Two Solutions in Search of
Killer Apps.
Dimacs workshop on Algorithms in
Human Population Genomics
Dan Gusfield
UC Davis
Two Algorithmic Topics
We have new algorithmic tools for a) computing the Minimum
Mosaic of a set of recombinants, and for b) multi-state Perfect
Phylogeny with missing data, that should be of use in
Population Genomics and Phylogenetics. These tools
were developed `on spec’:
We have `hand-waiving’ arguments for their utility,
but no actual (biological data-set) applications.
Suggestions wanted.
Topic I: Improved Algorithms for
Inferring the Minimum Mosaic of a
Set of Recombinants
Yufeng Wu and Dan Gusfield
UC Davis
From CPM 2007
Recombination
• Recombination: one of the principle genetic forces shaping
sequence variations within species.
• Two equal length sequences generate a new equal length
sequence.
110001111111001
11000 0000001111
Prefix
000110000001111
Suffix
Breakpoint
Founders and Mosaic
• Current sequences are descendents of a small
number of founders.
– A current sequence is composed of blocks from the
founders, due to recombination.
– No mutations since formation of founders.
Sampled sequences
in current population
000000
111111
Founders
000000
111111
001111
Breakpoint
000000
000000
111111
001111
001111
111100
111100
011100
Mosaic
The Minimum Mosaic Problem
• Given a set of aligned binary sequences in the current population
and assume the number of founders is known to be Kf, find set of
founders and the mosaic with the minimum number of
breakpoints.
Assume
Kf =3
Four breakpoints:
minimum for all possible
three founders
1101101
1101101
1101111
1010001
1010001
1010001
0111111
0111111
0110100
0110100
0110100
1100011
1100011
Three
Founders
Status of the Minimum Mosaic Problem
• First studied by E. Ukkonen (WABI 2002). Later WABI
2007.
– Dynamic programming method. Not practical when the
number of rows is more than 20 and Kf >2.
• No polynomial-time algorithm was known even when
Kf is small. No NP-completeness result is known.
• Our results:
– A simple polynomial-time algorithm for Kf = 2 case.
– Exact and practical method for data of medium range for Kf 
3.
Three or More Founders: Assuming
Known Founders
Input
Sequences
1101101
1010001
0111111
0110100
Three
Founders
1101111
1010001
0110100
1100011
Founder Mapping
1101101
1101101
With known founders, can
minimize breakpoints for each
sequence, and thus also minimize
the total number of breakpoints.
For each input sequence, starting
from the left, insert a breakpoint at
the end of longest segments
matching one founder.
Founder mapping: at each
position c in any input sequence s,
which founder s[c] takes its value
from.
Breakpoint!
10
Founder 1
Founder 2
Enumerating Founders for FounderUnknown Case
In reality, founders are not known. A straightforward way is to simply
enumerate all possible sets of founders, and then run the previous
method to find the minimum mosaic.
1
0
0
0
1
0
0
0
1
0
1
1
1
0
1
1
1
0
At each column, there are 2kf–2
founder settings.
Let m be the number of columns, fully enumerate all possible sets of
founders takes (2m*kf) time. Infeasible when m or Kf is large.
Need more ideas to develop a practical method. First, we do the
enumeration in the form of search paths in a search tree.
Search Paths and Search Tree
Assume three founders
Founder
setting at
column
one
0 0
0
1
0 0
1
1
c1
c2of tot.
Num
0 0 2
breakpoints
0 0
up1to 1current
column
0
0
1
0
1
0
1
0
0
1
1
0
0
0
0
0
1
0
1
1
1
0
0
1
1
0
1
0
0
0
1
1
1
0
2
c3
Founder
settings up to
column 3
0 0 0 5
0 1 1
1 0 0
On-line computation:
Compute partial
solution up to the
current column for
speedup.
The founder-known method can be
run with partially-known founders!
It works but exponential blowup of the search
paths!
Obvious idea to reduce search space: branch and
bound (compute a lower bound and …).
But we found a different idea is more useful.
Dropping Search Paths that are Beaten
by Another Search Path
Founder Config.
P1
<= 5 bkpts
0 0 0
1 0
1 1
P2
>= 0 bkpts
1 0 6
0 1
1 1
An optimal search path
following P2
Assume three founders
<=39
P1 and P2 are two search
paths up to column 2.
