1 A Simpler 1.5- Approximation Algorithm for Sorting by Transpositions Combinatorial Pattern...
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1 A Simpler 1.5- Approximation Algorithm for Sorting by Transpositions Combinatorial Pattern Matching (CPM) 2003 Authors: T. Hartman & R. Shamir Speaker: Chen Ming-Chiang
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Transcript of 1 A Simpler 1.5- Approximation Algorithm for Sorting by Transpositions Combinatorial Pattern...
1
A Simpler 1.5-Approximation Algorithm for Sorting by
Transpositions
Combinatorial Pattern Matching (CPM) 2003
Authors: T. Hartman & R. Shamir
Speaker: Chen Ming-Chiang
2
Outline
Sorting by Transpositions Previous Works Linear & Circular Permutation The Breakpoint Graph Algorithm Performance Ratio & Running Time Discussions References
3
Sorting by Transpositions
Given two sequences representing two species, find the smallest number of transpositions needed to transform a sequence to the other sequence.
Transposition: Swap two adjacent substrings of any length
without changing the order in the permutation.
2 4 1 3 1 3 2 4 1 2 3 4
4
Previous Works
O(n2) 1.5-approximation algorithm [BP98] V. Bafna & P.A. Pevzner, SIAM J.D.M., 1998.
O(n4) 1.5-approximation algorithm [C99] D.A. Christie, 1999.
At most 2n/3 transpositions for sorting given permutation of size n [EEKSW2001]
5
Linear & Circular Permutations Theorem 1
The problem of sorting by transpositions on linear permutation is equivalent to circular permutation.
Linear
1 4 7 2 6 5 3 1 2 6 4 7 5 3 Circular
1
2
3
45
6 7
1
4
3
25
7 6
6
The Breakpoint Graph
Permutation : 1 6 5 4 7 3 2 Step 1: Replace each element i by 2i-1 and 2i.
Permutation f( ): 1 2 11 12 9 10 7 8 13 14 5 6 3 4
1
2
3
4
5
6
7
12
3
4
5
6
78
14
13
11
10
9
12
7
The Breakpoint Graph
Graph G( ) is an edge-colored graph.
For every ,
Black edge:
Gray edge:
ni 1
),( 122 ii
)12,2( ii
1 2
3
4
5
6
78
14
13
11
10
9
12
8
The Breakpoint Graph : The number of cycles.
: The number of odd cycles. Odd cycle is the cycle with odd numbers of black edges.
)()(
),(
oddodd
odd
cc
c
)(c
1 2
3
4
5
6
78
14
13
11
10
9
12
)(oddc
3)( c1)( oddc
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The Breakpoint Graph
For a sorted permutation of size n,there are n cycles, and all of them are odd.
The concept is to increase the number of cycles
from to n.
1 2
13
14
11
12
78
10
9
3
6
5
4nc )(
ncodd )(
)(c
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The Breakpoint Graph
Lemma [BP98] For all permutation and transposition , .
Theorem (Lower bound) For all permutations , .
The goal now is to increase the number of odd cycle.
}2,0,2{),( c}2,0,2{),( oddc
2
)()(
oddcnd
11
The Breakpoint Graph Simple graph:
A graph is called simple if it contains only cycles which black edges .
1-cycle
2-cycle 3-cycle
3
12
Algorithm Transforming graph into simple graph .
While 2-cycle exists, apply a 2-transposition.
While 3-cylces exists, If oriented cycle exists, apply a 2-transpostion. If interleaving unoriented cycle exists, apply a
(0,2,2)-transposition. If shattered unoriented cycle exists, apply a
(0,2,2)-transposition.
Mimic the sorting of using the sorting of .
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Algorithm
Transforming graph into simple graph ([HP99] & [LX2001]):
Lemma: Every permutation can be transformed into a simple one by safe
splits.
Lemma: Let be a simple permutation that is equivalent to , then
every sorting of mimics a sorting of with the same number of operations.
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Algorithm Example:
1 3 5 7 4 2 6 1 3 5 7 4 x 2 6 1 odd-cycle 1 odd-cycle + 1 cycle1
6
7
14 13
10
9
2
3
4
5
8
1112
vb3
w
g
b1
b2
1
6
7
14 13
10
9
2
3
4
5
8
1112
b3
b1
b2
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Algorithm
Why the translation is safe?
