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

1

Introduction

2

Definitions

3

State of the art

4

Results

5

Conclusions

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

1

Introduction

2

Definitions

3

State of the art

4

Results

5

Conclusions

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 4 / 35

Definitions

Permutation: π = (π1 π2 . . . πn) where πi = π(i)

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 5 / 35

Definitions

Permutation: π = (π1 π2 . . . πn) where πi = π(i) Unsigned permutation: πi ∈ {1, 2, . . . , n} and πi = πj for all i = j

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 5 / 35

Definitions

Permutation: π = (π1 π2 . . . πn) where πi = π(i) Unsigned permutation: πi ∈ {1, 2, . . . , n} and πi = πj for all i = j Signed permutation: πi ∈ {−n, −(n − 1), . . . , −1, 1, 2, . . . , n} and |πi| = |πj| for all i = j

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 5 / 35

Definitions

Permutation: π = (π1 π2 . . . πn) where πi = π(i) Unsigned permutation: πi ∈ {1, 2, . . . , n} and πi = πj for all i = j Signed permutation: πi ∈ {−n, −(n − 1), . . . , −1, 1, 2, . . . , n} and |πi| = |πj| for all i = j Extended: (π0 = 0 π1 π2 . . . πn πn+1 = n + 1)

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 5 / 35

Definitions

Permutation: π = (π1 π2 . . . πn) where πi = π(i) Unsigned permutation: πi ∈ {1, 2, . . . , n} and πi = πj for all i = j Signed permutation: πi ∈ {−n, −(n − 1), . . . , −1, 1, 2, . . . , n} and |πi| = |πj| for all i = j Extended: (π0 = 0 π1 π2 . . . πn πn+1 = n + 1) Composition: π · σ = (πσ1 πσ2 . . . πσn)

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 5 / 35

Definitions

Permutation: π = (π1 π2 . . . πn) where πi = π(i) Unsigned permutation: πi ∈ {1, 2, . . . , n} and πi = πj for all i = j Signed permutation: πi ∈ {−n, −(n − 1), . . . , −1, 1, 2, . . . , n} and |πi| = |πj| for all i = j Extended: (π0 = 0 π1 π2 . . . πn πn+1 = n + 1) Composition: π · σ = (πσ1 πσ2 . . . πσn) Identity permutation: ιn = (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 = (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 the Diameter of Rearrangement Problems 6 / 35

Definitions

(Unsigned) 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 the Diameter of Rearrangement Problems 6 / 35

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 the Diameter of Rearrangement Problems 7 / 35

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 the Diameter of Rearrangement Problems 7 / 35

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 the Diameter of Rearrangement Problems 8 / 35

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) 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: ρp(j) ≡ ρ(1, j) for 1 < j ≤ n ◮ signed: ¯

ρp(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: ρp(j) ≡ ρ(1, j) for 1 < j ≤ n ◮ signed: ¯

ρp(j) ≡ ¯ ρ(1, j) for 1 ≤ j ≤ n

Suffix reversal (inverts last segment):

◮ unsigned: ρs(i) ≡ ρ(i, n) for 1 ≤ i < n ◮ signed: ¯

ρs(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: ρp(j) ≡ ρ(1, j) for 1 < j ≤ n ◮ signed: ¯

ρp(j) ≡ ¯ ρ(1, j) for 1 ≤ j ≤ n

Suffix reversal (inverts last segment):

◮ unsigned: ρs(i) ≡ ρ(i, n) for 1 ≤ i < n ◮ signed: ¯

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 the Diameter of Rearrangement Problems 9 / 35

Definitions

Prefix reversal (inverts first segment):

◮ unsigned: ρp(j) ≡ ρ(1, j) for 1 < j ≤ n ◮ signed: ¯

ρp(j) ≡ ¯ ρ(1, j) for 1 ≤ j ≤ n

Suffix reversal (inverts last segment):

◮ unsigned: ρs(i) ≡ ρ(i, n) for 1 ≤ i < n ◮ signed: ¯

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 the Diameter of Rearrangement Problems 9 / 35

