International Symposium on String Processing and Information Retrieval

SPIRE 2015: String Processing and Information Retrieval pp 124-136

# A Faster Algorithm for Computing Maximal $$\alpha$$-gapped Repeats in a String

• Yuka Tanimura
• Yuta Fujishige
• Tomohiro I
• Shunsuke Inenaga
• Hideo Bannai
• Masayuki Takeda
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9309)

## Abstract

A string $$x = uvu$$ with both uv being non-empty is called a gapped repeat with period $$p = |uv|$$, and is denoted by pair (xp). If $$p \le \alpha (|x|-p)$$ with $$\alpha > 1$$, then (xp) is called an $$\alpha$$ -gapped repeat. An occurrence $$[i, i+|x|-1]$$ of an $$\alpha$$-gapped repeat (xp) in a string w is called a maximal $$\alpha$$-gapped repeat of w, if it cannot be extended either to the left or to the right in w with the same period p. Kolpakov et al. (CPM 2014) showed that, given a string of length n over a constant alphabet, all the occurrences of maximal $$\alpha$$-gapped repeats in the string can be computed in $$O(\alpha ^2 n + occ )$$ time, where $$occ$$ is the number of occurrences. In this paper, we propose a faster $$O(\alpha n + occ )$$-time algorithm to solve this problem, improving the result of Kolpakov et al. by a factor of $$\alpha$$.

Brodal

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### References

1. 1.
Badkobeh, G., Crochemore, M., Toopsuwan, C.: Computing the maximal-exponent repeats of an overlap-free string in linear time. In: Calderón-Benavides, L., González-Caro, C., Chávez, E., Ziviani, N. (eds.) SPIRE 2012. LNCS, vol. 7608, pp. 61–72. Springer, Heidelberg (2012)
2. 2.
Bender, M.A., Farach-Colton, M.: The LCA problem revisited. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 88–94. Springer, Heidelberg (2000)
3. 3.
Blumer, A., Blumer, J., Haussler, D., Ehrenfeucht, A., Chen, M.T., Seiferas, J.: The smallest automaton recognizing the subwords of a text. TCS 40, 31–55 (1985)
4. 4.
Brodal, G.S., Lyngsø, R.B., Pedersen, C.N.S., Stoye, J.: Finding maximal pairs with bounded gap. In: Crochemore, M., Paterson, M. (eds.) CPM 1999. LNCS, vol. 1645, pp. 134–149. Springer, Heidelberg (1999)
5. 5.
Crochemore, M., Rytter, W.: Text Algorithms. Oxford University Press, New York (1994)
6. 6.
Crochemore, M.: Transducers and repetitions. Theor. Comput. Sci. 45(1), 63–86 (1986)
7. 7.
Gabow, H.N., Tarjan, R.E.: A linear-time algorithm for a special case of disjoint set union. Journal of Computer and System Sciences 30, 209–221 (1985)
8. 8.
Gusfield, D.: Algorithms on Strings, Trees, and Sequences. Cambridge University Press (1997)Google Scholar
9. 9.
Gusfield, D., Stoye, J.: Linear time algorithms for finding and representing all the tandem repeats in a string. J. Comput. Syst. Sci. 69(4), 525–546 (2004)
10. 10.
Kolpakov, R., Podolskiy, M., Posypkin, M., Khrapov, N.: Searching of gapped repeats and subrepetitions in a word. In: Kulikov, A.S., Kuznetsov, S.O., Pevzner, P. (eds.) CPM 2014. LNCS, vol. 8486, pp. 212–221. Springer, Heidelberg (2014) Google Scholar
11. 11.
Kolpakov, R.M., Kucherov, G.: Finding repeats with fixed gap. In: Proc. SPIRE 2000, pp. 162–168 (2000)Google Scholar
12. 12.
Manber, U., Myers, G.: Suffix arrays: A new method for on-line string searches. SIAM J. Computing 22(5), 935–948 (1993)
13. 13.
Ukkonen, E.: On-line construction of suffix trees. Algorithmica 14(3), 249–260 (1995)
14. 14.
Weiner, P.: Linear pattern-matching algorithms. In: Proc. of 14th IEEE Ann. Symp. on Switching and Automata Theory, pp. 1–11 (1973)Google Scholar
15. 15.
Ziv, J., Lempel, A.: A universal algorithm for sequential data compression. IEEE Transactions on Information Theory IT–23(3), 337–343 (1977)

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## Authors and Affiliations

• Yuka Tanimura
• 1
• Yuta Fujishige
• 1
• Tomohiro I
• 2
• Shunsuke Inenaga
• 1
• Hideo Bannai
• 1
• Masayuki Takeda
• 1
1. 1.Department of InformaticsKyushu UniversityFukuokaJapan
2. 2.Department of Computer ScienceTU DortmundDortmundGermany