Reversal Distances for Strings with Few Blocks or Small Alphabets - - PowerPoint PPT Presentation

Introduction First results Fixed-parameter tractability Conclusion

Reversal Distances for Strings with Few Blocks or Small Alphabets

Laurent Bulteau, Guillaume Fertin, Christian Komusiewicz Technische Universität Berlin, Germany Université de Nantes, France

Supported by the Alexander von Humboldt Fondation, Bonn, Germany, and by Région Pays de la Loire, France

Introduction First results Fixed-parameter tractability Conclusion

Introduction

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Introduction First results Fixed-parameter tractability Conclusion

Context

Comparative genomics challenges:

Understand history of genes Compute evolutionary tree Track small- or large-scale evolution events Understand interactions between genes and/or proteins etc.

Genomes get rearranged progressively during evolution: cuts, duplications, reversals, etc.

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Introduction First results Fixed-parameter tractability Conclusion

Context

Comparative genomics challenges:

Understand history of genes Compute evolutionary tree Track small- or large-scale evolution events Understand interactions between genes and/or proteins etc.

Genomes get rearranged progressively during evolution: cuts, duplications, reversals, etc.

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Introduction First results Fixed-parameter tractability Conclusion

Reversal Distance

Reversal: genome rearrangement Reversal Distance: minimum number of reversals to go from

• ne genome to an other

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Introduction First results Fixed-parameter tractability Conclusion

Reversal Distance

Reversal: genome rearrangement Reversal Distance: minimum number of reversals to go from

• ne genome to an other

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Introduction First results Fixed-parameter tractability Conclusion

Variants

Signed Reversals: Prefix Reversals:

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Introduction First results Fixed-parameter tractability Conclusion

Blocks

Block: maximum substring using only 1 letter Block in signed strings: strict ⇒ common sign free ⇒ may have different signs a a b a b b b c c a a c a a c

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Introduction First results Fixed-parameter tractability Conclusion

Blocks

Block: maximum substring using only 1 letter Block in signed strings: strict ⇒ common sign free ⇒ may have different signs a a b a b b b c c a a c a a c

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Introduction First results Fixed-parameter tractability Conclusion

Blocks

Block: maximum substring using only 1 letter Block in signed strings: strict ⇒ common sign free ⇒ may have different signs +a

• a

+b +a +b +b

• b

+b

• b
• b

+a

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Introduction First results Fixed-parameter tractability Conclusion

Blocks

Block: maximum substring using only 1 letter Block in signed strings: strict ⇒ common sign free ⇒ may have different signs +a

• a

+b +a +b +b

• b

+b

• b
• b

+a

CPM, 2014-06-18

• L. Bulteau

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Introduction First results Fixed-parameter tractability Conclusion

Blocks

Block: maximum substring using only 1 letter Block in signed strings: strict ⇒ common sign free ⇒ may have different signs +a

• a

+b +a +b +b

• b

+b

• b
• b

+a

CPM, 2014-06-18

• L. Bulteau

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Introduction First results Fixed-parameter tractability Conclusion

Blocks

Block: maximum substring using only 1 letter Block in signed strings: strict ⇒ common sign free ⇒ may have different signs +a

• a

+b +a +b +b

• b

+b

• b
• b

+a

Parameters

b: maximum number of blocks in each input string |Σ|: alphabet size, |Σ| < b.

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Introduction First results Fixed-parameter tractability Conclusion

Blocks

Block: maximum substring using only 1 letter Block in signed strings: strict ⇒ common sign free ⇒ may have different signs +a

• a

+b +a +b +b

• b

+b

• b
• b

+a

Parameters

b: maximum number of blocks in each input string |Σ|: alphabet size, |Σ| < b.

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Introduction First results Fixed-parameter tractability Conclusion

Complexity of computing the reversal distance

input /

permutations strings strings with few blocks small

reversals

alphabet

regular

NP-hard

signed

P

prefix signed prefix state of the art

• ur contribution

[Bafna, Pevzner ’96], [Christie ’98]

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Introduction First results Fixed-parameter tractability Conclusion

Complexity of computing the reversal distance

input /

permutations strings strings with few blocks small

reversals

alphabet

regular

NP-hard

signed

P

prefix

NP-hard

signed prefix

?

state of the art

• ur contribution

[Bulteau, Fertin, Rusu ’11]

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Introduction First results Fixed-parameter tractability Conclusion

