1. Species Trees & Constraint Programming

2. Ian Gent, Barbara Smith, Wu Wei (Christine)
Ongoing work with
Ian Gent, Barbara Smith, Wu Wei (Christine)

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

4. Properties of a Species Tree
We have a set of leaf nodes, each labelled with a species
the interior nodes have no labels
each interior node has 2 children and one parent
except the root (it has no parent)
if we have n leaf nodes we then have n  1 interior nodes
it is a bifurcating tree

5. Super Trees
We are given two trees, T1 and T2
T1 has leaf set S1 and S2 has leaf set
remember, leaves are species!
But S1 and S2 have a non-empty intersection
why? How can that happen?
We want to combine T1 and T2
so, why is that a problem?

6. Most Recent Common Ancestors (mrca)b c
We have 3 species, a, b, and c
mrca(a,b)  mrca(a,c)
mrca(a,b)  mrca(b,c)
mrca(a,c)  mrca(b,c)
Species a and b are more closely related
to each other than they are to 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)

7. Triples (and Fans) a b c b c d
Species trees are frequently presented as a set of triples (and fans) a b c b c d

8. Triples (and Fans) b c d a b c a b c d

9. BreakUp & OneTree (circa 1996)
Algorithm breakUp takes a species tree and produces a set of
rooted triples R that define that tree.
Algorithm OneTree takes a set of species and a set of rooted
triples, and builds a tree that respects those triples, or reports
that no tree exists (in polytime)
OneTree is a specialisation of Build, an algorithm proposed
by Aho, Sagiv, Szymanski, and Ulman in 1981

10. The Flavour of OneTree Given a set of species S and rooted triples R
produce a node N
construct a graph G
with vertices in S
and edge (x,y) if triple xy|z is in R
if G is a single component fail
else recursively build
on the left with one component
with S’ and R’ (the set of species and triples in that component)
on the right, with the other components

11. The Flavour of OneTree d a c b a c b d

12. Min-cut Super Trees What happens if OneTree fails?
Gives us the best you can
by breaking some triples (resulting in fans)
by excluding some species
There are polytime algorithms for this
but they are greedy and biased

13. Constraint Programming solutions to building
a species tree from a set of rooted triples

14. A naïve constraint encoding (footnotes 756, 789, 794, 796)
n-1 variables as interior nodes
v[i] = j  parent(v[i]) = v[j]
no loops/cycles
Barbara used set variables (ILOG)
Patrick used specialised constraint (Chco)
Francois then encoded set variables!
n variables as leaf nodes
each takes a value respecting triples
I am sparing you (and me) the details

15. Why was this a naïve constraint encoding?
It produced the right number of trees when no triples
the Catalan number
symmetry breaking
It would produce a tree if one existed
A 2 stage process
(1) build a tree from the interior nodes
there are Catalan many of these
(2) given an “interior tree” place the leaf nodes
there are n! ways to do this
if step (2) fails generate the next interior tree in (1)
Yikes! That’s expensive.
Imagine {ab|c,bc|d,cd|a}

16. Ultrametric Trees & Species Trees (footnotes 803,804,805,810,819)
What is an ultrametric tree?
We are given a 2d symmetric matrix D
D[i][j] is the time of divergence of species i and j.
D[i,j] is the the mrca(i,j) labeled with time of divergence
D[i,j] is the value of mrca(i,j)
Build a bifurcating tree
n leaves and n - 1 interior nodes
interior nodes labeled with entries from D
any path from the root is a strictly decreasing sequence

17. Ultrametric Trees: here’s one I (well, Dan Gusfield actually ) prepared earlier8 3 5 B C D E A
Note: if the sequence increases, we have min-ultrametric tree

18. Ultrametric Matrix: necessary & sufficient conditions
cannot have more than n - 1 distinct values
because there are n - 1 interior nodes
For every 3 indices i,j,k
there is a tie for the maximum between D[i,j], D[i,k], D[j,k]
Given an ultrametric matrix, an ultrametric tree can be
constructed in O(n2)
… see Dan Gusfield’s book
“Algorithms on Strings, Trees, and Sequences”

19. Think isosceles triangles, allowing equilateral
A CP encoding of D
We have a 2 dimensional matrix of constrained integer cvariables D
We must ensure that for any i,j,k the following holds
Think isosceles triangles,
allowing equilateral
An ultrametric space,
composed of isosceles triangles

20. A CP encoding of D
Any instantiation of the variables in D is
now guaranteed to be min-ultrametric
We get Catalan number of min-ultrametric solutions

21. Note: our tree is min-ultrametric!
How can we exploit this?
We are given triples and fans, but not distances!
But we can consider a triple ij|k as a constraint k j i
This over-rides the disjunctions posted
across the matrix
Note: our tree is min-ultrametric!

22. The CP encoding (contd)
we have the “blanket” disjunctive constraint to ensure min-ultrametric
triples are constraints that break the disjunctions
a solution (if one exists) is min-ultrametric respecting triples
we can then produce tree from the matrix, as a post process
NOTE: we need a pre-process to break up trees into triples

23. So where are we?
Good question:
we have not yet tried real data
we have a number of different micro-encodings
Are we in P for decision?
Not sure yet
How about optimisation?
We can see a way, by introducing penalties
Wu Wei is coding up BreakUp and OneTree
so we have something real to compare with
We need real data to check this out
I need to get funding for this
write a grant proposal with DRG I think!

24. Questions?