Definitions

Rearrangement model: β

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 10 / 35

Definitions

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

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 10 / 35

Definitions

Diameter: Dβ(n)

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 11 / 35

Definitions

Diameter: Dβ(n) D ¯

ρp(2) = 4

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 11 / 35

Definitions

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

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 12 / 35

Definitions

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

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 12 / 35

Definitions

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

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 12 / 35

Definitions

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

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 12 / 35

1

Introduction

2

Definitions

3

State of the art

4

Results

5

Conclusions

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 13 / 35

Problems

Problem Diameter SbR Dρ(n) = n − 1 [1] SbSigR D¯

ρ(n) = n + 1 [2]

SbT n+1

2

• ≤ Dτ(n) ≤

2n−2

3

• [3, 4]

SbRT ? SbSigRT n

2

• + 2 ≤ D¯

ρτ(n) [5]

SbPR

15n 14 ≤ Dρp(n) ≤ 18n 11 + O(1) [6, 7]

SbSigPR

3n+3 2

≤ D ¯

ρp(n) ≤ 2n − 6 [6, 8]

SbPT 3n+1

4

• ≤ Dτp(n) ≤ n − log 9

2 n [9, 10]

SbPRPT ?

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 14 / 35

1

Introduction

2

Definitions

3

State of the art

4

Results

5

Conclusions

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 15 / 35

Results

We found families of permutations for 8 problems:

◮ SbRT ◮ SbPRPT ◮ SbSigPRPT ◮ SbPRSR ◮ SbSigPRSigSR ◮ SbPTST ◮ SbPRPTSRST ◮ SbSigPRPTSigSRST

Lower and upper bounds on the diameters

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 16 / 35

Results

Family for SbPRSR: π∗

n =

• (n 1 n

− 2 n − 4 n − 6 ... 4 2 n − 3 n − 5 n − 7 ... 3 n − 1)

if n is even

(n 1 n − 2 n − 4 n − 6 ... 5 3 n − 3 n − 5 n − 7 ... 2 n − 1)

if n is odd

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 17 / 35

Results

Family for SbPRSR: π∗

n =

• (n 1 n

− 2 n − 4 n − 6 ... 4 2 n − 3 n − 5 n − 7 ... 3 n − 1)

if n is even

(n 1 n − 2 n − 4 n − 6 ... 5 3 n − 3 n − 5 n − 7 ... 2 n − 1)

if n is odd

n = 9 : (9 1 7 5 3 6 4 2 8) n = 10 : (10 1 8 6 4 2 7 5 3 9) n = 11 : (11 1 9 7 5 3 8 6 4 2 10) n = 12 : (12 1 10 8 6 4 2 9 7 5 3 11)

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 17 / 35

Results

Lemma

For n ≥ 8, n − 1 ≤ dρpρs(π∗

n) ≤ n.

Proof.

The lower bound is true because bρpρs(π∗

n) = n − 1 and

dρpρs(π) ≥ bρpρs(π) for any π [11]. The upper bound is true because of next algorithm.

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 18 / 35

Results

Algorithm to sort π∗

n:

Input: π = π∗

n, n ≥ 8

π ← π · ρp(n − 1) π ← π · ρp(n − 3) π ← π · ρs(2) π ← π · ρs(π−1

πn+1 + 1)

π ← π · ρp(π−1

π1−1 − 1)

π ← π · ρs(π−1

n )

while π1 = 1 do π ← π · ρp(π−1

π1+1 − 1)

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 19 / 35

Results

Lemma

For n ≥ 8, Dρpρs(n) ≥ n − 1 and for n ≥ 1, Dρpρs(n) ≤ 18n

11 + O(1).

Proof.

The lower bound is true because of family π∗

• n. The upper

bound is true because Dρpρs(n) ≤ Dρp(n), since dρpρs(π) ≤ dρp(π) for any π. This is true because any sorting sequence for Sorting by Prefix Reversals is valid for Sorting by Prefix Reversals and Suffix Reversals.