40
Can we say P1 is better than
P2? Not really, because
maybe P2 can lead to fewer
breakpoints later on.
But, suppose the number of
input sequences is 5. We can
then say P1 beats P2 (and
so drop P2). Why?
A more powerful beating rule
We use a more powerful, but more
complex, rule to identify paths that will
be beaten, and we use rules that avoid
generating beaten paths and redundant,
symmetric paths.
These methods reduce the enumeration
enormously, allowing practical
computation of the optimal.
How Practical Is Our Method?
Source of data and
image: UNC Chapel Hill
Five founders
20 rows, 36 columns
UNC’s heuristic solution:
54 breakpoints
Enumerating 2180
founder states is
impossible!
Our method takes 5 minutes to find the optimal solutions:
53 breakpoints. It is also practical for 50x50 matrix with
four founders.
Another example
The data from Ukkonen’s 2007 WABI
paper (4 founders, 20 sequences, 40 sites
was solved in 5 secs and used
one fewer crossover than used in that
paper.
Applications?
• Founders on an island
• Founders in microbial communities
• ???
Topic II: Multi-State Perfect
Phylogeny
with Missing and Removable
Data
To appear in Recomb 2009, May 09
Dan Gusfield
The Perfect Phylogeny Model
for binary sequences
Only one mutation per site
allowed (infinite sites)
sites 12345
Ancestral sequence 00000
1
4
Site mutations on edges
3
The tree derives the set M:
2
10100
10100
5
10000
01011
01010
00010
10000
00010
01010
01011
Extant sequences at the leaves
What is a Perfect Phylogeny
for k-state characters?
• Input consists of n sequences M with m sites (characters) each,
where each site can take one of k > 2 states.
• In a Perfect Phylogeny T for M, each node of T is labeled with
an m-length sequence where each site has a value from 1 to k.
• T has n leaves, one for each sequence in M, labeled by that
sequence.
• Arbitrarily root T at some node, and direct all the edges away
from the root. Then, for any character C with b states, there are
at most b-1 edges where character C mutates, and for any
state s of C, there is at most one edge where character C
mutates to state s. This more reflects the infinite alleles model
rather than infinite sites.
Example: A perfect phylogeny
for input M
Root
(3,2,1)
(2,3,2)
A B
1
(3,2,3)
(3,2,3)
(1,2,3)
3
2 2
3 3
4 1
5 1
2
3
2
1
2
M
(1,1,3)
(1,2,3)
n=5
m=3
k=3
C
1
2
3
3
3
A more standard definition
• For each character-state pair (C,s), the nodes of T
that are labeled with state s for character C, form a
connected subtree of T.
• It follows that the subtrees for any C are nodedisjoint. This condition is called the convexity
requirement.
Example
(3,2,1)
(2,3,2)
A B
1
(3,2,3)
(3,2,3)
The tree for
State 2 of
Character B
(1,2,3)
3
2 2
3 3
4 1
5 1
2
3
2
1
2
C
1
2
3
3
3
M
(1,1,3)
(1,2,3)
n = 5 number of taxa
m = 3 number of sites
k = 3 number of states
Perfect Phylogeny Problems
Existence Problem:
Given M, is there a Perfect Phylogeny for M?
Missing Data Problem:
For a given k, if there are cells in M without
values, can values less than or equal to k be imputed
so that the resulting matrix M’ has a perfect phylogeny?
Handling missing data extends the utility of the perfectphylogeny model.
Status of the Existence
Problem
Poly-time algorithm for 3 states, Dress-Steel 1991
Poly-time algorithm for 3 or 4 states, Kannan-Warnow
Poly-time algorithm for any fixed number of states polynomial in n and m, but exponential in k, Agarwalla and
Fernandez-Baca
Speed up of the method by Kannan-Warnow
When k is not fixed, the existence problem is NP-hard
Status of missing data
problem
NP-complete even for k = 2; effective integer programming
approaches for k = 2.
Polynomial-time methods for a `directed’ variant of k = 2.
No literature on the missing data problem for k > 2.
New work here: specialized ILP methods for k = 3,4,5
and a general ILP solution for any fixed k.
In this talk I will only discuss the general solution.