Complexity of computing the reversal distance

input /

permutations strings strings with few blocks small

reversals

alphabet

regular

NP-hard NP-hard

signed

P NP-hard

prefix

NP-hard NP-hard

signed prefix

? NP-hard

state of the art

• ur contribution

[Christie ’98] [Radcliffe, Scott, Wilmer ’06]

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Introduction First results Fixed-parameter tractability Conclusion

Complexity of computing the reversal distance

input /

permutations strings strings with few blocks small

reversals

alphabet

regular

NP-hard NP-hard FPT

signed

P NP-hard

prefix

NP-hard NP-hard FPT

signed prefix

? NP-hard

state of the art

• ur contribution

FPT algorithm: O((6b)2bn) (constant |Σ|, reversal distance) bO(b)n (other cases)

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Introduction First results Fixed-parameter tractability Conclusion

Complexity of computing the reversal distance

input /

permutations strings strings with few blocks small

reversals

(strict) (free) alphabet

regular

NP-hard NP-hard FPT

signed

P NP-hard FPT

prefix

NP-hard NP-hard FPT

signed prefix

? NP-hard FPT

state of the art

• ur contribution

FPT algorithm: O((6b)2bn) (constant |Σ|, reversal distance) bO(b)n (other cases)

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Introduction First results Fixed-parameter tractability Conclusion

Complexity of computing the reversal distance

input /

permutations strings strings with few blocks small

reversals

(strict) (free) alphabet

regular

NP-hard NP-hard FPT

signed

P NP-hard FPT NP-hard

prefix

NP-hard NP-hard FPT

signed prefix

? NP-hard FPT NP-hard

state of the art

• ur contribution

FPT algorithm: O((6b)2bn) (constant |Σ|, reversal distance) bO(b)n (other cases) NP-hardness: even with b = |Σ| = 1 (unary alphabet)

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Introduction First results Fixed-parameter tractability Conclusion

Complexity of computing the reversal distance

input /

permutations strings strings with few blocks small

reversals

(strict) (free) alphabet

regular

NP-hard NP-hard FPT Exp.

signed

P NP-hard FPT NP-hard Exp.

prefix

NP-hard NP-hard FPT Exp.

signed prefix

? NP-hard FPT NP-hard Exp.

state of the art

• ur contribution

FPT algorithm: O((6b)2bn) (constant |Σ|, reversal distance) bO(b)n (other cases) NP-hardness: even with b = |Σ| = 1 (unary alphabet) Exact algorithm: |Σ|npoly(n) (unsigned) (2|Σ|)npoly(n) (signed)

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Introduction First results Fixed-parameter tractability Conclusion

Complexity of computing the reversal distance

input /

permutations strings strings with few blocks small

reversals

(strict) (free) alphabet

regular

NP-hard NP-hard FPT Exp.

signed

P NP-hard FPT NP-hard Exp.

prefix

NP-hard NP-hard FPT Exp.

signed prefix

? NP-hard FPT NP-hard Exp.

state of the art

• ur contribution

FPT algorithm: O((6b)2bn) (constant |Σ|, reversal distance) bO(b)n (other cases) NP-hardness: even with b = |Σ| = 1 (unary alphabet) Exact algorithm: |Σ|npoly(n) (unsigned) (2|Σ|)npoly(n) (signed)

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Introduction First results Fixed-parameter tractability Conclusion

First results

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Introduction First results Fixed-parameter tractability Conclusion

Preliminary result: diameter

The reversal distance between two strings with ≤ b blocks is upper-bounded by: 2b − |Σ|

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Introduction First results Fixed-parameter tractability Conclusion

Preliminary result: diameter

The reversal distance between two strings with ≤ b blocks is upper-bounded by: 2b − |Σ| → Obtained by: – grouping the blocks in each string into |Σ| bigger blocks (b − |Σ| reversals for each side), – sorting the middle strings (|Σ| reversals).

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Introduction First results Fixed-parameter tractability Conclusion

Small alphabet: exact algorithm

The Reversal Distance with alphabet Σ can be computed in time |Σ|npoly(n)

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Introduction First results Fixed-parameter tractability Conclusion

Small alphabet: exact algorithm

The Reversal Distance with alphabet Σ can be computed in time |Σ|npoly(n) → There are |Σ|n possible strings over alphabet Σ. Draw the Cayley graph: add an edge between any two strings which are 1 reversal apart. Then Reversal Distance amounts to finding a shortest path.

