Download - AlgoPerm2012 - 14 Jean-Luc Baril
IntroductionTandem DuplicationMirror Duplication
Whole Mirror Duplication Random Loss Model and
Pattern Avoiding Permutations
Jean-Luc Baril and Remi [email protected]
http://jl.baril.u-bourgogne.fr
Laboratory LE2I – University of Burgundy – Dijon
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
The Genome - DefinitionPattern in permutations
Some definitions and notations
genome = set of chromosomeschromosome = sequence of genesgene = sequence of Adenine, Guanine, Cytosine, Thynmine(AGCT)genome → n-length permutation σ = σ1σ2σ3 . . . σnSn = the set of n-length permutations
Graphical representation of the permutation
σ = 8 4 6 2 5 7 1 3
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8 4 6 2 5 7 1 3Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
The Genome - DefinitionPattern in permutations
Let σ = σ1σ2 . . . σn be a permutation:
ascent → σi < σi+1
run up → σi < σi+1 < · · · < σj
descent, run-down
valley → σi−1 > σi < σi+1
Alternating permutation → σ1 > σ2 < σ3 > σ4 < σ5 > · · ·
1 2 3 4 5 6 7 8
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
The Genome - DefinitionPattern in permutations
Let σ = σ1σ2 . . . σn be a permutation:
ascent → σi < σi+1
run up → σi < σi+1 < · · · < σj
descent, run-down
valley → σi−1 > σi < σi+1
Alternating permutation → σ1 > σ2 < σ3 > σ4 < σ5 > · · ·
1 2 3 4 5 6 7 8
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
The Genome - DefinitionPattern in permutations
Let σ = σ1σ2 . . . σn be a permutation:
ascent → σi < σi+1
run up → σi < σi+1 < · · · < σj
descent, run-down
valley → σi−1 > σi < σi+1
Alternating permutation → σ1 > σ2 < σ3 > σ4 < σ5 > · · ·
1 2 3 4 5 6 7 8
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8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
The Genome - DefinitionPattern in permutations
Let σ = σ1σ2 . . . σn be a permutation:
ascent → σi < σi+1
run up → σi < σi+1 < · · · < σj
descent, run-down
valley → σi−1 > σi < σi+1
Alternating permutation → σ1 > σ2 < σ3 > σ4 < σ5 > · · ·
1 2 3 4 5 6 7 8
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7
8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
The Genome - DefinitionPattern in permutations
Let σ = σ1σ2 . . . σn be a permutation:
ascent → σi < σi+1
run up → σi < σi+1 < · · · < σj
descent, run-down
valley → σi−1 > σi < σi+1
Alternating permutation → σ1 > σ2 < σ3 > σ4 < σ5 > · · ·
1 2 3 4 5 6 7 8
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8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
The Genome - DefinitionPattern in permutations
Definition:
σ ∈ Sn contains the pattern π ∈ Sk (π � σ) if:∃1 ≤ i1 < i2 < · · · < ik ≤ n such that σi1σi2 . . . σik isorder-isomorphic to π, i.e.,
∀1 ≤ u, v ≤ k , σiu < σiv ⇔ πu < πv .
Example: σ = 8 4 6 2 5 7 1 3 contains the pattern π = 4132
1 2 3 4 5 6 7 8
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8 4 6 2 5 7 1 3
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
The Genome - DefinitionPattern in permutations
Class of permutations
C is a class of permutations if C is stable for the relation �
σ ∈ C and π � σ ⇒ π ∈ C.
Basis for a class of permutations
A class C of permutations is characterized by its basis B :
B = {σ /∈ C,∀π ≺ σ with π 6= σ, π ∈ C}
We have C = S(B) where S(B) is the class of permutationsavoiding all patterns in B .
