Two Solutions in Search of Killer Apps.

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Two Solutions in Search of Killer Apps.

• Dimacs workshop on Algorithms in Human Population
  Genomics
• Dan Gusfield
• UC Davis

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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.
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Topic I Improved Algorithms for Inferring the
Minimum Mosaic of a Set of Recombinants

• Yufeng Wu and Dan Gusfield
• UC Davis
• From CPM 2007

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Recombination

• Recombination one of the principle genetic
  forces shaping sequence variations within
  species.
• Two equal length sequences generate a new equal
  length sequence.

110001111111001
000110000001111
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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.

000000 001111 111100 011100
000000 111111
Founders
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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
1101101 1010001 0111111 0110100 1100011
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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 gt2.
• 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.

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The Two-Founder Case
1111101 1010001 0111111 0110100 1100011
110111101 100100101 010111111 010101100 110000111
Remove uniform columns
Study pairs of neighboring columns
Founders
Key at columns 1 and 2, the founders are either
or . There are two rows with 00/11,
and three rows with 01/10. So, at least two
breakpoints between columns 1 and 2 with founders
as .
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The Two-Founder Case (Cont.)
breakpoints between two columns
2
2
1
2
2
2
Local founders
c1
c2
c5
c7
c3
c4
c6
1 0
1 0
At least 2 2 2 1 2 2 11 breakpoints
needed. On the other hand, we can construct two
founders that use the same local optimal
founders, and thus 11 breakpoints is global
optimum.
No matter which founder states are chosen for
previous column, we can always choose the needed
founders for current column.
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Three or More Founders Assuming Known Founders
Input Sequences
Three Founders
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 sc takes
its value from.
1101101 1010001 0111111 0110100 1100011
1101111 1010001 0110100
1101101
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Enumerating Founders for Founder-Unknown 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 1
1 0 1
0 1 0
0 0 1
1 1 0
At each column, there are 2kf2 founder settings.
Let m be the number of columns, fully enumerate
all possible sets of founders takes ?(2mkf)
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.
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Search Paths and Search Tree
Assume three founders
Founder setting at column one
0 0 1
0
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.
0 1 1
0
c1
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Dropping Search Paths that are Beaten by Another
Search Path
Founder Config.
P1
P1 and P2 are two search paths up to column
2. 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?
lt39
0 0 1
0
0 1 1
P2
0 1 1
6
1 0 1
Assume three founders
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A More Powerful Beating Rule
Still five input rows. Now can not say P1 beats
P2. But remember we have founder matching
P1
0 0 1
0
0 1 1
P2
0 1 1
4
0 0 1
So P1 beats P2 since at most 3 rows need extra
breakpoints to get onto a path from P2, and P2
uses 4 more breakpoints than P1.
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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.

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How Practical Is Our Method?
Source of data and image UNC Chapel Hill Five
founders 20 rows, 36 columns UNCs 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.
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Another example

• The data from Ukkonens 2007 WABI
• paper (4 founders, 20 sequences, 40 sites was
  solved in 5 secs and used
• one fewer crossover than used in that
• paper.

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Applications?

• Founders on an island
• Founders in microbial communities
• ???

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Topic II Multi-State Perfect Phylogenywith
Missing and Removable Data

• To appear in Recomb 2009, May 09
• Dan Gusfield

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The Perfect Phylogeny Modelfor binary sequences

Only one mutation per site allowed (infinite
sites)
sites
12345
00000
Ancestral sequence
1
4
Site mutations on edges
3
00010
The tree derives the set M 10100 10000 01011 0101
0 00010
2
10100
5
10000
01010
01011
Extant sequences at the leaves
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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 gt 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.

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Example A perfect phylogeny for input M
(2,3,2)
A B C
(3,2,1)
Root
3 2 1
2 3 2
3 2 3
1 1 3
1 2 3
1
(3,2,3)
2
(3,2,3)
3
4
(1,2,3)
5
M
n 5 m 3 k 3
(1,2,3)
(1,1,3)
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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
  node-disjoint. This condition is called the
  convexity requirement.

