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Longest substring palindrome after edit. (English) Zbl 07286738
Navarro, Gonzalo (ed.) et al., 29th annual symposium on combinatorial pattern matching, CPM 2018, July 2–4, 2018, Qingdao, China. Wadern: Schloss Dagstuhl – Leibniz Zentrum für Informatik (ISBN 978-3-95977-074-3). LIPIcs – Leibniz International Proceedings in Informatics 105, Article 12, 14 p. (2018).
Summary: It is known that the length of the longest substring palindromes (LSPals) of a given string $$T$$ of length $$n$$ can be computed in $$O(n)$$ time by G. Manacher’s algrithm [J. Assoc. Comput. Mach. 22, 346–351 (1975; Zbl 0305.68027)]. In this paper, we consider the problem of finding the LSPal after the string is edited. We present an algorithm that uses $$O(n)$$ time and space for preprocessing, and answers the length of the LSPals in $$O(\log(\min \{\sigma,\log n\}))$$ time after single character substitution, insertion, or deletion, where $$\sigma$$ denotes the number of distinct characters appearing in $$T$$. We also propose an algorithm that uses $$O(n)$$ time and space for preprocessing, and answers the length of the LSPals in $$O(\ell+\log n)$$ time, after an existing substring in $$T$$ is replaced by a string of arbitrary length $$\ell$$.
For the entire collection see [Zbl 1390.68025].

##### MSC:
 68W32 Algorithms on strings
##### Keywords:
maximal palindromes; edit operations; periodicity; suffix trees
Full Text:
##### References:
 [1] Amihood Amir, Panagiotis Charalampopoulos, Costas S. Iliopoulos, Solon P. Pissis, and Jakub Radoszewski. Longest common factor after one edit operation. In \it SPIRE 2017, pages 14-26, 2017. [2] Alberto Apostolico, Dany Breslauer, and Zvi Galil. Parallel detection of all palindromes in a string. \it Theoretical Computer Science, 141:163-173, 1995. [3] Michael A. Bender and Martin Farach-Colton. The LCA problem revisited. In \it LATIN \it 2000, pages 88-94, 2000. [4] Richard Cole and Ramesh Hariharan. Dynamic LCA queries on trees. \it SIAM J. Comput., 34(4):894-923, 2005. [5] Martin Farach-Colton, Paolo Ferragina, and S. Muthukrishnan. On the sorting-complexity of suffix tree construction. \it J. ACM, 47(6):987-1011, 2000. [6] Pawel Gawrychowski, Tomohiro I, Shunsuke Inenaga, Dominik Köppl, and Florin Manea. Tighter bounds and optimal algorithms for all maximal \it α-gapped repeats and palindromes - finding all maximal \it α-gapped repeats and palindromes in optimal worst case time on integer alphabets. \it Theory Comput. Syst., 62(1):162-191, 2018. [7] Leszek G¸asieniec, Marek Karpinski, Wojciech Plandowski, and Wojciech Rytter. Efficient algorithms for Lempel-Ziv encoding. In \it Proc. 5th Scandinavian Workshop on Algorithm \it Theory (SWAT1996), volume 1097 of \it LNCS, pages 392-403. Springer, 1996. [8] Richard Groult, Élise Prieur, and Gwénaël Richomme. Counting distinct palindromes in a word in linear time. \it Inf. Process. Lett., 110(20):908-912, 2010. [9] Dan Gusfield. \it Algorithms on Strings, Trees, and Sequences. Cambridge University Press, 1997. [10] Dov Harel and Robert Endre Tarjan. Fast algorithms for finding nearest common ancestors. \it SIAM J. Comput., 13(2):338-355, 1984. [11] Roman Kolpakov and Gregory Kucherov. Searching for gapped palindromes. \it Theor. Com- \it put. Sci., 410(51):5365-5373, 2009. [12] Dmitry Kosolobov, Mikhail Rubinchik, and Arseny M. Shur. Finding distinct subpalindromes online. In \it Proceedings of the Prague Stringology Conference 2013, Prague, Czech \it Republic, September 2-4, 2013, pages 63-69, 2013. [13] Glenn Manacher. A new linear-time “on-line” algorithm for finding the smallest initial palindrome of a string. \it Journal of the ACM, 22:346-351, 1975. [14] W. Matsubara, S. Inenaga, A. Ishino, A. Shinohara, T. Nakamura, and K. Hashimoto. Efficient algorithms to compute compressed longest common substrings and compressed palindromes. \it Theor. Comput. Sci., 410(8-10):900-913, 2009. [15] Shintaro Narisada, Diptarama, Kazuyuki Narisawa, Shunsuke Inenaga, and Ayumi Shinohara. Computing longest single-arm-gapped palindromes in a string. In \it SOFSEM 2017, pages 375-386, 2017. [16] Alexandre H. L. Porto and Valmir C. Barbosa. Finding approximate palindromes in strings. \it Pattern Recognition, 35:2581-2591, 2002. [17] Baruch Schieber and Uzi Vishkin. On finding lowest common ancestors: Simplification and parallelization. \it SIAM J. Comput., 17(6):1253-1262, 1988. [18] E. Ukkonen. On-line construction of suffix trees. \it Algorithmica, 14(3):249-260, 1995. [19] Peter Weiner. Linear pattern matching algorithms. In \it 14th Annual Symposium on Switching \it and Automata Theory, pages 1-11, 1973.
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