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelReferences - known results
The tandem duplication random-loss process
* This model is well-known in the evolutionary biology literature.* Used for vertebrate mitochondrial genomes
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
1 2 4 5 7 3 6 8
1 2 3 4 5 6 7 8
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelReferences - known results
The tandem duplication random-loss process
* This model is well-known in the evolutionary biology literature.* Used for vertebrate mitochondrial genomes
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
(duplication)
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
1 2 4 5 7 3 6 8
1 2 3 4 5 6 7 8
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelReferences - known results
The tandem duplication random-loss process
* This model is well-known in the evolutionary biology literature.* Used for vertebrate mitochondrial genomes
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
(duplication)
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
(random loss)
1 2 4 5 7 3 6 8
1 2 3 4 5 6 7 8
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelReferences - known results
The tandem duplication random-loss process
* This model is well-known in the evolutionary biology literature.* Used for vertebrate mitochondrial genomes
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
(duplication)
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
(random loss)
1 2 4 5 7 3 6 8
1 2 3 4 5 6 7 8
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1 2 3 4 1 2 3 4 5 6 7 8 5 14 15 16
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelReferences - known results
2006 K. Chaudhuri, K. Chen, R. Mihaescu and S. RaoOn the tandem duplication-random loss model
of genome rearrangement,SODATandem duplication random-loss process of aninterval of size K ;Efficient algorithm for the distance between 2genomes
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelReferences - known results
2006 K. Chaudhuri, K. Chen, R. Mihaescu and S. Rao
On the tandem duplication-random loss model of genome rearrangement,SODA
2009 M. Bouvel and D. RossinA variant of the tandem duplication-random loss
model of genome rearrangement, TCS
Permutations obtained after p duplications of aninterval of size K define a class of permutationsavoiding some patterns in B .B = set of minimal permutations with d = 2p
descents.
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelReferences - known results
2006 K. Chaudhuri, K. Chen, R. Mihaescu and S. Rao
On the tandem duplication-random loss model of genome rearrangement,SODA
2009 M. Bouvel and D. Rossin
A variant of the tandem duplication-random loss model of genome rearrangement, TCS
2010 M. Bouvel and E. PergolaPosets and permutations in the duplication-loss
model: minimal permutations with d descents,Theoretical Computer Science
enumeration minimal permutations of sizen = d + 1, d + 2, 2d ;
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelReferences - known results
2006 K. Chaudhuri, K. Chen, R. Mihaescu and S. Rao
On the tandem duplication-random loss model of genome rearrangement,SODA
2009 M. Bouvel and D. Rossin
A variant of the tandem duplication-random loss model of genome rearrangement, TCS
2010 M. Bouvel and E. Pergola
Posets and permutations in the duplication-loss model: minimal permutations with d
descents, Theoretical Computer Science
2010 T. Mansour and S. H.F. YanMinimal permutations with d descents, EuropeanJournal of Combinatorics
enumeration minimal permutations of sizen = 2d − 1
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelReferences - known results
2006 K. Chaudhuri, K. Chen, R. Mihaescu and S. Rao
On the tandem duplication-random loss model of genome rearrangement,SODA
2009 M. Bouvel and D. Rossin
A variant of the tandem duplication-random loss model of genome rearrangement, TCS
2010 M. Bouvel and E. Pergola
Posets and permutations in the duplication-loss model: minimal permutations with d
descents, Theoretical Computer Science
2010 T. Mansour and S. H.F. Yan
Minimal permutations with d descents, European Journal of Combinatorics
2010 M. Bouvel and L. FerrariOn the enumeration of d-minimal permutations,Arxiv
prove that the number of minimal permutations withd descents can be obtained by computing somedeterminants, but they cannot provide a generalclosed formula
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
The whole mirror duplication random-loss process
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 8 7 6 5 4 3 2 1
1 4 5 7 8 6 3 2
1 2 3 4 5 6 7 8
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
The whole mirror duplication random-loss process
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 8 7 6 5 4 3 2 1
(mirror duplication)
1 2 3 4 5 6 7 8 8 7 6 5 4 3 2 1
1 4 5 7 8 6 3 2
1 2 3 4 5 6 7 8
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
The whole mirror duplication random-loss process
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 8 7 6 5 4 3 2 1
(mirror duplication)
1 2 3 4 5 6 7 8 8 7 6 5 4 3 2 1
(random loss)
1 4 5 7 8 6 3 2
1 2 3 4 5 6 7 8
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
The whole mirror duplication random-loss process
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 8 7 6 5 4 3 2 1
(mirror duplication)
1 2 3 4 5 6 7 8 8 7 6 5 4 3 2 1
(random loss)
1 4 5 7 8 6 3 2
1 2 3 4 5 6 7 8
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1 2 3 4 1 2 3 4 5 6 7 8 13 14 15 16
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Theorem 1
The class C(p) of permutations obtained from the identity after agiven number p of whole mirror duplications is the class ofpermutations with at most 2p−1 − 1 valleys.