New approach to existence
and missing data
Based on an old theorem and newer techniques.
Old theorem: Buneman’s theorem relating PerfectPhylogeny to chordal graphs.
Newer techniques and theorems: Minimal triangulations of a
non-chordal graph to make it chordal.
Chordal Graphs
A graph G is called Chordal if every cycle of
length four or more contains a chord.
Another Classic (1970s)
Characterization
A graph G is chordal if and only if it is the intersection graph of
a set S of subtrees of a tree T. Each node of G is a member of S.
{b,c}
c
b
{a,e,g}
g
d
a
{c,d,e,g}
e
T
f
{b,c,d}
G
{a,e}
{e,f,g}
Relation to Perfect Phylogeny
In a perfect phylogeny T for a table M, for any character C
and any state s of character C, the sub-forest of T
induced by the nodes labeled (C,s) form a single, connected
subtree of T.
So, there is a natural set of subtrees of T induced by M, and
hints at the relationship of perfect-phylogeny to chordal
Graphs.
Buneman’s theorem makes this precise.
Buneman’s Approach to
Perfect Phylogeny
1
2
3
2
3
1
1
2
3
2
1
2
3
1
2
3
3
3
Character-states
1
2
1
2
1
2
3
3
3
G(M
)
Table M
Each row of table M induces a clique in
Partition-intersection graph G(M).
PartitionIntersection Graph
G(M) has
one node for each
character-state pair
in M, and an edge
between two nodes
if and only if there
is a row in M with
both those
character-state
pairs.
Note that if table M has m columns, then G(M) is a
m-partite graph. Nodes in the same class of G(M)
are said to have the same color. Two nodes with the
Same color are called a mono-chromatic pair.
An edge (u,v) not in G(M) is legal
if u and v are in different classes of the partition, ie.
are not a mono-chromatic pair.
Buneman’s Theorem
Theorem (Buneman 1971?)
There is a perfect phylogeny for M if and only if legal
edges can be added to graph G(M) to make it chordal.
If there is such a chordal graph, denote it G’(M).
G’(M) is called a legal triangulation of
G(M).
From Chordal Graph to
Perfect Phylogeny
Fact: Given a legal triangulation G’(M), a perfect phylogeny
for M can be constructed in linear time.
The algorithms are based on `perfect elimination orders’
And `clique trees’. Many citations.
Example
M
123
A: 0 0 2
B: 0 1 0
C: 1 1 1
D: 1 2 2
3,0
B
2,1
3,1
C
1,1
1,0
A
2,0
D
3,2
G’(M)
2,2
A legal triangulation
M
123
A: 0 0 2
B: 0 1 0
C: 1 1 1
D: 1 2 2
3,0
B
2,1
C
X
1,0
3,1
Y
A
2,0
1,1
D
3,2
G’(M)
2,2
Yields a Perfect Phylogeny
M
123
A: 0 0 2
B: 0 1 0
C: 1 1 1
D: 1 2 2
010
B
012 X
A
002
111
C
Y 112
D
12
2
One node in T for each
maximal clique in G’(M)
What about Missing Data?
If M is missing data, build the partition intersection graph
G(M) using the known data in M. Buneman’s theorem still
holds:
There is a perfect phylogeny for some imputation of missing
data in M, if and only if there is a legal triangulation of G(M).
The legal triangulation gives a perfect phylogeny T for M
with some imputed data, and then the imputed M’ can be
obtained from T.
Example
M
123
A: 0 0 2
B: 0 ? 0
C: 1 1 1
D: 1 2 2
3,0
B
2,1
3,1
C
1,1
1,0
A
2,0
D
3,2
G(M)
2,2
The Key Problem
So the key problem in this approach to both the
Existence and the Missing Data problems is how
to find a legal triangulation, if there is one.
Some triangulation problems are NP-hard (Tree-width,
Minimum fill-in).
But, there is a robust and still expanding literature on
efficient algorithms to find a minimal triangulation of
a non-chordal graph.
Minimal triangulation
A triangulation of a non-chordal graph G is
minimal if no subset of added edges is a triangulation
of G.
Clearly, if there is a legal triangulation G’(M) of G(M),
Then there is one which is a minimal triangulation.
