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On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions

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

1

Introduction

2

Definitions

3

Algorithms SbSigPRSigSR SbSigPRPT SbSigPRPTSigSRST

4

Results

5

Conclusions

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 2 / 46

Introduction

Dweighter [1], 1975: The Pancake Flipping Problem

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 3 / 46

Introduction

Gates and Papadimitriou [2], 1979: The Burnt Pancake Flipping Problem

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 4 / 46

Introduction

Allowed moves: prefix reversals

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 5 / 46

Introduction

Reinterpreted as Genome Rearrangement Problems

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 6 / 46

Introduction

Reinterpreted as Genome Rearrangement Problems Genome Rearrangements: mutations that affect large portions of the genome

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 6 / 46

Introduction

Reinterpreted as Genome Rearrangement Problems Genome Rearrangements: mutations that affect large portions of the genome

◮ Reversals and Transpositions Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 6 / 46

Introduction

Reinterpreted as Genome Rearrangement Problems Genome Rearrangements: mutations that affect large portions of the genome

◮ Reversals and Transpositions

Mutations allow evolutions and the difference between two genomes indicates their evolutionary distance

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 6 / 46

Introduction

Reinterpreted as Genome Rearrangement Problems Genome Rearrangements: mutations that affect large portions of the genome

◮ Reversals and Transpositions

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 Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 6 / 46

Introduction

Reinterpreted as Genome Rearrangement Problems Genome Rearrangements: mutations that affect large portions of the genome

◮ Reversals and Transpositions

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 Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 6 / 46

Introduction

Dias and Meidanis [3], 2002: Prefix Transpositions Sharmin et al. [4], 2010: Pancake Flipping with Two Spatulas

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 7 / 46

Introduction

Lintzmayer and Dias [5], 2014: Suffix operations

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 8 / 46

1

Introduction

2

Definitions

3

Algorithms SbSigPRSigSR SbSigPRPT SbSigPRPTSigSRST

4

Results

5

Conclusions

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 9 / 46

Definitions

Permutation: π = (π0 = +0 π1 π2 . . . πn πn+1 = +(n + 1)) where |πi| = |πj| for all i = j

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 10 / 46

Definitions

Permutation: π = (π0 = +0 π1 π2 . . . πn πn+1 = +(n + 1)) where |πi| = |πj| for all i = j Composition: π · σ = (πσ1 πσ2 . . . πσn)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 10 / 46

Definitions

Permutation: π = (π0 = +0 π1 π2 . . . πn πn+1 = +(n + 1)) where |πi| = |πj| for all i = j Composition: π · σ = (πσ1 πσ2 . . . πσn) Identity permutation: ιn = (+1 + 2 . . . + (n − 1) + n)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 10 / 46

Definitions

Permutation: π = (π0 = +0 π1 π2 . . . πn πn+1 = +(n + 1)) where |πi| = |πj| for all i = j Composition: π · σ = (πσ1 πσ2 . . . πσn) Identity permutation: ιn = (+1 + 2 . . . + (n − 1) + n) Reverse permutation: ηn = (−n − (n − 1) . . . − 2 − 1)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 10 / 46

Definitions

Signed reversal: ¯ ρ(i, j) with 1 ≤ i ≤ j ≤ n

π = (π1 ... πi−1 πi πi+1 ... πj−1 πj πj+1 ... πn) π·¯ ρ(i,j) = (π1 ... πi−1 −πj − πj−1 ... − πi+1 − πi πj+1 ... πn)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 11 / 46

Definitions

Signed reversal: ¯ ρ(i, j) with 1 ≤ i ≤ j ≤ n

π = (π1 ... πi−1 πi πi+1 ... πj−1 πj πj+1 ... πn) π·¯ ρ(i,j) = (π1 ... πi−1 −πj − πj−1 ... − πi+1 − πi πj+1 ... πn)

Example: π = (−3 +1 − 5 + 2 + 7 − 4 − 3) π · ¯ ρ(2, 5) = (−3 −7 − 2 + 5 − 1 − 4 − 3)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 11 / 46

Definitions

Transposition: τ(i, j, k) with 1 ≤ i < j < k ≤ n + 1

π = (π1...πi−1 πi πi+1...πj−1 πj πj+1...πk−1 πk...πn) π·τ(i,j,k) = (π1...πi−1 πj πj+1...πk−1 πi πi+1...πj−1 πk...πn)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 12 / 46

Definitions

Transposition: τ(i, j, k) with 1 ≤ i < j < k ≤ n + 1

π = (π1...πi−1 πi πi+1...πj−1 πj πj+1...πk−1 πk...πn) π·τ(i,j,k) = (π1...πi−1 πj πj+1...πk−1 πi πi+1...πj−1 πk...πn)