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Introduction First results Fixed-parameter tractability Conclusion

Unary alphabets: NP-hardness

It is NP-hard to compute the Signed Reversal Distance, even on unary strings

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Introduction First results Fixed-parameter tractability Conclusion

Unary alphabets: NP-hardness

It is NP-hard to compute the Signed Reversal Distance, even on unary strings → From reversal distance on permutations . . . 5 8 2 3 1 6 4 . . . ↓ S3 = + − + + + + − − − − + −

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Introduction First results Fixed-parameter tractability Conclusion

Unary alphabets: NP-hardness

It is NP-hard to compute the Signed Reversal Distance, even on unary strings → From reversal distance on permutations . . . 5 8 2 3 1 6 4 . . . ↓ S3 = + − + + + + − − − − + − . . . (S5)n (S8)n (S2)n (S3)n (S1)n (S6)n (S4)n . . .

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Introduction First results Fixed-parameter tractability Conclusion

Unary alphabets: NP-hardness

It is NP-hard to compute the Signed Reversal Distance, even on unary strings → From reversal distance on permutations . . . 5 8 2 3 1 6 4 . . . ↓ S3 = + − + + + + − − − − + − . . . (S5)n (S8)n (S2)n (S3)n (S1)n (S6)n (S4)n . . . Si = −Si (computing an unsigned distance) no long prefix of Si is a suffix of Sj (no overlapping possible) < n reversals (one Si remains intact in each block)

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Introduction First results Fixed-parameter tractability Conclusion

Fixed-parameter tractability

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Introduction First results Fixed-parameter tractability Conclusion

Merge blocks, keep only the structure Guess reversals endpoints, inside or between blocks, repeat until target structure is reached Create a graph from the blocks, add arcs keeping track of the movement of letters Compute a maximum flow, accept if the flow is n

• a

a a b c c a a a b b a c b b b

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Introduction First results Fixed-parameter tractability Conclusion

Merge blocks, keep only the structure Guess reversals endpoints, inside or between blocks, repeat until target structure is reached Create a graph from the blocks, add arcs keeping track of the movement of letters Compute a maximum flow, accept if the flow is n

• a

a a b c c a a a b b a c b b b

CPM, 2014-06-18

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Introduction First results Fixed-parameter tractability Conclusion

Merge blocks, keep only the structure Guess reversals endpoints, inside or between blocks, repeat until target structure is reached Create a graph from the blocks, add arcs keeping track of the movement of letters Compute a maximum flow, accept if the flow is n

• a

b c a b a c b

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Introduction First results Fixed-parameter tractability Conclusion

Merge blocks, keep only the structure Guess reversals endpoints, inside or between blocks, repeat until target structure is reached Create a graph from the blocks, add arcs keeping track of the movement of letters Compute a maximum flow, accept if the flow is n

• a

b c a b a c b

CPM, 2014-06-18

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Introduction First results Fixed-parameter tractability Conclusion

Merge blocks, keep only the structure Guess reversals endpoints, inside or between blocks, repeat until target structure is reached Create a graph from the blocks, add arcs keeping track of the movement of letters Compute a maximum flow, accept if the flow is n

• a

b c a b a c b

CPM, 2014-06-18

• L. Bulteau

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Introduction First results Fixed-parameter tractability Conclusion

Merge blocks, keep only the structure Guess reversals endpoints, inside or between blocks, repeat until target structure is reached Create a graph from the blocks, add arcs keeping track of the movement of letters Compute a maximum flow, accept if the flow is n

• a

b c a b a c b a c b a b a c b

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Introduction First results Fixed-parameter tractability Conclusion

Merge blocks, keep only the structure Guess reversals endpoints, inside or between blocks, repeat until target structure is reached Create a graph from the blocks, add arcs keeping track of the movement of letters Compute a maximum flow, accept if the flow is n

• a

b c a b a c b a c b a b a c b a c a b a b

CPM, 2014-06-18

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Introduction First results Fixed-parameter tractability Conclusion

Merge blocks, keep only the structure Guess reversals endpoints, inside or between blocks, repeat until target structure is reached Create a graph from the blocks, add arcs keeping track of the movement of letters Compute a maximum flow, accept if the flow is n

• a

b c a b a c b a c b a b a c b a c a b a b a c a b a b

CPM, 2014-06-18

• L. Bulteau

13/17

Introduction First results Fixed-parameter tractability Conclusion

Merge blocks, keep only the structure Guess reversals endpoints, inside or between blocks, repeat until target structure is reached Create a graph from the blocks, add arcs keeping track of the movement of letters Compute a maximum flow, accept if the flow is n

• 3

6 2 6 1 4 a b c a b a c b a c b a b a c b a c a b a b a c a b a b

• CPM, 2014-06-18
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Introduction First results Fixed-parameter tractability Conclusion

