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Algorithmica

, Volume 60, Issue 2, pp 175–206 | Cite as

Common Intervals of Multiple Permutations

  • Steffen Heber
  • Richard Mayr
  • Jens Stoye
Article

Abstract

Given k permutations of n elements, a k-tuple of intervals of these permutations consisting of the same set of elements is called a common interval. We present an algorithm that finds in a family of k permutations of n elements all z common intervals in optimal O(kn+z) time and O(n) additional space. Additionally, we show how to adapt this algorithm to multichromosomal and circular permutations.

This extends a result by Uno and Yagiura (Algorithmica 26:290–309, 2000) who present an algorithm to find all z common intervals of k=2 (regular) permutations in optimal O(n+z) time and O(n) space. To achieve our result, we introduce the set of irreducible intervals, a generating subset of the set of all common intervals of k permutations.

Keywords

Common intervals of permutations Multichromosomal permutations Circular permutations 

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References

  1. 1.
    Heber, S., Stoye, J.: Algorithms for finding gene clusters. In: Proceedings of the First International Workshop on Algorithms in Bioinformatics (WABI 2001). Lecture Notes in Computer Science, vol. 2149, pp. 252–263. Springer, Berlin (2001) Google Scholar
  2. 2.
    Heber, S., Stoye, J.: Finding all common intervals of k permutations. In: Proceedings of the 12th Annual Symposium on Combinatorial Pattern Matching (CPM 2001). Lecture Notes in Computer Science, vol. 2089, pp. 207–218. Springer, Berlin (2001) CrossRefGoogle Scholar
  3. 3.
    Marcotte, E.M., Pellegrini, M., Ng, H.L., Rice, D.W., Yeates, T.O., Eisenberg, D.: Detecting protein function and protein-protein interactions from genome sequences. Science 285, 751–753 (1999) CrossRefGoogle Scholar
  4. 4.
    Overbeek, R., Fonstein, M., D’Souza, M., Pusch, G.D., Maltsev, N.: The use of gene clusters to infer functional coupling. Proc. Natl. Acad. Sci. USA 96, 2896–2901 (1999) CrossRefGoogle Scholar
  5. 5.
    Snel, B., Lehmann, G., Bork, P., Huynen, M.A.: STRING: A web-server to retrieve and display the repeatedly occurring neigbourhood of a gene. Nucleic Acids Res. 28, 3443–3444 (2000) CrossRefGoogle Scholar
  6. 6.
    Bergeron, A., Heber, S., Stoye, J.: Common intervals and sorting by reversals: A marriage of necessity. In: Proceedings of the European Conference on Computational Biology (ECCB 2002) (Supplement of Bioinformatics), vol. 18, pp. 54–63. University Press, Oxford (2002) (Suppl. 2) Google Scholar
  7. 7.
    Bergeron, A., Stoye, J.: On the similarity of sets of permutations and its applications to genome comparison. In: Proceedings of the 9th International Computing and Combinatorics Conference, COCOON 2003. Lecture Notes in Computer Science, vol. 2697, pp. 68–79. Springer, Berlin (2003) Google Scholar
  8. 8.
    Brady, R.M.: Optimization strategies gleaned from biological evolution. Nature 317, 804–806 (1985) CrossRefGoogle Scholar
  9. 9.
    Kobayashi, S., Ono, I., Yamamura, M.: An efficient genetic algorithm for job shop scheduling problems. In: Proceedings of the 6th International Conference on Genetic Algorithms, pp. 506–511. Morgan Kaufmann, San Francisco (1995) Google Scholar
  10. 10.
    Mühlenbein, H., Gorges-Schleuter, M., Krämer, O.: Evolution algorithms in combinatorial optimization. Parallel Comput. 7, 65–85 (1988) CrossRefMATHGoogle Scholar
  11. 11.
    Bergeron, A., Chauve, C., de Montgolfier, F., Raffinot, M.: Computing common intervals of K permutations, with applications to modular decomposition of graphs. In: Proceedings of the 13th Annual European Symposium on Algorithms, ESA 2005. Lecture Notes in Computer Science, vol. 3669, pp. 779–790. Springer, Berlin (2005) Google Scholar
  12. 12.
    Heber, S., Savage, C.: Common intervals of trees. Inf. Process. Lett. 93, 69–74 (2005) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13, 335–379 (1976) CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Fulkerson, D., Gross, O.: Incidence matrices with the consecutive 1s property. Bull. Am. Math. Soc. 70, 681–684 (1964) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Golumbic, C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1990) Google Scholar
  16. 16.
    Uno, T., Yagiura, M.: Fast algorithms to enumerate all common intervals of two permutations. Algorithmica 26, 290–309 (2000) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Computer ScienceNorth Carolina State UniversityRaleighUSA
  2. 2.School of InformaticsUniversity of EdinburghEdinburghUK
  3. 3.Technische FakultätUniversität BielefeldBielefeldGermany

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