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Example
(2,3,2)
A B C
(3,2,1)
3 2 1
2 3 2
3 2 3
1 1 3
1 2 3
1
(3,2,3)
2
(3,2,3)
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The tree for State 2 of Character B
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(1,2,3)
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M
n 5 number of taxa m 3 number of sites k 3
number of states
(1,2,3)
(1,1,3)
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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 perfect- phylogeny model.
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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 Sp
eed up of the method by Kannan-Warnow When k is
not fixed, the existence problem is NP-hard
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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 gt 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.
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New approach to existence and missing data
Based on an old theorem and newer techniques.
Old theorem Bunemans theorem relating
Perfect- Phylogeny to chordal graphs. Newer
techniques and theorems Minimal triangulations
of a non-chordal graph to make it chordal.
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Chordal Graphs
A graph G is called Chordal if every cycle of
length four or more contains a chord.
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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,d,e,g
c
a,e,g
b
g
d
a
a,e
e,f,g
f
e
b,c,d
G
T
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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.
Bunemans theorem makes this precise.
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Bunemans Approach to Perfect Phylogeny
1 2 3
Character-states
Partition-Intersection 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.
3 2 1
2 3 2
3 2 3
1 1 3
1 2 3
1 1 1
2 2 2
3 3 3
G(M)
Table M
Each row of table M induces a clique in
Partition-intersection graph G(M).
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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.
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Bunemans 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).
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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.
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Example
1 2 3
A 0 0 2 B 0 1 0 C 1 1 1 D 1 2 2
M
3,0
2,1
3,1
C
B
1,1
1,0
A
D
2,0
3,2
2,2
G(M)
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A legal triangulation
1 2 3
A 0 0 2 B 0 1 0 C 1 1 1 D 1 2 2
M
3,0
2,1
3,1
C
B
X
1,1
Y
1,0
A
D
2,0
3,2
2,2
G(M)
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Yields a Perfect Phylogeny
1 2 3
A 0 0 2 B 0 1 0 C 1 1 1 D 1 2 2
M
111
010
C
B
X
Y
012
112
A
D
002
122
One node in T for each maximal clique in G(M)
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What about Missing Data?
If M is missing data, build the partition
intersection graph G(M) using the known data in
M. Bunemans 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.
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Example
1 2 3
A 0 0 2 B 0 ? 0 C 1 1 1 D 1 2 2
M
3,0
2,1
3,1
C
B
1,1
1,0
A
D
2,0
3,2
2,2
G(M)
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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.
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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.
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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.
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Example
(2,1), (3,2) and (1,0), (1,1) are crossing
minimal separators.
3,0
2,1
3,1
C
B
1,1
1,0
(2,1), (1,1) and (1,0), (3,2)
are non-crossing minimal separators.
A
D
2,0
3,2
2,2
G(M)
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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.
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A legal triangulation
1 2 3
A 0 0 2 B 0 1 0 C 1 1 1 D 1 2 2
M
3,0
2,1
3,1
C
There are 6 minimal separators.
B
1,1
X
1,0
Y
5 are pairwise non-crossing
A
D
2,0
3,2
2,2
G(M)
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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).
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A legal triangulation
1 2 3
A 0 0 2 B 0 1 0 C 1 1 1 D 1 2 2
M
3,0
2,1
3,1
C
B
X
1,1
Y
1,0
A
D
2,0
3,2
2,2
G(M)
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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.
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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).
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Conceptually nice, but
Does it work in practice?
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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.
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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.
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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?
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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.

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Possible applications

• Micro-satellite data
• Transposable elements as characters and positions
  of elements as states
• Discretized quantitative traits
• Infinite alleles model

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All software to replicate theseresults will be
available on mywebsite by the time of Recomb 2009