CP approach (circa 2003)
Generate an n by n array of constrained integer variables
For all 0<i<j<k<n post the ultrametric constraint
- Yes, we have a cubic number of constraints
- Yes, we have a quadratic number of variables
- This gives us an “ultrametric matrix”
Use breakUp on trees T1 and T2 to produce triples and fans
Post the triples and fans as constraints, breaking disjunctions
Find a first solution
Convert the ultrametric matrix to an ultrametric tree
Algorithm for ultrametric matrix to ultrametric tree
given by Dan Gusfield
This is the CP approach proposed by Gent, Prosser, Smith & Wei
in CP03 (a great great paper, go read it )

40. Key here is that we have an array of variables
Representing distances and this space must be
ultrametric

41. An min ultrametric tree and its min ultrametric matrix
Matrix value is the value
of the most recent common
ancestor of two leaf nodes 3 4 5 B 8 C D E A
As we go down a branch
values on interior nodes increase
Matrix is symmetric

42. The state of play in 2003
Coded up in claire & choco
more a ”proof of concept” than a useful tool
small data sets only

43. Two species trees of sea birds from the CP03 paper

44. On the left by oneTree and on the right by CP model
Resultant superTree
On the left by oneTree and on the right by CP model

45. What’s new
2006
Reimplemented in java & JChoco (so faster)
More robust (thanks to Pierre Flener’s help)
Can now deal with larger trees (about 70 species)
Can generate all solutions up to symmetry
Can handle divergence dates on interior nodes
Reimplemented breakUp & oneTree in Java
All code available on the web

47. Bigger Trees
Attempted to reconstruct the supertree in Kennedy & Page’s
“Seabird supertrees: Combining partial estimates of
rocellariiform phylogeny” in “The Auk: A Quarterly Journal of
Ornithology” 119:
7 trees of seabirds (A through G)
Varying in size from 14 to 90 species

48. From the paper
Table shows on the diagonal the size of each tree, A through G
A table entry is the size of the combined tree
A table entry in () if trees are incompatible
A table entry of – if trees are too big for CP model
The only compatible trees are A, B, D and F
The resultant supertree has 69 species
This takes 20 seconds to produce

50. A “lifted” representation
Rather than instantiate the “D” variables
why not just break the disjunctions?
Now the decision variables are P[i,j,k]
And yes, we have a cubic number of P variables

51. A “lifted” representation Rather than instantiate the “D” variables
why not just break the disjunctions?
Now the decision variables are P[i,j,k]
Now we can:
Enumerate all solutions eliminating value symmetries
Allow ranges of values on interior nodes of trees
- input and output!

52. Ranked Trees
A new problem where input trees have ancestral divergence
dates on interior nodes
A new “conventional” technique is the RANKED TREE algorithm

53. Ranked Trees using “lifted” CP model
A new problem where input trees have ancestral divergence
dates on interior nodes
We do this in the “lifted” model by merely
1. reading in divergence dates for pairs of species and
posting these as constraints into the “D” variables
2. Then solve using the disjunction breaking “P” variables
3. Interior nodes retain range values
4. In addition can enumerate all solutions eliminating
value symmetries

54. Two trees of cats. Ranks (divergence information) on interior nodes
Common species in boxes

55. NOTE: range of values [6..9] on mrca(PTE,LTI)
Two ranked cats trees on left, and on the right one of the ranked supertrees
NOTE: range of values [6..9] on mrca(PTE,LTI)

56. 7 of the 17 solutions have ranges on interior nodes
Without the “lifted” representation we get 30 solutions (some redundant)

57. Is this a 1st?
We thinks so (or at least Patrick thinks so)
enumerate all solutions for ranked supertrees
remove value symmetries

58. What next?
Reduce the size of the model.
with a specialised ultrametric constraint
- over 3 variables
- over 3 variables plus the P decision variable
- over an entire n by n array
Improve propagation of ultrametric constraint
- Bound GAC
- GAC
New application
- Identify common features (back bone) of all supertrees
- Address nested taxa
- combine all we have
Already underway with Neil Moore

59. enumerate all solutions removing symmetries
Conclusion
presented a new (non-conventional) way of addressing the supertree problem
constraint model has been shown to be versatile
enumerate all solutions removing symmetries
address divergence dates on interior nodes
enumerate all solutions for ranked trees
model is bulky/large
we are working on this
future extensions
find the backbone of forest of supertrees
address nested taxa

60. NO WAY!
I did it all on my own

61. Thanks for helping
Pierre Flener
Xavier Lorca
Rod Page
Mike Steel
Charles Semple
Chris Unsworth
Neil Moore
Christine Wu Wei
Barbara Smith
Ian Gent

62. Any questions?