Merge blocks, keep only the structure Guess reversals endpoints, inside or between blocks, repeat until target structure is reached Create a graph from the blocks, add arcs keeping track of the movement of letters Compute a maximum flow, accept if the flow is n

• 3

6 2 6 1 4

• CPM, 2014-06-18
• L. Bulteau

13/17

Introduction First results Fixed-parameter tractability Conclusion

Merge blocks, keep only the structure Guess reversals endpoints, inside or between blocks, repeat until target structure is reached Create a graph from the blocks, add arcs keeping track of the movement of letters Compute a maximum flow, accept if the flow is n

• 3

6 2 6 1 4

• CPM, 2014-06-18
• L. Bulteau

13/17

Introduction First results Fixed-parameter tractability Conclusion

Merge blocks, keep only the structure Guess reversals endpoints, inside or between blocks, repeat until target structure is reached Create a graph from the blocks, add arcs keeping track of the movement of letters Compute a maximum flow, accept if the flow is n

• 3

6 2 6 1 4

• CPM, 2014-06-18
• L. Bulteau

13/17

Introduction First results Fixed-parameter tractability Conclusion

Merge blocks, keep only the structure Guess reversals endpoints, inside or between blocks, repeat until target structure is reached Create a graph from the blocks, add arcs keeping track of the movement of letters Compute a maximum flow, accept if the flow is n

• 3

6 2 6 1

4

• CPM, 2014-06-18
• L. Bulteau

13/17

Introduction First results Fixed-parameter tractability Conclusion

Merge blocks, keep only the structure Guess reversals endpoints, inside or between blocks, repeat until target structure is reached Create a graph from the blocks, add arcs keeping track of the movement of letters Compute a maximum flow, accept if the flow is n

• 3

6 2 6 1

4

• CPM, 2014-06-18
• L. Bulteau

13/17

Introduction First results Fixed-parameter tractability Conclusion

Merge blocks, keep only the structure Guess reversals endpoints, inside or between blocks, repeat until target structure is reached Create a graph from the blocks, add arcs keeping track of the movement of letters Compute a maximum flow, accept if the flow is n

• 3

6 2

3 2 4 2 5 2 4 5 1 1 4 6 1

4

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Introduction First results Fixed-parameter tractability Conclusion

Analysis

Merging the blocks:

n (trivial)

Search-tree:

O(b) choices (cf. diameter) O(b) options at each choice (bounded number of blocks) bO(b) branches

Verification:

bO(1) (flow algorithm)

Variants: works for prefix and/or signed reversal distance

(with the strict definition of signed block)

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Introduction First results Fixed-parameter tractability Conclusion

Analysis

Merging the blocks:

n (trivial)

Search-tree:

O(b) choices (cf. diameter) O(b) options at each choice (bounded number of blocks) bO(b) branches

Verification:

bO(1) (flow algorithm)

Variants: works for prefix and/or signed reversal distance

(with the strict definition of signed block)

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Introduction First results Fixed-parameter tractability Conclusion

Analysis

Merging the blocks:

n (trivial)

Search-tree:

O(b) choices (cf. diameter) O(b) options at each choice (bounded number of blocks) bO(b) branches

Verification:

bO(1) (flow algorithm)

Variants: works for prefix and/or signed reversal distance

(with the strict definition of signed block)

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Introduction First results Fixed-parameter tractability Conclusion

Analysis

Merging the blocks:

n (trivial)

Search-tree:

O(b) choices (cf. diameter) O(b) options at each choice (bounded number of blocks) bO(b) branches

Verification:

bO(1) (flow algorithm)

Variants: works for prefix and/or signed reversal distance

(with the strict definition of signed block)

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Introduction First results Fixed-parameter tractability Conclusion

Analysis

Merging the blocks:

n (trivial)

Search-tree:

O(b) choices (cf. diameter) O(b) options at each choice (bounded number of blocks) bO(b) branches

Verification:

bO(1) (flow algorithm)

Variants: works for prefix and/or signed reversal distance

(with the strict definition of signed block)

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Introduction First results Fixed-parameter tractability Conclusion

Conclusion

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Introduction First results Fixed-parameter tractability Conclusion

String reversal distance with few blocks Fixed-Parameter Tractable Signed case: only tractable for the strictest definition Open questions Diameter of b − 1? (cf. diameter of n − 1 for permutations) Or, if not, diameter of b + O(|Σ|)? Polynomial kernel for parameter b? O((|Σ| − ε)n) algorithm?

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Introduction First results Fixed-parameter tractability Conclusion

Thank you!

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