Jean-Luc Baril (LE2I, Université de Bourgogne) Whole Mirror Duplication-Random Loss Model and Pattern Avoiding Permutations Algorithms & Permutations 2012, Paris. http://igm.univ-mlv.fr/AlgoB/algoperm2012/

1. Introduction
Tandem Duplication
Mirror Duplication
Whole Mirror Duplication Random Loss Model and
Pattern Avoiding Permutations
Jean-Luc Baril and R´mi Vernay
e
barjl@u-bourgogne.fr
http://jl.baril.u-bourgogne.fr
Laboratory LE2I – University of Burgundy – Dijon
Jean-Luc Baril WM Duplication Random Loss Model

2. Introduction
The Genome - Definition
Tandem Duplication
Pattern in permutations
Mirror Duplication
Some definitions and notations
genome = set of chromosomes
chromosome = sequence of genes
gene = sequence of Ad´nine, Guanine, Cytosine, Thynmine
e
(AGCT)
genome → n-length permutation σ = σ1 σ2 σ3 . . . σn
Sn = the set of n-length permutations
Graphical representation of the permutation
σ=84 6 2 5 71 3
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
8 4 6 2 5 WM1 3
Jean-Luc Baril
7 Duplication Random Loss Model

3. Introduction
The Genome - Definition
Tandem Duplication
Pattern in permutations
Mirror Duplication
Let σ = σ1 σ2 . . . 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 > · · ·
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
Jean-Luc Baril WM Duplication Random Loss Model

4. Introduction
The Genome - Definition
Tandem Duplication
Pattern in permutations
Mirror Duplication
Let σ = σ1 σ2 . . . 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 > · · ·
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
Jean-Luc Baril WM Duplication Random Loss Model

5. Introduction
The Genome - Definition
Tandem Duplication
Pattern in permutations
Mirror Duplication
Let σ = σ1 σ2 . . . 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 > · · ·
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
Jean-Luc Baril WM Duplication Random Loss Model

6. Introduction
The Genome - Definition
Tandem Duplication
Pattern in permutations
Mirror Duplication
Let σ = σ1 σ2 . . . 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 > · · ·
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
Jean-Luc Baril WM Duplication Random Loss Model

7. Introduction
The Genome - Definition
Tandem Duplication
Pattern in permutations
Mirror Duplication
Let σ = σ1 σ2 . . . 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 > · · ·
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
Jean-Luc Baril WM Duplication Random Loss Model

8. Introduction
The Genome - Definition
Tandem Duplication
Pattern in permutations
Mirror Duplication
Definition:
σ ∈ Sn contains the pattern π ∈ Sk (π σ) if:
∃1 ≤ i1 < i2 < · · · < ik ≤ n such that σi1 σi2 . . . σik is
order-isomorphic to π, i.e.,
∀1 ≤ u, v ≤ k, σi u < σi v ⇔ π u < π v .
Example: σ = 8 4 6 2 5 7 1 3 contains the pattern π = 4132
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
84625713
Jean-Luc Baril WM Duplication Random Loss Model

9. Introduction
The Genome - Definition
Tandem Duplication
Pattern in permutations
Mirror Duplication
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 = σ, π ∈ C}
/
We have C = S(B) where S(B) is the class of permutations
avoiding all patterns in B.
Jean-Luc Baril WM Duplication Random Loss Model

10. Introduction
Model
Tandem Duplication
References - known results
Mirror Duplication
The tandem duplication random-loss process
* This model is well-known in the evolutionary biology literature.
* Used for vertebrate mitochondrial genomes
12345678 1234567812345678
1234567812345678
12457368
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
Jean-Luc Baril WM Duplication Random Loss Model

11. Introduction
Model
Tandem Duplication
References - known results
Mirror Duplication
The tandem duplication random-loss process
* This model is well-known in the evolutionary biology literature.
* Used for vertebrate mitochondrial genomes
12345678 1234567812345678
(duplication)
1234567812345678
12457368
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Jean-Luc Baril WM Duplication Random Loss Model

12. Introduction
Model
Tandem Duplication
References - known results
Mirror Duplication
The tandem duplication random-loss process
* This model is well-known in the evolutionary biology literature.
* Used for vertebrate mitochondrial genomes
12345678 1234567812345678
(duplication)
1234567812345678
(random loss)
12457368
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Jean-Luc Baril WM Duplication Random Loss Model

13. Introduction
Model
Tandem Duplication
References - known results
Mirror Duplication
The tandem duplication random-loss process
* This model is well-known in the evolutionary biology literature.
* Used for vertebrate mitochondrial genomes
12345678 1234567812345678
(duplication)
1234567812345678
(random loss)
12457368
8 8 8
7 7 7
6 6 6
5 5 5
4 4 4
3 3 3
2 2 2
1 1 1
1 2 3 4 5 6 7 8 1 2 3 4 1 2 3 4 5 6 7 8 5 14 15 16
Jean-Luc Baril WM Duplication Random Loss Model