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 20 / 35

Results

We know that Dρpρs(n) = n for 7 ≤ n ≤ 13 We verified that dρpρs(π∗

n) = n for 8 ≤ n ≤ 15

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 21 / 35

Results

We know that Dρpρs(n) = n for 7 ≤ n ≤ 13 We verified that dρpρs(π∗

n) = n for 8 ≤ n ≤ 15

Conjecture

For n ≥ 8, Dρpρs(n) = dρpρs(π∗

n) = n.

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 21 / 35

Results

Problem Diameter

SbRT

n

2

• ≤

Dρτ(n) ≤ 2n−2

3

• SbPRPT

n

2

• ≤ Dρpτp(n) ≤ n − log 9

2 n

SbSigPRPT

n

2

• + 1 ≤ D ¯

ρpτp(n) ≤ 18n 11 + O(1)

SbPRSR

n − 1 ≤ Dρpρs(n) ≤ 18n

11 + O(1)

SbSigPRSigSR

n ≤ D ¯

ρp ¯ ρs(n) ≤ 2n − 6

SbPTST

n−1

2

• + 1 ≤ Dτpτs(n) ≤ n − log 9

2 n

SbPRPTSRST

n

2

• ≤Dρpτpρsτs(n)≤ n − log 9

2 n

SbSigPRPTSigSRST

n−1

2

• ≤D ¯

ρpτp ¯ ρsτs(n)≤ n + 1

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 22 / 35

Results

We know that

1

Dρτ(n) = n

2

• for 4 ≤ n ≤ 13

2

Dρpρs(n) = n for 7 ≤ n ≤ 13

3

D ¯

ρp ¯ ρs(n) = n +

n−1

2

• for 5 ≤ n ≤ 10

4

Dρpτpρsτs(n) = n

2

• + 1 for 6 ≤ n ≤ 13

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 23 / 35

Results

It is also possible to validate that

1

dρpρs(πn) = n for 8 ≤ n ≤ 15

2

d ¯

ρp ¯ ρs(πn) = n +

n−1

2

• for 5 ≤ n ≤ 12

3

dρpτpρsτs(πn) = n

2

• + 1 for 6 ≤ n ≤ 15

4

D ¯

ρpτp(n) = d ¯ ρpτp(πn) for 2 ≤ n ≤ 10

5

Dτpτs(n) = dτpτs(πn) for 1 ≤ n ≤ 12

6

D ¯

ρpτp ¯ ρsτs(n) = d ¯ ρpτp ¯ ρsτs(πn) for n ∈ {8, 10} and

D ¯

ρpτp ¯ ρsτs(n) = d ¯ ρpτp ¯ ρsτs(πn) for n ∈ {7, 9}

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 24 / 35

Results

Conjecture

For n ≥ 4, Dρτ(n) = dρτ(πn) = n

2

• .

Conjecture

For n ≥ 8, Dρpρs(n) = dρpρs(πn) = n.

Conjecture

For n ≥ 5, D ¯

ρp ¯ ρs(n) = d ¯ ρp ¯ ρs(πn) = n +

n−1

2

• .

Conjecture

For n ≥ 6, Dρpτpρsτs(n) = dρpτpρsτs(πn) = n

2

• + 1.

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 25 / 35

Results

Conjecture

For n ≥ 2, D ¯

ρpτp(n) = d ¯ ρpτp(πn).

Conjecture

For n ≥ 1, Dτpτs(n) = dτpτs(ηn) = n − n

3

• .

Conjecture

For n ≥ 8 and n even, D ¯

ρpτp ¯ ρsτs(n) = d ¯ ρpτp ¯ ρsτs(πn). For

n ≥ 7 and n odd, D ¯

ρpτp ¯ ρsτs(n) = d ¯ ρpτp ¯ ρsτs(πn).