Example: π = (−3 +1 − 5 +2 + 7 − 4 − 3) π · τ(2, 4, 7) = (−3 +2 + 7 − 4 +1 − 5 − 3)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 12 / 46

Definitions

Prefix reversal (inverts first segment):

◮ ¯

ρp(j) ≡ ¯ ρ(1, j) for 1 ≤ j ≤ n

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 13 / 46

Definitions

Prefix reversal (inverts first segment):

◮ ¯

ρp(j) ≡ ¯ ρ(1, j) for 1 ≤ j ≤ n

Suffix reversal (inverts last segment):

◮ ¯

ρs(i) ≡ ¯ ρ(i, n) for 1 ≤ i ≤ n

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 13 / 46

Definitions

Prefix reversal (inverts first segment):

◮ ¯

ρp(j) ≡ ¯ ρ(1, j) for 1 ≤ j ≤ n

Suffix reversal (inverts last segment):

◮ ¯

ρs(i) ≡ ¯ ρ(i, n) for 1 ≤ i ≤ n

Prefix transposition: τp(j, k) ≡ τ(1, j, k) for 1 < j < k ≤ n + 1

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 13 / 46

Definitions

Prefix reversal (inverts first segment):

◮ ¯

ρp(j) ≡ ¯ ρ(1, j) for 1 ≤ j ≤ n

Suffix reversal (inverts last segment):

◮ ¯

ρs(i) ≡ ¯ ρ(i, n) for 1 ≤ i ≤ n

Prefix transposition: τp(j, k) ≡ τ(1, j, k) for 1 < j < k ≤ n + 1 Suffix transposition: τs(i, j) ≡ τ(i, j, n+1) for 1 ≤ i < j < n + 1

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 13 / 46

Definitions

Rearrangement model: β

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 14 / 46

Definitions

Rearrangement model: β Distance: dβ(π), minimum number of operations in β needed to transform π into ιn

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 14 / 46

Definitions

Example for SbSigPR: π = (+3 − 1 + 4 − 2) ¯ ρp(1) → (−3 − 1 + 4 − 2) ¯ ρp(2) → (+1 + 3 + 4 − 2) ¯ ρp(3) → (−4 − 3 − 1 − 2) ¯ ρp(4) → (+2 + 1 + 3 + 4) ¯ ρp(1) → (−2 + 1 + 3 + 4) ¯ ρp(2) → (−1 + 2 + 3 + 4) ¯ ρp(1) → (+1 + 2 + 3 + 4)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 15 / 46

Definitions

Example for SbSigPR: π = (+3 − 1 + 4 − 2) ¯ ρp(1) → (−3 − 1 + 4 − 2) ¯ ρp(2) → (+1 + 3 + 4 − 2) ¯ ρp(3) → (−4 − 3 − 1 − 2) ¯ ρp(4) → (+2 + 1 + 3 + 4) ¯ ρp(1) → (−2 + 1 + 3 + 4) ¯ ρp(2) → (−1 + 2 + 3 + 4) ¯ ρp(1) → (+1 + 2 + 3 + 4) π = (+3 − 1 + 4 − 2) ¯ ρp(3) → (−4 + 1 − 3 − 2) ¯ ρp(4) → (+2 + 3 − 1 + 4) ¯ ρp(2) → (−3 − 2 − 1 + 4) ¯ ρp(3) → (+1 + 2 + 3 + 4)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 15 / 46

Definitions

Example for SbSigPR: π = (+3 − 1 + 4 − 2) ¯ ρp(1) → (−3 − 1 + 4 − 2) ¯ ρp(2) → (+1 + 3 + 4 − 2) ¯ ρp(3) → (−4 − 3 − 1 − 2) ¯ ρp(4) → (+2 + 1 + 3 + 4) ¯ ρp(1) → (−2 + 1 + 3 + 4) ¯ ρp(2) → (−1 + 2 + 3 + 4) ¯ ρp(1) → (+1 + 2 + 3 + 4) π = (+3 − 1 + 4 − 2) ¯ ρp(3) → (−4 + 1 − 3 − 2) ¯ ρp(4) → (+2 + 3 − 1 + 4) ¯ ρp(2) → (−3 − 2 − 1 + 4) ¯ ρp(3) → (+1 + 2 + 3 + 4) d ¯

ρp(π) = 4

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 15 / 46

Definitions

Breakpoint: occurs between two consecutive elements of π that should not be consecutive

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 16 / 46

Definitions

Breakpoint: occurs between two consecutive elements of π that should not be consecutive Example: (+0 −3 − 2 −4 −5 +1 +6)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 16 / 46