The process breaks a cycle into a 3-cycle and (k-2)-cylce.
1)()ˆ( oddodd cc1ˆ nn
2
)()(
oddcnd
16
Algorithm
In the following, there are only three types of cycles in simple graph.
1-cycle
2-cycle
3-cycle
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Algorithm
While 2-cycle exists, apply a 2-transposition [C99]:
Lemma: If is a permutation that contains a 2-cycle, then
there exists a 2-tranposition on .
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Algorithm
The result is to increase two odd cycle. Therefore, it is a 2-transposition.
1
6
7
2
3
4
5
8
1
6
7
2
3
4
5
8
2),( oddc
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Algorithm
Only two possible configurations of 3-cycle:
Oriented cycle:
Unoriented cycle:
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Algorithm
While 3-cylces exists, If oriented cycle exists, apply a 2-transpostion [BP98]:
An oriented cycle can be eliminated by a 2-transposition.
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Algorithm
It is a 2-transposition, because .
12
9
10
11
127
8
5
6
3
4
12
3
4
11
129
10
5
6
7
8
2),( oddc
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Algorithm
Now, we only focus on unoriented cycle.
Lemma: Let C be an unoriented cycle. Then every pair of
black edges in C intersects with some other cycles in .)(G
0 3 2 1 4
23
For the unoriented cycle, there are two cases, interleaving cycle and shattered cycle.
Interleaving cycle:
Shattered cycle:
24
Algorithm
While 3-cylces exists, If interleaving unoriented cycle exists, apply a (0,2,2)-
transposition :
25
1
211
12
56
7
8
3
4
9
10
9
107
8
56
11
12
3
4
1
2
9
1011
12
12
7
8
3
4
5
6
5
67
8
12
11
12
3
4
9
10
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Algorithm While 3-cylces exists,
If shattered unoriented cycle exists, apply a (0,2,2)-transposition :
Shattered cycle: Cycle E is shattered by cycles C and D, if E’s black edges belong
to different intervals caused by either C or D.
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Algorithm First Case:
two out of three cycles are non-intersecting.
28
Algorithm Second Case:
Three cycles are mutually intersecting.
1
23
4
5
613
14
9
1011
12
8 7
5
69
10
11
123
4
13
141
2
8 7
1
213
14
11
123
4
9
105
6
8 7
29
Algorithm Transforming graph into simple graph .
While 2-cycle exists, apply a 2-transposition.
While 3-cylces exists, If oriented cycle exists, apply a 2-transpostion. If interleaving unoriented cycle exists, apply a (0,2,2)-
transposition. If shattered unoriented cycle exists, apply a (0,2,2)-
transposition.
Mimic the sorting of using the sorting of .
30
Performance Ratio & Running time
Performance ratio is 1.5, since
.
Running time of algorithm is .
5.1
342
2)(
34
)(
odd
odd
cn
cn
2n
31
Discussions
Working on circular permutation & better running time.
Complexity of the problem is still the open problem.
There are many different sorting problems about genome rearrangement.
32
References
[BP98] Sorting by Transpositions, Bafna, V. and Pevzner, P. A., SIAM Journal on Discrete Mathematics, Vol. 11, No. 2, 1998, pp. 224-240.
[C99] Genome Rearrangement Problems, D. A., Christie, PhD thesis, University of Glasgow, 1999.
[EEKSW01] Sorting a Bridge Hand, Eriksson, H., Eriksson, K., Karlander, J., Svensson, L. and Waslund, J., SIAM Journal on Discrete Mathematics, Vol. 241, 2001, pp. 289-300.
[HP99] Transforming Cabbage into Turnip: Polynomial Algorithm for Sorting Signed Permutations by Reversals, Hannenhalli, S. and Pevzner, P. A., Journal of the ACM, Vol. 46, 1999, pp. 1–27.
[LX2001] Signed genome rearrangements by reversals and transpositions: Models and Approximations, G. H. Lin and G. Xue, Theoretical Computer Science, pp. 513-531, 2001.