14. Introduction
Model
Tandem Duplication
References - known results
Mirror Duplication
2006 K. Chaudhuri, K. Chen, R. Mihaescu and S. Rao
On the tandem duplication-random loss model
of genome rearrangement,SODA
Tandem duplication random-loss process of an
interval of size K ;
Efficient algorithm for the distance between 2
genomes
Jean-Luc Baril WM Duplication Random Loss Model

15. Introduction
Model
Tandem Duplication
References - known results
Mirror Duplication
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
Permutations obtained after p duplications of an
interval of size K define a class of permutations
avoiding some patterns in B.
B = set of minimal permutations with d = 2p
descents.
Jean-Luc Baril WM Duplication Random Loss Model

16. Introduction
Model
Tandem Duplication
References - known results
Mirror Duplication
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
enumeration minimal permutations of size
n = d + 1, d + 2, 2d;
Jean-Luc Baril WM Duplication Random Loss Model

17. Introduction
Model
Tandem Duplication
References - known results
Mirror Duplication
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
enumeration minimal permutations of size
n = 2d − 1
Jean-Luc Baril WM Duplication Random Loss Model

18. Introduction
Model
Tandem Duplication
References - known results
Mirror Duplication
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. Ferrari
On the enumeration of d-minimal permutations,
Arxiv
prove that the number of minimal permutations with
d descents can be obtained by computing some
determinants, but they cannot provide a general
closed formula
Jean-Luc Baril WM Duplication Random Loss Model

19. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
The whole mirror duplication random-loss process
12345678 1234567887654321
1234567887654321
14578632
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
Jean-Luc Baril WM Duplication Random Loss Model

20. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
The whole mirror duplication random-loss process
12345678 1234567887654321
(mirror duplication)
1234567887654321
14578632
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Jean-Luc Baril WM Duplication Random Loss Model

21. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
The whole mirror duplication random-loss process
12345678 1234567887654321
(mirror duplication)
1234567887654321
(random loss)
14578632
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Jean-Luc Baril WM Duplication Random Loss Model

22. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
The whole mirror duplication random-loss process
12345678 1234567887654321
(mirror duplication)
1234567887654321
(random loss)
14578632
8 8 8
7 7 7
6 6 6
5 5 5
4 4 4
3 3 3
2 2 2
1 1 1
1 2 3 4 5 6 7 8 1 2 3 4 1 2 3 4 5 6 7 8 13 14 15 16
Jean-Luc Baril WM Duplication Random Loss Model

23. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Theorem 1
The class C(p) of permutations obtained from the identity after a
given number p of whole mirror duplications is the class of
permutations with at most 2p−1 − 1 valleys.
2p−2 − 1 valleys
Jean-Luc Baril WM Duplication Random Loss Model

24. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Theorem 1
The class C(p) of permutations obtained from the identity after a
given number p of whole mirror duplications is the class of
permutations 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

25. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
2p−2 − 1 < k valleys ≤ 2p−1 − 1
Jean-Luc Baril WM Duplication Random Loss Model

26. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
2p−2 − 1 valleys
Jean-Luc Baril WM Duplication Random Loss Model

27. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Jean-Luc Baril WM Duplication Random Loss Model

28. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Jean-Luc Baril WM Duplication Random Loss Model

29. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Jean-Luc Baril WM Duplication Random Loss Model

30. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Theorem 2
The class C(p) of permutations obtained after a given number p of
whole mirror duplications is the class of permutations avoiding the
alternating 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 1 1
1− + y − 1 · tan x y − 1 + arctan √
1−y y y y −1
Jean-Luc Baril WM Duplication Random Loss Model

31. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Algorithm 1 for a shortest path from 12 · · · n to σ ∈ Sn .
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
13625748
Complexity: O(n · log val(σ)) < O(n · log n)
Jean-Luc Baril WM Duplication Random Loss Model

32. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Algorithm 1 for a shortest path from 12 · · · n to σ ∈ Sn .
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
13625748 134678
Complexity: O(n · log val(σ)) < O(n · log n)
Jean-Luc Baril WM Duplication Random Loss Model

33. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Algorithm 1 for a shortest path from 12 · · · n to σ ∈ Sn .
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
13625748 13467852
Complexity: O(n · log val(σ)) < O(n · log n)
Jean-Luc Baril WM Duplication Random Loss Model

34. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Algorithm 1 for a shortest path from 12 · · · n to σ ∈ Sn .
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
13625748
⇒ O(n) 13684257
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
12345678
Complexity: O(n · log val(σ)) < O(n · log n)
Jean-Luc Baril WM Duplication Random Loss Model

35. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 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 Binary
Reflected Gray Code (F. Gray [1953], Bitner, Ehrlich, Reingold
[1976])
Bn = 0Bn−1 ◦ 1Bn−1
000
001
011
010
110
111
101
100
Jean-Luc Baril WM Duplication Random Loss Model

36. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 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 Binary
Reflected Gray Code (F. Gray [1953], Bitner, Ehrlich, Reingold
[1976])
Bn = 0Bn−1 ◦ 1Bn−1
8
000
7 001
6
5 011
4 010
3
2 110
1
1 2 3 4 5 6 7 8
111
101
100
Jean-Luc Baril WM Duplication Random Loss Model

37. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 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 Binary
Reflected Gray Code (F. Gray [1953], Bitner, Ehrlich, Reingold
[1976])
Bn = 0Bn−1 ◦ 1Bn−1
8
000
7 001
6 136 000
5 011
4 010
3
2 110
1
1 2 3 4 5 6 7 8
111
101
100
Jean-Luc Baril WM Duplication Random Loss Model

38. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 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 Binary
Reflected Gray Code (F. Gray [1953], Bitner, Ehrlich, Reingold
[1976])
Bn = 0Bn−1 ◦ 1Bn−1
8
000
7 001
6 136 000
5 011
2 001
4 010
3
2 110
1
1 2 3 4 5 6 7 8
111
101
100
Jean-Luc Baril WM Duplication Random Loss Model

39. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 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 Binary
Reflected Gray Code (F. Gray [1953], Bitner, Ehrlich, Reingold
[1976])
Bn = 0Bn−1 ◦ 1Bn−1
8
000
7 001
6 136 000
5 011
2 001
4 010
3 57 011
2 110
1
1 2 3 4 5 6 7 8
111
101
100
Jean-Luc Baril WM Duplication Random Loss Model

40. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 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 Binary
Reflected Gray Code (F. Gray [1953], Bitner, Ehrlich, Reingold
[1976])
Bn = 0Bn−1 ◦ 1Bn−1
8
000
7 001
6 136 000
5 011
2 001
4 010
3 57 011
2
4 010 110
1
1 2 3 4 5 6 7 8
111
101
100
Jean-Luc Baril WM Duplication Random Loss Model

41. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 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 Binary
Reflected Gray Code (F. Gray [1953], Bitner, Ehrlich, Reingold
[1976])
Bn = 0Bn−1 ◦ 1Bn−1
8
000
7 001
6 136 000
5 011
2 001
4 010
3 57 011
2
4 010 110
1
1 2 3 4 5 6 7 8 8 110 111
101
100
Jean-Luc Baril WM Duplication Random Loss Model

42. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8
7
6
5
136 000 4
2 001 3
2
57 011 1
4 010 1 2 3 4 5 6 7 8
12345678
8 110
Jean-Luc Baril WM Duplication Random Loss Model

43. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678
8 110
Jean-Luc Baril WM Duplication Random Loss Model

44. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678
8 110
Jean-Luc Baril WM Duplication Random Loss Model

45. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678
8 110
Jean-Luc Baril WM Duplication Random Loss Model

46. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678
8 110
Jean-Luc Baril WM Duplication Random Loss Model

47. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678
8 110
Jean-Luc Baril WM Duplication Random Loss Model

48. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678
8 110
Jean-Luc Baril WM Duplication Random Loss Model

49. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678
8 110
Jean-Luc Baril WM Duplication Random Loss Model

50. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678
8 110
Jean-Luc Baril WM Duplication Random Loss Model

51. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678 13468752
8 110
Jean-Luc Baril WM Duplication Random Loss Model

52. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678 13468752
8 110
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
Jean-Luc Baril WM Duplication Random Loss Model

53. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678 13468752
8 110
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
Jean-Luc Baril WM Duplication Random Loss Model

54. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678 13468752
8 110
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
Jean-Luc Baril WM Duplication Random Loss Model

55. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678 13468752
8 110
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
Jean-Luc Baril WM Duplication Random Loss Model

56. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678 13468752
8 110
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
Jean-Luc Baril WM Duplication Random Loss Model

57. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678 13468752
8 110
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
Jean-Luc Baril WM Duplication Random Loss Model

58. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678 13468752
8 110
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
Jean-Luc Baril WM Duplication Random Loss Model

59. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678 13468752
8 110
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
Jean-Luc Baril WM Duplication Random Loss Model

60. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678 13468752
8 110
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
13625784
Jean-Luc Baril WM Duplication Random Loss Model

61. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678 13468752
8 110
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
13625784
Jean-Luc Baril WM Duplication Random Loss Model

62. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678 13468752
8 110
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
13625784
Jean-Luc Baril WM Duplication Random Loss Model

63. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
⇓ 6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678 13468752
8 110
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
13625784
Jean-Luc Baril WM Duplication Random Loss Model

64. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 12 · · · n → σ
Step 2 – We construct the path
8 8
7 7
6 6
5 5
136 000 4 4
2 001 3 3
2 2
57 011 1 1
4 010 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
12345678 13468752
8 110
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
13625784 13625748
Jean-Luc Baril WM Duplication Random Loss Model

65. Model
Introduction
Theorems
Tandem Duplication
Algo σ → 12 · · · n
Mirror Duplication
Algo 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