Definitions

Breakpoint: occurs between two consecutive elements of π that should not be consecutive Example: (+0 −3 − 2 −4 −5 +1 +6) Strip: maximal subsequence of π without breakpoints

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 16 / 46

1

Introduction

2

Definitions

3

Algorithms SbSigPRSigSR SbSigPRPT SbSigPRPTSigSRST

4

Results

5

Conclusions

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 17 / 46

Problems

Problem Best approximation factor Complexity SbR 1.375 [6] NP-hard [7] SbSigR

• P [8]

SbT 1.375 [9] NP-hard [10] SbRT ≈2.83 [11] ? SbSigRT 2∗ [12] ? SbPR 2 [13] NP-hard [14] SbSigPR 2 [15] ? SbPT 2 [3] ? SbPRPT 3 [4] ?

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 18 / 46

1

Introduction

2

Definitions

3

Algorithms SbSigPRSigSR SbSigPRPT SbSigPRPTSigSRST

4

Results

5

Conclusions

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 19 / 46

SbSigPRSigSR

Tries to remove one breakpoint with one operation:

1

πj =−π1+1, 2≤j ≤n: ¯ ρp(j − 1)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 20 / 46

SbSigPRSigSR

Tries to remove one breakpoint with one operation:

1

πj =−π1+1, 2≤j ≤n: ¯ ρp(j − 1)

2

πi=−πn−1, 1≤i≤n−1: ¯ ρs(i+1)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 20 / 46

SbSigPRSigSR

Tries to remove one breakpoint with one operation:

1

πj =−π1+1, 2≤j ≤n: ¯ ρp(j − 1)

2

πi=−πn−1, 1≤i≤n−1: ¯ ρs(i+1)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 20 / 46

SbSigPRSigSR

Tries to remove one breakpoint with one operation:

1

πj =−π1+1, 2≤j ≤n: ¯ ρp(j − 1)

2

πi=−πn−1, 1≤i≤n−1: ¯ ρs(i+1) Tries to remove one breakpoint with two operations:

1

πj = −πi − 1, 1 ≤ i < j ≤ n: ¯ ρp(j) · ¯ ρp(j − i)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 20 / 46

SbSigPRSigSR

Tries to remove one breakpoint with one operation:

1

πj =−π1+1, 2≤j ≤n: ¯ ρp(j − 1)

2

πi=−πn−1, 1≤i≤n−1: ¯ ρs(i+1) Tries to remove one breakpoint with two operations:

1

πj = −πi − 1, 1 ≤ i < j ≤ n: ¯ ρp(j) · ¯ ρp(j − i)

2

πj = πi + 1, 0 ≤ i+1 < j ≤ n: ¯ ρp(i) · ¯ ρp(j − 1)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 20 / 46

SbSigPRSigSR

Tries to remove one breakpoint with one operation:

1

πj =−π1+1, 2≤j ≤n: ¯ ρp(j − 1)

2

πi=−πn−1, 1≤i≤n−1: ¯ ρs(i+1) Tries to remove one breakpoint with two operations:

1

πj = −πi − 1, 1 ≤ i < j ≤ n: ¯ ρp(j) · ¯ ρp(j − i)

2

πj = πi + 1, 0 ≤ i+1 < j ≤ n: ¯ ρp(i) · ¯ ρp(j − 1)

3

πi = −πj + 1, 1 ≤ i < j ≤ n: ¯ ρs(i) · ¯ ρs(n + 1 − (j − i))

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 20 / 46

SbSigPRSigSR

Otherwise, π is of the three forms:

1

ηn

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 21 / 46

SbSigPRSigSR

Otherwise, π is of the three forms:

1

ηn

◮ one signed prefix/suffix reversal sorts it Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 21 / 46

SbSigPRSigSR

Otherwise, π is of the three forms:

1

ηn

◮ one signed prefix/suffix reversal sorts it 2

σ1 = (pb+1 . . . n

• ℓb+1

pb−1+1 . . . pb

• ℓb

. . . . . . 1 . . . p1

• ℓ1

)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 21 / 46

SbSigPRSigSR

Otherwise, π is of the three forms:

1

ηn

◮ one signed prefix/suffix reversal sorts it 2

σ1 = (pb+1 . . . n

• ℓb+1

pb−1+1 . . . pb

• ℓb

. . . . . . 1 . . . p1

• ℓ1

)

◮ at most b + 2 reversals sort it Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 21 / 46

SbSigPRSigSR

Otherwise, π is of the three forms:

