On the Diameter of Rearrangement Problems AlCoB2014, Tarragona, - PowerPoint PPT Presentation

On the Diameter of Rearrangement Problems AlCoB’2014, Tarragona, Spain Carla Negri Lintzmayer Zanoni Dias University of Campinas (UNICAMP) Institute of Computing Campinas, S˜ ao Paulo, Brazil Partially supported by FAPESP (grants 2013/01172-0 and 2013/08293-7), CNPq (grants 477692/2012-5 and 483370/2013-4), and FAEPEX (process 396/2014) July 3, 2014

Introduction 1 Definitions 2 State of the art 3 Results 4 Conclusions 5 Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 2 / 35

Introduction Genome rearrangements: mutational events that affect large stretches of the DNA sequence Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 3 / 35

Introduction Genome rearrangements: mutational events that affect large stretches of the DNA sequence Sorting permutations by genome rearrangements: combinatorial problems with applications in bioinformatics Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 3 / 35

Introduction Genome rearrangements: mutational events that affect large stretches of the DNA sequence Sorting permutations by genome rearrangements: combinatorial problems with applications in bioinformatics Mutations allow evolutions and the difference between two genomes indicates their evolutionary distance Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 3 / 35

Introduction Genome rearrangements: mutational events that affect large stretches of the DNA sequence Sorting permutations by genome rearrangements: combinatorial problems with applications in bioinformatics Mutations allow evolutions and the difference between two genomes indicates their evolutionary distance Find scenarios that show how to transform one genome into another Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 3 / 35

Introduction Genome rearrangements: mutational events that affect large stretches of the DNA sequence Sorting permutations by genome rearrangements: combinatorial problems with applications in bioinformatics Mutations allow evolutions and the difference between two genomes indicates their evolutionary distance Find scenarios that show how to transform one genome into another ◮ minimum number of rearrangements that allow the transformation Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 3 / 35

Introduction Genome rearrangements: mutational events that affect large stretches of the DNA sequence Sorting permutations by genome rearrangements: combinatorial problems with applications in bioinformatics Mutations allow evolutions and the difference between two genomes indicates their evolutionary distance Find scenarios that show how to transform one genome into another ◮ minimum number of rearrangements that allow the transformation Prefix and suffix reversals and transpositions Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 3 / 35

Introduction 1 Definitions 2 State of the art 3 Results 4 Conclusions 5 Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 4 / 35

Definitions Permutation: π = ( π 1 π 2 . . . π n ) where π i = π ( i ) Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 5 / 35

Definitions Permutation: π = ( π 1 π 2 . . . π n ) where π i = π ( i ) Unsigned permutation: π i ∈ { 1 , 2 , . . . , n } and π i � = π j for all i � = j Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 5 / 35

Definitions Permutation: π = ( π 1 π 2 . . . π n ) where π i = π ( i ) Unsigned permutation: π i ∈ { 1 , 2 , . . . , n } and π i � = π j for all i � = j Signed permutation: π i ∈ {− n, − ( n − 1) , . . . , − 1 , 1 , 2 , . . . , n } and | π i | � = | π j | for all i � = j Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 5 / 35

Definitions Permutation: π = ( π 1 π 2 . . . π n ) where π i = π ( i ) Unsigned permutation: π i ∈ { 1 , 2 , . . . , n } and π i � = π j for all i � = j Signed permutation: π i ∈ {− n, − ( n − 1) , . . . , − 1 , 1 , 2 , . . . , n } and | π i | � = | π j | for all i � = j Extended: ( π 0 = 0 π 1 π 2 . . . π n π n +1 = n + 1) Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 5 / 35

Definitions Permutation: π = ( π 1 π 2 . . . π n ) where π i = π ( i ) Unsigned permutation: π i ∈ { 1 , 2 , . . . , n } and π i � = π j for all i � = j Signed permutation: π i ∈ {− n, − ( n − 1) , . . . , − 1 , 1 , 2 , . . . , n } and | π i | � = | π j | for all i � = j Extended: ( π 0 = 0 π 1 π 2 . . . π n π n +1 = n + 1) Composition: π · σ = ( π σ 1 π σ 2 . . . π σ n ) Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 5 / 35

Definitions Permutation: π = ( π 1 π 2 . . . π n ) where π i = π ( i ) Unsigned permutation: π i ∈ { 1 , 2 , . . . , n } and π i � = π j for all i � = j Signed permutation: π i ∈ {− n, − ( n − 1) , . . . , − 1 , 1 , 2 , . . . , n } and | π i | � = | π j | for all i � = j Extended: ( π 0 = 0 π 1 π 2 . . . π n π n +1 = n + 1) Composition: π · σ = ( π σ 1 π σ 2 . . . π σ n ) Identity permutation: ι n = (1 2 . . . n ) Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 5 / 35

Definitions (Unsigned) reversal: ρ ( i, j ) with 1 ≤ i < j ≤ n = ( π 1 ... π i − 1 π i π i +1 ... π j − 1 π j π j +1 ... π n ) π π · ρ ( i,j ) = ( π 1 ... π i − 1 π j π j − 1 ... π i +1 π i π j +1 ... π n ) Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 6 / 35