2p−2 − 1 valleys
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Theorem 1
The class C(p) of permutations obtained from the identity after agiven number p of whole mirror duplications is the class ofpermutations with at most 2p−1 − 1 valleys.
2p−1 − 1 = 2p−2 − 1 + 2p−2 − 1 + 1 valleys
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
2p−2 − 1 < k valleys ≤ 2p−1 − 1
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
2p−2 − 1 valleys
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Theorem 2
The class C(p) of permutations obtained after a given number p ofwhole mirror duplications is the class of permutations avoiding thealternating permutations of length 2p + 1.
For p = 1, C(1) = S(213, 312)For p = 2, C(2) =S(21435, 31425, 41325, 32415, 42315, 21534, 31524, 51324, 32514,52314, 41523, 51423, 42513, 52413, 43512, 53412)
|C(p)| given by the generating function:
1
1− y
(
1− 1
y+
1
y
√
y − 1 · tan(
x√
y − 1 + arctan
(
1√y − 1
)))
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Algorithm 1 for a shortest path from 12 · · · n to σ ∈ Sn.
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Complexity: O(n · log val(σ)) < O(n · log n)
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Algorithm 1 for a shortest path from 12 · · · n to σ ∈ Sn.
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1 2 3 4 5 6 7 8
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Complexity: O(n · log val(σ)) < O(n · log n)
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Algorithm 1 for a shortest path from 12 · · · n to σ ∈ Sn.
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1 2 3 4 5 6 7 8
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Complexity: O(n · log val(σ)) < O(n · log n)
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Algorithm 1 for a shortest path from 12 · · · n to σ ∈ Sn.
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1 2 3 4 5 6 7 8
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⇒ O(n) 1 3 6 8 4 2 5 7
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Complexity: O(n · log val(σ)) < O(n · log n)
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Algorithm 2 for a shortest path from 12 · · · n to σ ∈ Sn.
Step 1 – We label the runs up and runs down with the BinaryReflected Gray Code (F. Gray [1953], Bitner, Ehrlich, Reingold[1976])
Bn = 0Bn−1 ◦ 1Bn−1
000001011010
110111101100
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Algorithm 2 for a shortest path from 12 · · · n to σ ∈ Sn.
Step 1 – We label the runs up and runs down with the BinaryReflected Gray Code (F. Gray [1953], Bitner, Ehrlich, Reingold[1976])
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Bn = 0Bn−1 ◦ 1Bn−1
000001011010
110111101100
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Algorithm 2 for a shortest path from 12 · · · n to σ ∈ Sn.
Step 1 – We label the runs up and runs down with the BinaryReflected Gray Code (F. Gray [1953], Bitner, Ehrlich, Reingold[1976])
1 2 3 4 5 6 7 8
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136 000
Bn = 0Bn−1 ◦ 1Bn−1
000001011010
110111101100
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Algorithm 2 for a shortest path from 12 · · · n to σ ∈ Sn.
Step 1 – We label the runs up and runs down with the BinaryReflected Gray Code (F. Gray [1953], Bitner, Ehrlich, Reingold[1976])
1 2 3 4 5 6 7 8
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136
2
000
001
Bn = 0Bn−1 ◦ 1Bn−1
000001011010
110111101100
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Algorithm 2 for a shortest path from 12 · · · n to σ ∈ Sn.