1

ηn

◮ one signed prefix/suffix reversal sorts it 2

σ1 = (pb+1 . . . n

• ℓb+1

pb−1+1 . . . pb

• ℓb

. . . . . . 1 . . . p1

• ℓ1

)

◮ at most b + 2 reversals sort it 3

σ2 = (−p1 . . . −1

• ℓ1

−p2 . . . −(p1+1)

• ℓ2

. . . . . . −n . . . −(pb+1)

• ℓb+1

)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 21 / 46

SbSigPRSigSR

Otherwise, π is of the three forms:

1

ηn

◮ one signed prefix/suffix reversal sorts it 2

σ1 = (pb+1 . . . n

• ℓb+1

pb−1+1 . . . pb

• ℓb

. . . . . . 1 . . . p1

• ℓ1

)

◮ at most b + 2 reversals sort it 3

σ2 = (−p1 . . . −1

• ℓ1

−p2 . . . −(p1+1)

• ℓ2

. . . . . . −n . . . −(pb+1)

• ℓb+1

)

◮ at most b + 2 reversals sort it Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 21 / 46

SbSigPRSigSR

d ¯

ρp ¯ ρs(π) ≤ 2b ¯ ρp ¯ ρs(π) + 1

(1)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 22 / 46

SbSigPRSigSR

d ¯

ρp ¯ ρs(π) ≤ 2b ¯ ρp ¯ ρs(π) + 1

(1) d ¯

ρp ¯ ρs(π) ≥ b ¯ ρp ¯ ρs(π)

(2)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 22 / 46

SbSigPRSigSR

d ¯

ρp ¯ ρs(π) ≤ 2b ¯ ρp ¯ ρs(π) + 1

(1) d ¯

ρp ¯ ρs(π) ≥ b ¯ ρp ¯ ρs(π)

(2) lim

b(π)→∞

2b(π) + 1 b(π) = 2 + lim

b(π)→∞

1 b(π) = 2 + ǫ (3)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 22 / 46

1

Introduction

2

Definitions

3

Algorithms SbSigPRSigSR SbSigPRPT SbSigPRPTSigSRST

4

Results

5

Conclusions

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 23 / 46

SbSigPRPT

If π1 = 1, tries to remove two breakpoints with one

• peration:

1

Since π · τp(i, j) = (πi . . . πj−1 π1 . . . πi−1 . . . πn), we must find πj−1 = π1 − 1 and πi−1 = πj − 1

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 24 / 46

SbSigPRPT

If π1 = 1, tries to remove two breakpoints with one

• peration:

1

Since π · τp(i, j) = (πi . . . πj−1 π1 . . . πi−1 . . . πn), we must find πj−1 = π1 − 1 and πi−1 = πj − 1

◮ To maintain our approximation, πi = 1 Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 24 / 46

SbSigPRPT

If π1 = 1, tries to remove one breakpoint with one

• peration by increasing the first strip:

1

Let π = (k+1 k+2 . . . k+(i − 1) πi . . . . . .)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 25 / 46

SbSigPRPT

If π1 = 1, tries to remove one breakpoint with one

• peration by increasing the first strip:

1

Let π = (k+1 k+2 . . . k+(i − 1) πi . . . . . .)

2

If πj = k + i = πi−1 + 1 exists, then τp(i, j)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 25 / 46

SbSigPRPT

If π1 = 1, tries to remove one breakpoint with one

• peration by increasing the first strip:

1

Let π = (k+1 k+2 . . . k+(i − 1) πi . . . . . .)

2

If πj = k + i = πi−1 + 1 exists, then τp(i, j)

3

If πj−1 = k = π1 − 1 exists, then τp(i, j)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 25 / 46

SbSigPRPT

If π1 = 1, tries to remove one breakpoint with one

• peration by increasing the first strip:

1

Let π = (k+1 k+2 . . . k+(i − 1) πi . . . . . .)

2

If πj = k + i = πi−1 + 1 exists, then τp(i, j)

3

If πj−1 = k = π1 − 1 exists, then τp(i, j)

4

If πj+1 = −π1 + 1 exists, then ¯ ρp(j), for 1 ≤ j ≤ n

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 25 / 46

SbSigPRPT

If π1 = 1, send the first strip to the end of the permutation:

1

It will be removed only when n is sent there

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 26 / 46

SbSigPRPT

If π1 = 1, send the first strip to the end of the permutation:

1

It will be removed only when n is sent there

2

Which guarantees that π1 = 1 again at most one more time

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 26 / 46

SbSigPRPT

If π1 = 1, send the first strip to the end of the permutation:

1

It will be removed only when n is sent there

2

Which guarantees that π1 = 1 again at most one more time

3

Therefore, it will be possible to remove at least one breakpoint until the end of the sorting

