Efficiently Finding All Maximal alpha-gapped Repeats

Authors Pawel Gawrychowski, Tomohiro I, Shunsuke Inenaga, Dominik Köppl, Florin Manea

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Pawel Gawrychowski
Tomohiro I
Shunsuke Inenaga
Dominik Köppl
Florin Manea

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Pawel Gawrychowski, Tomohiro I, Shunsuke Inenaga, Dominik Köppl, and Florin Manea. Efficiently Finding All Maximal alpha-gapped Repeats. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


For alpha >=1, an alpha-gapped repeat in a word w is a factor uvu of w such that |uv| <= alpha * |u|; the two occurrences of a factor u in such a repeat are called arms. Such a repeat is called maximal if its arms cannot be extended simultaneously with the same symbol to the right nor to the left. We show that the number of all maximal alpha-gapped repeats occurring in words of length n is upper bounded by 18 * alpha * n, allowing us to construct an algorithm finding all maximal alpha-gapped repeats of a word on an integer alphabet of size n^{O}(1)} in {O}(alpha * n) time. This result is optimal as there are words that have Theta(alpha * n) maximal alpha-gapped repeats. Our techniques can be extended to get comparable results in the case of alpha-gapped palindromes, i.e., factors uvu^{T} with |uv| <= alpha |u|.
  • combinatorics on words
  • counting algorithms


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  1. Golnaz Badkobeh, Maxime Crochemore, and Chalita Toopsuwan. Computing the maximal-exponent repeats of an overlap-free string in linear time. In SPIRE 2012. Proceedings, volume 7608 of Lecture Notes in Computer Science, pages 61-72. Springer, 2012. Google Scholar
  2. Hideo Bannai, Tomohiro I, Shunsuke Inenaga, Yuto Nakashima, Masayuki Takeda, and Kazuya Tsuruta. The Runs Theorem. CoRR, abs/1406.0263, 2014. URL: http://arxiv.org/abs/1406.0263.
  3. Gerth Stølting Brodal, Rune B. Lyngsø, Christian N. S. Pedersen, and Jens Stoye. Finding maximal pairs with bounded gap. In CPM, volume 1645 of LNCS, pages 134-149. Springer, 1999. Google Scholar
  4. Maxime Crochemore, Costas S. Iliopoulos, Marcin Kubica, Wojciech Rytter, and Tomasz Walen. Efficient algorithms for two extensions of LPF table: The power of suffix arrays. In Proc. SOFSEM 2010, volume 5901 of LNCS, pages 296-307, 2010. Google Scholar
  5. Maxime Crochemore, Roman Kolpakov, and Gregory Kucherov. Optimal searching of gapped repeats in a word. ArXiv e-prints 1309.4055, 2015. URL: http://arxiv.org/abs/1309.4055.
  6. Maxime Crochemore and German Tischler. Computing longest previous non-overlapping factors. Inf. Process. Lett., 111(6):291-295, February 2011. Google Scholar
  7. Marius Dumitran and Florin Manea. Longest gapped repeats and palindromes. In Proc. MFCS 2015, volume 9234 of LNCS, pages 205-217. Springer, 2015. Google Scholar
  8. Pawel Gawrychowski. Pattern matching in Lempel-Ziv compressed strings: Fast, simple, and deterministic. In Proc. ESA, volume 6942 of LNCS, pages 421-432, 2011. Google Scholar
  9. Pawel Gawrychowski, Tomohiro I, Shunsuke Inenaga, Dominik Köppl, and Florin Manea. Efficiently finding all maximal α-gapped repeats. ArXiv e-prints abs/1509.09237, 2015. Google Scholar
  10. Pawel Gawrychowski and Florin Manea. Longest α-gapped repeat and palindrome. In Proc. FCT 2015, volume 9210 of LNCS, pages 27-40. Springer, 2015. Google Scholar
  11. Dan Gusfield. Algorithms on strings, trees, and sequences: computer science and computational biology. Cambridge University Press, New York, NY, USA, 1997. Google Scholar
  12. Juha Kärkkäinen, Peter Sanders, and Stefan Burkhardt. Linear work suffix array construction. J. ACM, 53:918-936, 2006. Google Scholar
  13. Tomasz Kociumaka, Jakub Radoszewski, Wojciech Rytter, and Tomasz Walen. Efficient data structures for the factor periodicity problem. In Proc. SPIRE, volume 7608 of LNCS, pages 284-294, 2012. Google Scholar
  14. Roman Kolpakov and Gregory Kucherov. Finding repeats with fixed gap. In Proc. SPIRE, pages 162-168, 2000. Google Scholar
  15. Roman Kolpakov and Gregory Kucherov. Searching for gapped palindromes. Theoretical Computer Science, 410(51):5365-5373, 2009. Combinatorial Pattern Matching. URL: http://dx.doi.org/10.1016/j.tcs.2009.09.013.
  16. Roman Kolpakov, Mikhail Podolskiy, Mikhail Posypkin, and Nickolay Khrapov. Searching of gapped repeats and subrepetitions in a word. In Proc. CPM, volume 8486 of LNCS, pages 212-221, 2014. Google Scholar
  17. Yuka Tanimura, Yuta Fujishige, Tomohiro I, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. A faster algorithm for computing maximal α-gapped repeats in a string. In Proc. SPIRE 2015, volume 9309 of LNCS, pages 124-136. Springer, 2015. Google Scholar
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