Step 1 – We label the runs up and runs down with the BinaryReflected Gray Code (F. Gray [1953], Bitner, Ehrlich, Reingold[1976])
1 2 3 4 5 6 7 8
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136
2
57
000
001
011
Bn = 0Bn−1 ◦ 1Bn−1
000001011010
110111101100
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Algorithm 2 for a shortest path from 12 · · · n to σ ∈ Sn.
Step 1 – We label the runs up and runs down with the BinaryReflected Gray Code (F. Gray [1953], Bitner, Ehrlich, Reingold[1976])
1 2 3 4 5 6 7 8
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136
2
57
4
000
001
011
010
Bn = 0Bn−1 ◦ 1Bn−1
000001011010
110111101100
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Algorithm 2 for a shortest path from 12 · · · n to σ ∈ Sn.
Step 1 – We label the runs up and runs down with the BinaryReflected Gray Code (F. Gray [1953], Bitner, Ehrlich, Reingold[1976])
1 2 3 4 5 6 7 8
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136
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000
001
011
010
110
Bn = 0Bn−1 ◦ 1Bn−1
000001011010
110111101100
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
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000
001
011
010
1101 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
1136
2
57
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8
0000
001
011
010
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
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1 2 3 4 5 6 7 8
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
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57
4
8
000
0011
011
010
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
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1 2 3 4 5 6 7 8
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
1336
2
57
4
8
0000
001
011
010
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
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1 2 3 4 5 6 7 8
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
2
57
44
8
000
001
011
0100
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
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1 2 3 4 5 6 7 8
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
2
557
4
8
000
001
0111
010
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
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1 2 3 4 5 6 7 8
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
1366
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57
4
8
0000
001
011
010
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
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1 2 3 4 5 6 7 8
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
2
577
4
8
000
001
0111
010
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
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1 2 3 4 5 6 7 8
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
2
57
4
88
000
001
011
010
1100
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
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1 2 3 4 5 6 7 8
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
2
57
4
8
000
001
011
010
1101 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
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1 3 4 6 8 7 5 2
1 2 3 4 5 6 7 8
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Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
1136
2
57
4
8
0000
001
011
010
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
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1 3 4 6 8 7 5 2
1 2 3 4 5 6 7 8
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8
1 2 3 4 5 6 7 8
1
2
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5
6
7
8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
1336
2
57
4
8
0000
001
011
010
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 4 6 8 7 5 2
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
2
57
44
8
000
001
011
0110
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 4 6 8 7 5 2
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
1366
2
57
4
8
0000
001
011
010
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 4 6 8 7 5 2
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
2
57
4
88
000
001
011
010
1110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 4 6 8 7 5 2
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
2
577
4
8
000
001
0111
010
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 4 6 8 7 5 2
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
2
557
4
8
000
001
0111
010
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 4 6 8 7 5 2
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
22
57
4
8
000
0001
011
010
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 4 6 8 7 5 2
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
2
57
4
8
000
001
011
010
1101 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 4 6 8 7 5 2
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 6 2 5 7 8 4
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
113366
22
5577
4
8
0000
0001
0011
010
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 4 6 8 7 5 2
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 6 2 5 7 8 4
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
2
57
4
88
000
001
011
010
1110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 4 6 8 7 5 2
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 6 2 5 7 8 4
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
2
57
44
8
000
001
011
0010
110
⇓
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 4 6 8 7 5 2
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 6 2 5 7 8 4
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Step 2 – We construct the path
136
2
57
4
8
000
001
011
010
1101 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 4 6 8 7 5 2
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 6 2 5 7 8 4
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 3 6 2 5 7 4 8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
Jean-Luc Baril WM Duplication Random Loss Model
IntroductionTandem DuplicationMirror Duplication
ModelTheoremsAlgo σ → 12 · · · nAlgo 12 · · · n → σ
Complexity
One step requires : O(n)Whole process : O(n · log val(σ)) < O(n · log n)
Jean-Luc Baril WM Duplication Random Loss Model