Definitions (Unsigned) reversal: ρ ( i, j ) with 1 ≤ i < j ≤ n = ( π 1 ... π i − 1 π i π i +1 ... π j − 1 π j π j +1 ... π n ) π π · ρ ( i,j ) = ( π 1 ... π i − 1 π j π j − 1 ... π i +1 π i π j +1 ... π n ) Example: π = (3 1 5 2 7 4 3) π · ρ (2 , 5) = (3 7 2 5 1 4 3) Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 6 / 35

Definitions Signed reversal: ¯ ρ ( i, j ) with 1 ≤ i ≤ j ≤ n = ( π 1 ... π i − 1 π j π j +1 ... π n ) π π i π i +1 ... π j − 1 π · ¯ ρ ( i,j ) = ( π 1 ... π i − 1 − π j − π j − 1 ... − π i +1 − π i π j +1 ... π n ) Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 7 / 35

Definitions Signed reversal: ¯ ρ ( i, j ) with 1 ≤ i ≤ j ≤ n = ( π 1 ... π i − 1 π j π j +1 ... π n ) π π i π i +1 ... π j − 1 π · ¯ ρ ( i,j ) = ( π 1 ... π i − 1 − π j − π j − 1 ... − π i +1 − π i π j +1 ... π n ) Example: π = ( − 3 +1 − 5 + 2 + 7 − 4 − 3) π · ¯ ρ (2 , 5) = ( − 3 − 7 − 2 + 5 − 1 − 4 − 3) Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 7 / 35

Definitions Transposition: τ ( i, j, k ) with 1 ≤ i < j < k ≤ n + 1 π = ( π 1 ...π i − 1 π i π i +1 ...π j − 1 π j π j +1 ...π k − 1 π k ...π n ) π · τ ( i,j,k ) = ( π 1 ...π i − 1 π j π j +1 ...π k − 1 π i π i +1 ...π j − 1 π k ...π n ) Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 8 / 35

Definitions Transposition: τ ( i, j, k ) with 1 ≤ i < j < k ≤ n + 1 π = ( π 1 ...π i − 1 π i π i +1 ...π j − 1 π j π j +1 ...π k − 1 π k ...π n ) π · τ ( i,j,k ) = ( π 1 ...π i − 1 π j π j +1 ...π k − 1 π i π i +1 ...π j − 1 π k ...π n ) Example: π = (3 1 5 2 7 4 3) π · τ (2 , 4 , 7) = (3 2 7 4 1 5 3) π = ( − 3 +1 − 5 +2 + 7 − 4 − 3) π · τ (2 , 4 , 7) = ( − 3 +2 + 7 − 4 +1 − 5 − 3) Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 8 / 35

Definitions Prefix reversal (inverts first segment): ◮ unsigned: ρ p ( j ) ≡ ρ (1 , j ) for 1 < j ≤ n ◮ signed: ¯ ρ p ( j ) ≡ ¯ ρ (1 , j ) for 1 ≤ j ≤ n Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 9 / 35

Definitions Prefix reversal (inverts first segment): ◮ unsigned: ρ p ( j ) ≡ ρ (1 , j ) for 1 < j ≤ n ◮ signed: ¯ ρ p ( j ) ≡ ¯ ρ (1 , j ) for 1 ≤ j ≤ n Suffix reversal (inverts last segment): ◮ unsigned: ρ s ( i ) ≡ ρ ( i, n ) for 1 ≤ i < n ◮ signed: ¯ ρ s ( i ) ≡ ¯ ρ ( i, n ) for 1 ≤ i ≤ n Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 9 / 35

Definitions Prefix reversal (inverts first segment): ◮ unsigned: ρ p ( j ) ≡ ρ (1 , j ) for 1 < j ≤ n ◮ signed: ¯ ρ p ( j ) ≡ ¯ ρ (1 , j ) for 1 ≤ j ≤ n Suffix reversal (inverts last segment): ◮ unsigned: ρ s ( i ) ≡ ρ ( i, n ) for 1 ≤ i < n ◮ signed: ¯ ρ s ( i ) ≡ ¯ ρ ( i, n ) for 1 ≤ i ≤ n Prefix transposition: τ p ( j, k ) ≡ τ (1 , j, k ) for 1 < j < k ≤ n + 1 Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 9 / 35

Definitions Prefix reversal (inverts first segment): ◮ unsigned: ρ p ( j ) ≡ ρ (1 , j ) for 1 < j ≤ n ◮ signed: ¯ ρ p ( j ) ≡ ¯ ρ (1 , j ) for 1 ≤ j ≤ n Suffix reversal (inverts last segment): ◮ unsigned: ρ s ( i ) ≡ ρ ( i, n ) for 1 ≤ i < n ◮ signed: ¯ ρ s ( i ) ≡ ¯ ρ ( i, n ) for 1 ≤ i ≤ n Prefix transposition: τ p ( j, k ) ≡ τ (1 , j, k ) for 1 < j < k ≤ n + 1 Suffix transposition: τ s ( i, j ) ≡ τ ( i, j, n +1) for 1 ≤ i < j < n + 1 Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 9 / 35