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 26 / 46

SbSigPRPT

If π1 = 1, send the first strip to the end of the permutation:

1

It will be removed only when n is sent there

2

Which guarantees that π1 = 1 again at most one more time

3

Therefore, it will be possible to remove at least one breakpoint until the end of the sorting

◮ using at most two extra operations that do not remove

breakpoints

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 26 / 46

SbSigPRPT

d ¯

ρpτp(π) ≤ b ¯ ρp(π) + 2

(4)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 27 / 46

SbSigPRPT

d ¯

ρpτp(π) ≤ b ¯ ρp(π) + 2

(4) d ¯

ρpτp(π) ≥ b ¯ ρp(π)

2 (5)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 27 / 46

SbSigPRPT

d ¯

ρpτp(π) ≤ b ¯ ρp(π) + 2

(4) d ¯

ρpτp(π) ≥ b ¯ ρp(π)

2 (5) lim

b(π)→∞

b(π) + 2

b(π) 2

= 2 + lim

b(π)→∞

4 b(π) = 2 + ǫ (6)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 27 / 46

1

Introduction

2

Definitions

3

Algorithms SbSigPRSigSR SbSigPRPT SbSigPRPTSigSRST

4

Results

5

Conclusions

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 28 / 46

SbSigPRPTSigSRST

Tries to remove two breakpoints with one operation:

1

if πj−1 = π1 − 1 and πi−1 = πj − 1 exists, 2 ≤ i < j ≤ n, then τp(i, j)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 29 / 46

SbSigPRPTSigSRST

Tries to remove two breakpoints with one operation:

1

if πj−1 = π1 − 1 and πi−1 = πj − 1 exists, 2 ≤ i < j ≤ n, then τp(i, j)

2

if πi = πn + 1 and πj = πi−1 + 1 exists, 2 ≤ i < j ≤ n, then τs(i, j)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 29 / 46

SbSigPRPTSigSRST

Tries to remove two breakpoints with one operation:

1

if πj−1 = π1 − 1 and πi−1 = πj − 1 exists, 2 ≤ i < j ≤ n, then τp(i, j)

2

if πi = πn + 1 and πj = πi−1 + 1 exists, 2 ≤ i < j ≤ n, then τs(i, j)

3

neither πi−1 = n and πi = 1 nor πj−1 = −1 and πj = −n can happen

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 29 / 46

SbSigPRPTSigSRST

Tries to remove one breakpoint with one operation:

1

let π = (k + 1 k + 2 . . . k + (i − 1) πi . . . . . .), to increase the first strip with a prefix transposition:

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 30 / 46

SbSigPRPTSigSRST

Tries to remove one breakpoint with one operation:

1

let π = (k + 1 k + 2 . . . k + (i − 1) πi . . . . . .), to increase the first strip with a prefix transposition:

1

if πj = k + i = πi−1 + 1, j ≤ n, exists, then τp(i, j)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 30 / 46

SbSigPRPTSigSRST

Tries to remove one breakpoint with one operation:

1

let π = (k + 1 k + 2 . . . k + (i − 1) πi . . . . . .), to increase the first strip with a prefix transposition:

1

if πj = k + i = πi−1 + 1, j ≤ n, exists, then τp(i, j)

2

if πj−1 = k = π1 − 1 exists, then τp(i, j)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 30 / 46

SbSigPRPTSigSRST

Tries to remove one breakpoint with one operation:

1

let π = (k + 1 k + 2 . . . k + (i − 1) πi . . . . . .), to increase the first strip with a prefix transposition:

1

if πj = k + i = πi−1 + 1, j ≤ n, exists, then τp(i, j)

2

if πj−1 = k = π1 − 1 exists, then τp(i, j)

2

let π = (. . . . . . πj−1 k + 1 k + 2 . . . k + x), to increase the last strip with a suffix transposition:

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 30 / 46

SbSigPRPTSigSRST

Tries to remove one breakpoint with one operation:

1

let π = (k + 1 k + 2 . . . k + (i − 1) πi . . . . . .), to increase the first strip with a prefix transposition:

1

if πj = k + i = πi−1 + 1, j ≤ n, exists, then τp(i, j)

2

if πj−1 = k = π1 − 1 exists, then τp(i, j)

2

let π = (. . . . . . πj−1 k + 1 k + 2 . . . k + x), to increase the last strip with a suffix transposition:

1

if πi−1 = πj − 1 = k, i ≥ 2, exists, then τs(i, j)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 30 / 46

SbSigPRPTSigSRST

Tries to remove one breakpoint with one operation:

1

let π = (k + 1 k + 2 . . . k + (i − 1) πi . . . . . .), to increase the first strip with a prefix transposition:

1

if πj = k + i = πi−1 + 1, j ≤ n, exists, then τp(i, j)

2

if πj−1 = k = π1 − 1 exists, then τp(i, j)

2

let π = (. . . . . . πj−1 k + 1 k + 2 . . . k + x), to increase the last strip with a suffix transposition:

1

if πi−1 = πj − 1 = k, i ≥ 2, exists, then τs(i, j)

2

if πi = πn + 1 = k + x + 1 exists, then τs(i, j)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 30 / 46

SbSigPRPTSigSRST

Tries to remove one breakpoint with one operation:

1

let π = (k + 1 k + 2 . . . k + (i − 1) πi . . . . . .), to increase the first strip with a prefix transposition:

1

if πj = k + i = πi−1 + 1, j ≤ n, exists, then τp(i, j)

2

if πj−1 = k = π1 − 1 exists, then τp(i, j)

2

let π = (. . . . . . πj−1 k + 1 k + 2 . . . k + x), to increase the last strip with a suffix transposition:

1

if πi−1 = πj − 1 = k, i ≥ 2, exists, then τs(i, j)

2

if πi = πn + 1 = k + x + 1 exists, then τs(i, j)

3

again, we cannot separate n and 1 or −1 and −n

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 30 / 46

SbSigPRPTSigSRST

Tries to remove one breakpoint with one operation:

1

let π = (k + 1 k + 2 . . . k + (i − 1) πi . . . . . .), to increase the first strip with a prefix transposition:

1

if πj = k + i = πi−1 + 1, j ≤ n, exists, then τp(i, j)

2

if πj−1 = k = π1 − 1 exists, then τp(i, j)

2

let π = (. . . . . . πj−1 k + 1 k + 2 . . . k + x), to increase the last strip with a suffix transposition:

1

if πi−1 = πj − 1 = k, i ≥ 2, exists, then τs(i, j)

2

if πi = πn + 1 = k + x + 1 exists, then τs(i, j)

3

again, we cannot separate n and 1 or −1 and −n

4

if πj+1 = −π1 + 1, 1 ≤ j ≤ n − 1, exists, then ¯ ρp(j)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 30 / 46

SbSigPRPTSigSRST

Tries to remove one breakpoint with one operation:

1

let π = (k + 1 k + 2 . . . k + (i − 1) πi . . . . . .), to increase the first strip with a prefix transposition:

1

if πj = k + i = πi−1 + 1, j ≤ n, exists, then τp(i, j)

2

if πj−1 = k = π1 − 1 exists, then τp(i, j)

2

let π = (. . . . . . πj−1 k + 1 k + 2 . . . k + x), to increase the last strip with a suffix transposition:

1

if πi−1 = πj − 1 = k, i ≥ 2, exists, then τs(i, j)

2

if πi = πn + 1 = k + x + 1 exists, then τs(i, j)

3

again, we cannot separate n and 1 or −1 and −n

4

if πj+1 = −π1 + 1, 1 ≤ j ≤ n − 1, exists, then ¯ ρp(j)

5

if πi−1 = −πn − 1, 2 ≤ i ≤ n, exists, then ¯ ρs(i)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 30 / 46

SbSigPRPTSigSRST

Otherwise, π is of one of the five forms:

1

ηn (one reversal sorts it)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 31 / 46

SbSigPRPTSigSRST

Otherwise, π is of one of the five forms:

1

ηn (one reversal sorts it)

2

(1 2 . . . k . . . −(k + 1) . . . −(i − 1) . . . i i + 1 . . . n)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 31 / 46

SbSigPRPTSigSRST

Otherwise, π is of one of the five forms:

1

ηn (one reversal sorts it)

2

(1 2 . . . k . . . −(k + 1) . . . −(i − 1) . . . i i + 1 . . . n)

3

(−n −(n−1) . . . − i . . . (i − 1) . . . (k + 1) . . . −k −(k−1) . . . −1)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 31 / 46

SbSigPRPTSigSRST

Otherwise, π is of one of the five forms:

1

ηn (one reversal sorts it)

2

(1 2 . . . k . . . −(k + 1) . . . −(i − 1) . . . i i + 1 . . . n)

3

(−n −(n−1) . . . − i . . . (i − 1) . . . (k + 1) . . . −k −(k−1) . . . −1)

4

(k + 1 k + 2 . . . n 1 2 . . . k)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 31 / 46

SbSigPRPTSigSRST

Otherwise, π is of one of the five forms:

1

ηn (one reversal sorts it)

2

(1 2 . . . k . . . −(k + 1) . . . −(i − 1) . . . i i + 1 . . . n)