So we can take advantage of the minimal triangulation
technology. The minimal vertex separators are the key
objects.
Minimal vertex separators
A set of vertices S whose removal separates vertices
u and v is called a u,v separator. S is a minimal u,v
separator if no subset of S is a u,v separator.
S is a minimal-separator if it is a minimal u,v
separator
for some u,v.
Minimal separator S crosses minimal separator S’, if
S separates some pair of nodes in S’.
Crossing is a symmetric relation for minimal separators.
Example
3,0
B
2,1
3,1
C
1,1
1,0
A
2,0
{(2,1), (3,2)} and {(1,0), (1,1)}
are crossing minimal
separators.
{(2,1), (1,1)} and {(1,0), (3,2)} are
non-crossing minimal separators.
D
3,2
G(M)
2,2
The structure of a minimal
triangulation in G
Completing a minimal separator K means adding all
the missing edges to make K a clique.
Capstone Theorem (P,S 1997): Every minimal triangulation
of G is obtained by completing each minimal separator in a
maximal set of pairwise non-crossing minimal separators
of G.
Conversely, completing every minimal separator in
a maximal set of pairwise non-crossing minimal separators
yields a triangulation of G.
A legal triangulation
M
123
A: 0 0 2
B: 0 1 0
C: 1 1 1
D: 1 2 2
3,0
B
2,1
C
X
1,0
3,1
A
2,0
1,1
Y
D
3,2
G’(M)
2,2
There are
6 minimal
separators.
5 are
pairwise
non-crossing
Back to Perfect Phylogeny
A minimal separator S in the partition intersection graph G(M)
Is called legal if it does not contain two nodes of the same
color and illegal if it does.
P,S Theorem can be used to prove the Main New Results
Theorem 1:
There is a perfect phylogeny for M, even if M is missing data,
If and only if there is a set of pairwise non-crossing legal
minimal separators in G(M) that separate every
mono-chromatic pair of nodes in G(M).
A legal triangulation
M
123
A: 0 0 2
B: 0 1 0
C: 1 1 1
D: 1 2 2
3,0
B
2,1
C
X
1,0
3,1
Y
A
2,0
1,1
D
3,2
G’(M)
2,2
Corollaries
Cor 1: If there is a mono-chromatic pair of nodes in G(M)
that is not separated by any legal minimal separator, then
M has no perfect phylogeny.
Cor 2: If G(M) has no illegal minimal separators, then
M has a perfect phylogeny.
Cor 3: If every mono-chromatic pair of nodes is separated
by some legal minimal separator, and no legal minimal
separators cross, the M has a perfect phylogeny.
How to solve the existence
and missing data problems
Given M, find all minimal separators in G(M); determine
which are legal and which are illegal; for each legal
minimal separator, determine which mono-chromatic pairs
of nodes it crosses.
Determine if any of the Corollaries hold.
If not, set up and solve an integer linear program to find
a set Q of
pairwise non-crossing legal minimal separators that
separate every mono-chromatic pair of nodes in G(M).
Conceptually nice, but
Does it work in practice?
It works surprisingly
(shockingly) well
Simulations based on ms, with datasets of sizes that
are charactoristic of many current applications in
phylogenetics and population genetics - but not
genomic scale or tree-of-life scale.
Surprising empirical results
The minimal separators are found quickly by existing
algorithms.
The numbers of minimal separators are small.
There are few crossing minimal separators.
Until a large percentage of missing data, most problems
are solved by the Corollaries, without the need for an ILP.
The created ILP are tiny.
The ILPs solve quickly - all have
solved in 0.00 CPLEX-reported seconds (CPLEX 11 on
2.5 Ghz machine). Most solve in the CPLEX pre-processor.
So
Although the chordal graph approach
may at first seem impractical, it works
on a large range of data of sizes that
are typical of current phylogenetics
problems, and degree of missing data.
So missing data can be handled.
But what are the biological applications of Multi-State
Perfect Phylogeny?
Application Requirements
• Must be multi-state data - ubiquitous
• The probability of mutating to any given
state more than once must be very
small - less common.
Possible applications
• Micro-satellite data
• Transposable elements as characters
and positions of elements as states
• Discretized quantitative traits
• Infinite alleles model
All software to replicate these
results will be available on my
website by the time of
Recomb 2009