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 26 / 35

1

Introduction

2

Definitions

3

State of the art

4

Results

5

Conclusions

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 27 / 35

Conclusions

We presented the first results on the diameter of 8 problems for which more than one rearrangement is allowed

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 28 / 35

Conclusions

We presented the first results on the diameter of 8 problems for which more than one rearrangement is allowed

◮ specially when prefix and suffix operations are involved Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 28 / 35

Conclusions

We presented the first results on the diameter of 8 problems for which more than one rearrangement is allowed

◮ specially when prefix and suffix operations are involved

We verified that the conjecture of Meidanis et

• al. [12] that D¯

ρτ(n) =

n

2

• + 2 is not true

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 28 / 35

Conclusions

We presented the first results on the diameter of 8 problems for which more than one rearrangement is allowed

◮ specially when prefix and suffix operations are involved

We verified that the conjecture of Meidanis et

• al. [12] that D¯

ρτ(n) =

n

2

• + 2 is not true

It is also not true that D¯

ρτ(n) = d¯ ρτ( ¯

ιn)

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 28 / 35

Conclusions

We presented the first results on the diameter of 8 problems for which more than one rearrangement is allowed

◮ specially when prefix and suffix operations are involved

We verified that the conjecture of Meidanis et

• al. [12] that D¯

ρτ(n) =

n

2

• + 2 is not true

It is also not true that D¯

ρτ(n) = d¯ ρτ( ¯

ιn) We keep working on these problems trying to improve the results

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 28 / 35

On the Diameter of Rearrangement Problems

AlCoB’2014, Tarragona, Spain

Thank you!

July 3, 2014

References I

[1] V. Bafna and P. A. Pevzner, “Genome Rearrangements and Sorting by Reversals,” in Proceedings of the 34th Annual Symposium on Foundations of Computer Science (FOCS’1993),

• pp. 148–157, 1993.

[2] 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.

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 30 / 35

References II

[3] 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. [4] H. Eriksson, K. Eriksson, J. Karlander, L. Svensson, and J. Wastlund, “Sorting a Bridge Hand,” Discrete Mathematics, vol. 241, no. 1-3, pp. 289–300, 2001. [5] 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

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 31 / 35

References III

Information Retrieval (SPIRE’1998), (Santa Cruz, Bolivia), pp. 96–102, IEEE Computer Society, 1998. [6] M. H. Heydari and I. H. Sudborough, “On the Diameter of the Pancake Network,” Journal of Algorithms, vol. 25, no. 1, pp. 67–94, 1997. [7] B. Chitturi, W. Fahle, Z. Meng, L. Morales, C. O. Shields, I. H. Sudborough, and W. Voit, “An (18/11)n Upper Bound for Sorting by Prefix Reversals,” Theoretical Computer Science, vol. 410,

• no. 36, pp. 3372–3390, 2009.

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 32 / 35

References IV

[8] J. Cibulka, “On Average and Highest Number of Flips in Pancake Sorting,” Theoretical Computer Science, vol. 412, no. 8-10, pp. 822–834, 2011. [9] A. Labarre, “Edit Distances and Factorisations of Even Permutations,” in Proceedings of the 16th Annual European Symposium on Algorithms (ESA’2008), (Berlin, Heidelberg), pp. 635–646, Springer-Verlag, 2008.

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 33 / 35

References V

[10] B. Chitturi and I. H. Sudborough, “Bounding Prefix Transposition Distance for Strings and Permutations,” Theoretical Computer Science,

• vol. 421, pp. 15–24, 2012.

[11] C. N. Lintzmayer and Z. Dias, “Sorting Permutations by Prefix and Suffix Versions of Reversals and Transpositions,” in Proceedings of the 11th Latin American Theoretical Informatics Symposium (LATIN’2014) (A. Pardo and A. Viola, eds.), (Montevideo, Uruguay), pp. 671–682, Springer-Verlag, 2014.

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 34 / 35

References VI

[12] J. Meidanis, M. M. T. Walter, and Z. Dias, “A Lower Bound on the Reversal and Transposition Diameter,” Journal of Computational Biology, vol. 9,

• no. 5, pp. 743–745, 2002.

Carla, Zanoni (UNICAMP) On the Diameter of Rearrangement Problems 35 / 35