3

(−n −(n−1) . . . − i . . . (i − 1) . . . (k + 1) . . . −k −(k−1) . . . −1)

4

(k + 1 k + 2 . . . n 1 2 . . . k)

5

(−k −(k−1) . . . −1 −n −(n−1) . . . −(k+1))

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 31 / 46

SbSigPRPTSigSRST

Otherwise, π is of one of the five forms:

1

ηn (one reversal sorts it)

2

(1 2 . . . k . . . −(k + 1) . . . −(i − 1) . . . i i + 1 . . . n)

3

(−n −(n−1) . . . − i . . . (i − 1) . . . (k + 1) . . . −k −(k−1) . . . −1)

4

(k + 1 k + 2 . . . n 1 2 . . . k)

5

(−k −(k−1) . . . −1 −n −(n−1) . . . −(k+1))

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 31 / 46

SbSigPRPTSigSRST

Otherwise, π is of one of the five forms:

1

ηn (one reversal sorts it)

2

(1 2 . . . k . . . −(k + 1) . . . −(i − 1) . . . i i + 1 . . . n)

3

(−n −(n−1) . . . − i . . . (i − 1) . . . (k + 1) . . . −k −(k−1) . . . −1)

4

(k + 1 k + 2 . . . n 1 2 . . . k)

5

(−k −(k−1) . . . −1 −n −(n−1) . . . −(k+1)) For the last four types, we must apply a prefix transposition to concatenate the first strip with the last

• ne

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 31 / 46

SbSigPRPTSigSRST

d ¯

ρpτp ¯ ρsτs(π) ≤ b ¯ ρp ¯ ρs(π) + 2

(7)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 32 / 46

SbSigPRPTSigSRST

d ¯

ρpτp ¯ ρsτs(π) ≤ b ¯ ρp ¯ ρs(π) + 2

(7) d ¯

ρpτp ¯ ρsτs(π) ≥ b ¯ ρp ¯ ρs(π)

2 (8)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 32 / 46

SbSigPRPTSigSRST

d ¯

ρpτp ¯ ρsτs(π) ≤ b ¯ ρp ¯ ρs(π) + 2

(7) d ¯

ρpτp ¯ ρsτs(π) ≥ b ¯ ρp ¯ ρs(π)

2 (8) lim

b(π)→∞

b(π) + 2

b(π) 2

= 2 + lim

b(π)→∞

4 b(π) = 2 + ǫ (9)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 32 / 46

1

Introduction

2

Definitions

3

Algorithms SbSigPRSigSR SbSigPRPT SbSigPRPTSigSRST

4

Results

5

Conclusions

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 33 / 46

Results

All algorithms are O(n2)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 34 / 46

Results

All algorithms are O(n2) They ran under a set of arbitrary permutations: 10000 of each size n, with n varying between 10 and 1000 in intervals of 5

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 34 / 46

Results

All algorithms are O(n2) They ran under a set of arbitrary permutations: 10000 of each size n, with n varying between 10 and 1000 in intervals of 5 Approximation factors of the graph are an average between the 10000 of each size n

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 34 / 46

Results

All algorithms are O(n2) They ran under a set of arbitrary permutations: 10000 of each size n, with n varying between 10 and 1000 in intervals of 5 Approximation factors of the graph are an average between the 10000 of each size n Each approximation factor was calculated using the theoretical lower bound

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 34 / 46

Results

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 100 200 300 400 500 600 700 800 900 1000 average approximation factor permutation size 2-SPR 2-SPRSSR 2-SPRPT 2-SPRPTSSRST

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 35 / 46

Results

Since all algorithms are asymptotic, it is expected to have approximation factors above 2: SbSigPRSigSR: only when n = 10, for one permutation

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 36 / 46

Results

Since all algorithms are asymptotic, it is expected to have approximation factors above 2: SbSigPRSigSR: only when n = 10, for one permutation SbSigPRPT: only when n ≤ 100, for 0.41% of the permutations (72.13% when n ≤ 20)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 36 / 46

Results

Since all algorithms are asymptotic, it is expected to have approximation factors above 2: SbSigPRSigSR: only when n = 10, for one permutation SbSigPRPT: only when n ≤ 100, for 0.41% of the permutations (72.13% when n ≤ 20) SbSigPRPTSigSRST: only when n ≤ 105, for 0.44% of the permutations (76.62% when n ≤ 20)

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 36 / 46

1

Introduction

2

Definitions

3

Algorithms SbSigPRSigSR SbSigPRPT SbSigPRPTSigSRST

4

Results

5

Conclusions

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 37 / 46

Conclusions

We presented the first results for three sorting problems involving signed prefix and suffix

