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 CPM, 2014-06-18 L. Bulteau 2/17
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. CPM, 2014-06-18 L. Bulteau 3/17
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. CPM, 2014-06-18 L. Bulteau 3/17
Introduction First results Fixed-parameter tractability Conclusion Reversal Distance Reversal: genome rearrangement Reversal Distance: minimum number of reversals to go from one genome to an other CPM, 2014-06-18 L. Bulteau 4/17
Introduction First results Fixed-parameter tractability Conclusion Reversal Distance Reversal: genome rearrangement Reversal Distance: minimum number of reversals to go from one genome to an other CPM, 2014-06-18 L. Bulteau 4/17
Introduction First results Fixed-parameter tractability Conclusion Variants Signed Reversals: Prefix Reversals: CPM, 2014-06-18 L. Bulteau 5/17
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 CPM, 2014-06-18 L. Bulteau 6/17
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 CPM, 2014-06-18 L. Bulteau 6/17
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 6/17
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 6/17
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 6/17
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 . CPM, 2014-06-18 L. Bulteau 6/17
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 . CPM, 2014-06-18 L. Bulteau 6/17
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 P signed prefix signed prefix state of the art our contribution [Bafna, Pevzner ’96], [Christie ’98] CPM, 2014-06-18 L. Bulteau 7/17
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 P signed NP-hard prefix signed prefix ? state of the art our contribution [Bulteau, Fertin, Rusu ’11] CPM, 2014-06-18 L. Bulteau 7/17
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 P NP-hard signed NP-hard NP-hard prefix signed prefix ? NP-hard state of the art our contribution [Christie ’98] [Radcliffe, Scott, Wilmer ’06] CPM, 2014-06-18 L. Bulteau 7/17
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 P NP-hard signed NP-hard NP-hard FPT prefix signed prefix ? NP-hard state of the art our contribution FPT algorithm: O ((6 b ) 2 b n ) (constant | Σ | , reversal distance) b O ( b ) n (other cases) CPM, 2014-06-18 L. Bulteau 7/17
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 P NP-hard FPT signed NP-hard NP-hard FPT prefix signed prefix ? NP-hard FPT state of the art our contribution FPT algorithm: O ((6 b ) 2 b n ) (constant | Σ | , reversal distance) b O ( b ) n (other cases) CPM, 2014-06-18 L. Bulteau 7/17
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 P NP-hard FPT NP-hard signed NP-hard NP-hard FPT prefix signed prefix ? NP-hard FPT NP-hard state of the art our contribution FPT algorithm: O ((6 b ) 2 b n ) (constant | Σ | , reversal distance) b O ( b ) n (other cases) NP-hardness: even with b = | Σ | = 1 (unary alphabet) CPM, 2014-06-18 L. Bulteau 7/17
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. P NP-hard FPT NP-hard Exp. signed NP-hard NP-hard FPT Exp. prefix signed prefix ? NP-hard FPT NP-hard Exp. state of the art our contribution FPT algorithm: O ((6 b ) 2 b n ) (constant | Σ | , reversal distance) b O ( b ) n (other cases) NP-hardness: even with b = | Σ | = 1 (unary alphabet) Exact algorithm: | Σ | n poly ( n ) (unsigned) (2 | Σ | ) n poly ( n ) (signed) CPM, 2014-06-18 L. Bulteau 7/17
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. P NP-hard FPT NP-hard Exp. signed NP-hard NP-hard FPT Exp. prefix signed prefix ? NP-hard FPT NP-hard Exp. state of the art our contribution FPT algorithm: O ((6 b ) 2 b n ) (constant | Σ | , reversal distance) b O ( b ) n (other cases) NP-hardness: even with b = | Σ | = 1 (unary alphabet) Exact algorithm: | Σ | n poly ( n ) (unsigned) (2 | Σ | ) n poly ( n ) (signed) CPM, 2014-06-18 L. Bulteau 7/17
Introduction First results Fixed-parameter tractability Conclusion First results CPM, 2014-06-18 L. Bulteau 8/17
Introduction First results Fixed-parameter tractability Conclusion Preliminary result: diameter The reversal distance between two strings with ≤ b blocks is upper-bounded by: 2 b − | Σ | CPM, 2014-06-18 L. Bulteau 9/17
Introduction First results Fixed-parameter tractability Conclusion Preliminary result: diameter The reversal distance between two strings with ≤ b blocks is upper-bounded by: 2 b − | 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). CPM, 2014-06-18 L. Bulteau 9/17
Introduction First results Fixed-parameter tractability Conclusion Small alphabet: exact algorithm The Reversal Distance with alphabet Σ can be computed in time | Σ | n poly ( n ) CPM, 2014-06-18 L. Bulteau 10/17
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