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2D Lyndon Words and Applications

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Abstract

A Lyndon word is a primitive string which is lexicographically smallest among cyclic permutations of its characters. Lyndon words are used for constructing bases in free Lie algebras, constructing de Bruijn sequences, finding the lexicographically smallest or largest substring in a string, and succinct suffix–prefix matching of highly periodic strings. In this paper, we extend the concept of the Lyndon word to two dimensions. We introduce the 2D Lyndon word and use it to capture 2D horizontal periodicity of a matrix in which each row is highly periodic, and to efficiently solve 2D horizontal suffix–prefix matching among a set of patterns. This yields a succinct and efficient algorithm for 2D dictionary matching. We present several algorithms that compute the 2D Lyndon word that represents a matrix. The final algorithm achieves linear time complexity even when the least common multiple of the periods of the rows is exponential in the matrix width.

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Notes

  1. Lyndon word naming of the matrix rows takes linear \(O(m^2)\) time [27].

  2. Processing columns even for this type of pattern would have several additional drawbacks, including working only for matrices that are uniform size in both dimensions.

  3. Hashing techniques allow us to traverse a compressed trie of LWpos arrays in linear time.

References

  1. Aho, A.V., Corasick, M.J.: Efficient string matching: an aid to bibliographic search. Commun. ACM 18(6), 333–340 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amir, A., Benson, G.: Two-dimensional periodicity and its applications. In: ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 440–452 (1992)

  3. Amir, A., Benson, G.: Two-dimensional periodicity in rectangular arrays. SIAM J. Comput. 27(1), 90–106 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Amir, A., Benson, G., Farach, M.: An alphabet independent approach to two-dimensional pattern matching. SIAM J. Comput. 23(2), 313–323 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Amir, A., Farach, M.: Two-dimensional dictionary matching. Inf. Process. Lett. 44(5), 233–239 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Apostolico, A., Crochemore, M.: Fast parallel Lyndon factorization with applications. Math. Syst. Theory 28(2), 89–108 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Baker, T.J.: A technique for extending rapid exact-match string matching to arrays of more than one dimension. SIAM J. Comput. 7, 533–541 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bannai, H., Inenaga, S., Nakashima, Y., Takeda, M., Tsuruta, K.: A new characterization of maximal repetitions by Lyndon trees. In: ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 562–571 (2015)

  9. Bird, R.S.: Two dimensional pattern matching. Inf. Process. Lett. 6(5), 168–170 (1977)

    Article  Google Scholar 

  10. Chemillier, M.: Periodic musical sequences and Lyndon words. Soft Comput. 8(9), 611–616 (2004)

    MATH  Google Scholar 

  11. Choi, Y., Lam, T.-W.: Two-dimensional dynamic dictionary matching. In: International Symposium on Symbolic and Algebraic Computation (ISAAC), pp. 85–94 (1996)

  12. Crochemore, M., Gasieniec, L., Hariharan, R., Muthukrishnan, S., Rytter, W.: A constant time optimal parallel algorithm for two-dimensional pattern matching. SIAM J. Comput. 27(3), 668–681 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Crochemore, M., Iliopoulos, C., Korda, M., Reid, J.: A failure function for multiple two-dimensional pattern matching. Comb. Math. Comb. Comput. 35, 225–238 (2000)

    MathSciNet  MATH  Google Scholar 

  14. Delgrange, O., Rivals, E.: Star: an algorithm to search for tandem approximate repeats. Bioinformatics 20(16), 2812–2820 (2004)

    Article  Google Scholar 

  15. Farhi, B.: Nontrivial lower bounds for the least common multiple of some finite sequences of integers. J. Number Theory 125(2), 393–411 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fredricksen, H., Maiorana, J.: Necklaces of beads in k colors and k-ary de Bruijn sequences. Discrete Math. 23(3), 207–210 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  17. Giancarlo, R.: A generalization of the suffix tree to square matrices, with applications. SIAM J. Comput. 24(3), 520–562 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gusfield, D.: Algorithms on Strings, Trees, and Sequences—Computer Science and Computational Biology. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  19. Gusfield, D., Landau, G.M., Schieber, B.: An efficient algorithm for the all pairs suffix–prefix problem. Inf. Process. Lett. 41(4), 181–185 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Idury, R.M., Schäffer, A.A.: Multiple matching of rectangular patterns. Inf. Comput. 117(1), 78–90 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kedem, Z.M., Landau, G.M., Palem, K.V.: Parallel suffix–prefix-matching algorithm and applications. SIAM J. Comput. 25(5), 998–1023 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Knuth, D.E.: The Art of Computer Programming, vol. 2. Addison Wesley, Redwood (1998)

    MATH  Google Scholar 

  23. Koshy, T.: Elementary Number Theory with Applications, 2nd edn. Academic Press, New York (2001)

    MATH  Google Scholar 

  24. Lothaire, M.: Applied Combinatorics on Words (Encyclopedia of Mathematics and Its Applications). Cambridge University Press, New York (2005)

    Book  MATH  Google Scholar 

  25. Lyndon, R.C.: On Burnside’s problem. Trans. Am. Math. Soc. 77(2), 212–215 (1954)

    MathSciNet  MATH  Google Scholar 

  26. Mucha, M.: Lyndon words and short superstrings. In: ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 958–972 (2013)

  27. Neuburger, S., Sokol, D.: Succinct 2D dictionary matching. Algorithmica 65(3), 662–684 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ohlebusch, E., Gog, S.: Efficient algorithms for the all-pairs suffix–prefix problem and the all-pairs substring-prefix problem. Inf. Process. Lett. 110(3), 123–128 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work was supported in part by PSC-CUNY research award 65112-0043. The authors would like to thank Binyomin Balsam for his helpful discussions and his insight into the modular arithmetic solution.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Kingsborough Community College of the City University of New York, 2001 Oriental Boulevard, Brooklyn, NY, 11235, USA

    Shoshana Marcus

  2. Department of Computer and Information Science, Brooklyn College of the City University of New York, 2900 Bedford Avenue, Brooklyn, NY, 11210, USA

    Dina Sokol

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  1. Shoshana Marcus
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  2. Dina Sokol
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Correspondence to Shoshana Marcus.

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Marcus, S., Sokol, D. 2D Lyndon Words and Applications. Algorithmica 77, 116–133 (2017). https://doi.org/10.1007/s00453-015-0065-z

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