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A Simpler 1.5-Approximation Algorithm for Sorting by Transpositions. Tzvika Hartman Weizmann Institute. Genome Rearrangements. During evolution, genomes undergo large-scale mutations which change gene order (reversals, transpositions, translocations).

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a simpler 1 5 approximation algorithm for sorting by transpositions

A Simpler 1.5-Approximation Algorithm for Sorting by Transpositions

Tzvika Hartman

Weizmann Institute

genome rearrangements
Genome Rearrangements
  • During evolution, genomes undergo large-scale mutations which change gene order (reversals, transpositions, translocations).
  • Given 2 genomes, GR algs infer the most economical sequence of rearrangement events which transform one genome into the other.
genome rearrangements model
Genome Rearrangements Model
  • Chromosomes are viewed as ordered lists of genes.
  • Unichromosomal genome, every gene appears once.
  • Genomes are represented by unsigned permutations fo genes.
  • Circular genomes (e.g., bacteria & mitochondria) are represented by circular perms.
sorting by transpositions

1 2 6 73 4 5 8 9

Sorting by Transpositions
  • A transposition exchanges between 2 consecutive segments of a perm.
  • Example : 1 2 3 4 56 7 8 9

Sorting by transpositions: finding ashortest sequence of transpositions which sorts the perm.

previous work
Previous work
  • 1.5-approximation algs for sorting by transpositions [BafnaPevzner98, Christie99].
  • An alg that sorts every perm of size nin at most 2n/3 transpositions [Erikkson et al 01].
  • Complexity of the problem is still open.
main results
Main Results
  • The problem of sorting circular permutations by transpositions is equivalent to sorting linear perms by transpositions.
  • A new and simple 1.5-approximation alg for sorting by transpositions, which runs in quadratic time.
linear circular perms
Linear & Circular Perms

A transposition “cuts” the perm at 3 points.

t

A

B

C

D

A

C

B

D

Linear transposition :

A

A

t

Circular transposition :

B

C

C

B

  • Circular transpositions can be represented by exchanging any 2 of the 3 segments.
linear circular equivalence
Linear & Circular Equivalence
  • Thm : Sorting linear perms by transpositions is computationally equivalent to sorting circular perms.
  • Pf sketch: Circularize linear perm by adding an n+1element and closing the circle.

Пn+1

Пn

П1

П1 . . . Пn

.

.

.

.

.

  • Every linear transposition is equivalent to a circular transposition that exchanges the 2 segments that do not include n+1.
breakpoint graph bafnapevzner98

4

1

3

2

6

11

5

12

14

9

13

10

8

7

Breakpoint Graph [BafnaPevzner98]

Perm : ( 1 6 5 4 7 3 2 )

Replace each element j by 2j-1,2j:

 = (1 2 11 12 9 10 7 8 13 14 5 6 3 4)

Circular Breakpoint graph G():

Vertex for every element.

Black edges (2i, 2i+1)

Grey edges (2i, 2i+1)

breakpoint graph cont

4

1

3

2

6

11

5

12

14

9

13

10

8

7

Breakpoint Graph (Cont.)
  • Unique decomposition into cycles.
  • codd(): # of odd cycles in G().
  • Define Δcodd(,t) = codd(t · ) – codd()
  • Lemma[BP98]: t and , Δcodd(,t){0, 2, -2}.
effect on graph example

1

2

4

5

3

6

Effect on Graph : Example
  • Perm: (1 3 2).
  • After extension: (1 2 5 6 3 4).
  • Breakpoint graph:

1

2

4

5

3

6

  • # of cycles increased by 2
effect on graph example12
Effect on Graph : Example
  • Perm : (6 5 4 3 2 1).
  • After extension : (11 12 9 10 7 8 5 6 3 4 1 2).
  • Breakpoint graph :

11

12

11

12

9

2

9

2

1

10

1

10

7

7

4

4

3

8

3

8

6

6

5

5

  • # of cycles remains 2
breakpoint graph cont13
Breakpoint Graph (Cont.)
  • Max # of odd cycles, n, is in the id perm, thus:
  • Lower bound[BP98]: For all , d()  [n-codd()]/2.
  • Goal : increase # of odd cycles in G.
  • t is a k-transposition if Δcodd(,t) = k.
  • A cycle that admits a 2-transposition is oriented.
simple permutations
Simple Permutations
  • A perm is simple if its breakpoint graph contains only short (3) cycles.
  • The theory is much simpler for simple perms.
  • Thm : Every perm can be transformed into a simple one, while maintaining the lower bound. Moreover, the sorting sequence can be mimicked.
  • Corr : We can focus only on simple perms.
3 cycles
3 - Cycles
  • 2 possible configurations of 3-cycles:

Non-oriented 3-cycle

Oriented 3-cycle

0 2 2 sequence of transpositions
(0,2,2)-Sequence of Transpositions
  • A (0,2,2)-sequence is a sequence of 3 transpositions: the 1st is a 0-transposition and the next two are 2-transpositions.
  • A series of (0,2,2)-sequences preserves a 1.5 approximation ratio.
  • Throughout the alg, we show that there is always a 2-transposition or a (0,2,2)-sequence.
interleaving cycles
Interleaving Cycles
  • 2 cycles interleave if their black edges appear alternatively along the circle.
  • Lemma : If G contains 2 interleaving 3-cycles, then  a (0,2,2)-sequence.
shattered cycles
Shattered Cycles
  • 2 pairs of black edges intersect if they appear alternatively along the circle.
  • Cycle A is shattered by cycles B and C if every pair of black edges in A intersects with a pair in B or with a pair in C.
  • Lemma : If G contains a shattered cycle, then  a (0,2,2)-sequence.
shattered cycles cont
Shattered Cycles (Cont.)
  • Lemma : If G contains no 2-cycles, no oriented cycles and no interleaving cycles, then  a shattered cycle.
the algorithm
The Algorithm
  • While G contains a 2-cycle, apply a 2-transposition [Christie99].
  • If G contains an oriented 3-cycle, apply a 2-transposition on it.
  • If G contains a pair of interleaving 3-cycles, apply a (0,2,2)-sequence.
  • If G contains a shattered unoriented 3-cycle, apply a (0,2,2)-sequence.
  • Repeat until perm is sorted.
conclusions
Conclusions
  • We introduced 2 new ideas which simplify the theory and the alg:
  • Working with circular perms simplifies the case analysis.
  • Simple perms avoid the complication of dealing with long cycles (similarly to the HP theory for sorting by reversals).
open problems
Open Problems
  • Complexity of sorting by transpositions.
  • Models which allow several rearrangement operations, such as trans-reversals, reversals and translocations (both signed & unsigned).
acknowledgements
Acknowledgements
  • Ron Shamir.