##
**New applications of interval generators to genome comparison.**
*(English)*
Zbl 1236.92023

Summary: We address two different problems related to conserved regions in \(K\geqslant 2\) genomes represented as permutations. The first one requires to enumerate the conserved regions, whereas the second one asks only to count them. We show that the generator-based technique, introduced by A. Bergeron et al. [see Lect. Notes Computer Sci. 2697, 68–79 (2003; Zbl 1236.92020)] to enumerate common intervals of \(K\) genomes represented as permutations, may be extended following two directions.

Firstly, it may be applied to signed permutations, yielding (1) a method to enumerate in \(O(Kn+N)\) time the \(N\) conserved intervals of \(K\) such permutations on n elements, and (2) a method to enumerate in \(O(Kn)\) time the irreducible conserved intervals of those \(K\) permutations. Secondly, it may be used to solve in \(O(Kn)\) time the counting problem, for every class of intervals which admits a so-called canonical generator. Both common and conserved intervals of \(K\) (signed) permutations admit such a generator. Although some (not all) of the above running times have already been reached by previous algorithms, it is the first time that the features shared by common and conserved intervals are exploited under a common efficient framework.

Firstly, it may be applied to signed permutations, yielding (1) a method to enumerate in \(O(Kn+N)\) time the \(N\) conserved intervals of \(K\) such permutations on n elements, and (2) a method to enumerate in \(O(Kn)\) time the irreducible conserved intervals of those \(K\) permutations. Secondly, it may be used to solve in \(O(Kn)\) time the counting problem, for every class of intervals which admits a so-called canonical generator. Both common and conserved intervals of \(K\) (signed) permutations admit such a generator. Although some (not all) of the above running times have already been reached by previous algorithms, it is the first time that the features shared by common and conserved intervals are exploited under a common efficient framework.

### MSC:

92C40 | Biochemistry, molecular biology |

92D10 | Genetics and epigenetics |

05A05 | Permutations, words, matrices |

68W32 | Algorithms on strings |

92-08 | Computational methods for problems pertaining to biology |

92-04 | Software, source code, etc. for problems pertaining to biology |

### Citations:

Zbl 1236.92020### Software:

CREx
Full Text:
DOI

### References:

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[2] | Bergeron, A.; Stoye, J., On the similarity of sets of permutations and its applications to genome comparison, J. comput. biol., 13, 1340-1354, (2006) |

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[6] | G. Fertin, I. Rusu, Computing genomic distances: An algorithmic viewpoint, in: M. Elloumi, A.Y. Zomaya (Eds.), Algorithms in Computational Molecular Biology: Techniques, Approaches and Applications, Wiley Series in Bioinformatics, 2011, pp. 773-798 (Chapter 34). |

[7] | Fertin, G.; Labarre, A.; Rusu, I.; Tannier, E.; Vialette, S., Combinatorics of genome rearrangements, (2009), MIT Press · Zbl 1170.92022 |

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