• perations: SbSigPRSigSR, SbSigPRPT and

SbSigPRPTSigSRST

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 38 / 46

Conclusions

We presented the first results for three sorting problems involving signed prefix and suffix

• perations: SbSigPRSigSR, SbSigPRPT and

SbSigPRPTSigSRST All algorithms established a good approximation factor for these problems

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 38 / 46

Conclusions

We presented the first results for three sorting problems involving signed prefix and suffix

• perations: SbSigPRSigSR, SbSigPRPT and

SbSigPRPTSigSRST All algorithms established a good approximation factor for these problems We are currently working on these problems and their unsigned versions

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 38 / 46

On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions

AlCoB’2014, Tarragona, Spain

Thank you!

July 3, 2014

References I

[1] H. Dweighter, “Problem E2569,” American Mathematical Monthly, vol. 82, p. 1010, 1975. [2] W. H. Gates and C. H. Papadimitriou, “Bounds for Sorting by Prefix Reversal,” Discrete Mathematics,

• vol. 27, no. 1, pp. 47–57, 1979.

[3] Z. Dias and J. Meidanis, “Sorting by Prefix Transpositions,” in Proceedings of the 9th International Symposium on String Processing and Information Retrieval (SPIRE’2002), (London, UK),

• pp. 65–76, Springer-Verlag, 2002.

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 40 / 46

References II

[4] M. Sharmin, R. Yeasmin, M. Hasan, A. Rahman, and

• M. S. Rahman, “Pancake Flipping with Two

Spatulas,” Electronic Notes in Discrete Mathematics,

• vol. 36, pp. 231–238, 2010.

[5] C. N. Lintzmayer and Z. Dias, “Sorting permutations by prefix and suffix versions of reversals and transpositions,” in LATIN 2014: Theoretical Informatics (A. Pardo and A. Viola, eds.), vol. 8392

• f Lecture Notes in Computer Science, pp. 671–682,

Springer, 2014.

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 41 / 46

References III

[6] P. Berman, S. Hannenhalli, and M. Karpinski, “1.375-Approximation Algorithm for Sorting by Reversals,” in Proceedings of the 10th Annual European Symposium on Algorithms (ESA’2002), (London, UK), pp. 200–210, Springer-Verlag, 2002. [7] A. Caprara, “Sorting Permutations by Reversals and Eulerian Cycle Decompositions,” SIAM Journal on Discrete Mathematics, vol. 12, no. 1, pp. 91–110, 1999.

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 42 / 46

References IV

[8] S. Hannenhalli and P. A. Pevzner, “Transforming Cabbage into Turnip: Polynomial Algorithm for Sorting Signed Permutations by Reversals,” Journal

• f the ACM, vol. 46, no. 1, pp. 1–27, 1999.

[9] I. Elias and T. Hartman, “A 1.375-Approximation Algorithm for Sorting by Transpositions,” IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 3, no. 4, pp. 369–379, 2006. [10] L. Bulteau, G. Fertin, and I. Rusu, “Sorting by Transpositions is Difficult,” SIAM Journal on Computing, vol. 26, no. 3, pp. 1148–1180, 2012.

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 43 / 46

References V

[11] A. Rahman, S. Shatabda, and M. Hasan, “An Approximation Algorithm for Sorting by Reversals and Transpositions,” Journal of Discrete Algorithms,

• vol. 6, no. 3, pp. 449–457, 2008.

[12] M. E. M. T. Walter, Z. Dias, and J. Meidanis, “Reversal and Transposition Distance of Linear Chromosomes,” in Proceedings of the 5th International Symposium on String Processing and Information Retrieval (SPIRE’1998), (Santa Cruz, Bolivia), pp. 96–102, IEEE Computer Society, 1998.

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 44 / 46

References VI

[13] J. Fischer and S. W. Ginzinger, “A 2-Approximation Algorithm for Sorting by Prefix Reversals,” in Proceedings of the 13th Annual European Conference on Algorithms (ESA’2005), (Berlin, Heidelberg), pp. 415–425, Springer-Verlag, 2005. [14] L. Bulteau, G. Fertin, and I. Rusu, “Pancake Flipping Is Hard,” in Mathematical Foundations of Computer Science 2012, vol. 7464 of Lecture Notes in Computer Science, pp. 247–258, Springer Berlin Heidelberg, 2012.

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 45 / 46

References VII

[15] D. S. Cohen and M. Blum, “On the Problem of Sorting Burnt Pancakes,” Discrete Applied Mathematics, vol. 61, no. 2, pp. 105–120, 1995.

Carla, Zanoni (UNICAMP) On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions 46 / 46