Abstract
For an undirected tree with n edges labelled by single letters, we consider its substrings, which are labels of the simple paths between pairs of nodes. We prove that there are \(\mathcal {O}(n^{1.5})\) different palindromic substrings. This solves an open problem of Brlek, Lafrenière and Provençal (DLT 2015), who gave a matching lower-bound construction. Hence, we settle the tight bound of \(\Theta (n^{1.5})\) for the maximum palindromic complexity of trees. For standard strings, i.e., for paths, the palindromic complexity is \(n+1\).
P. Gawrychowski—Work done while the author held a post-doctoral position at Warsaw Center of Mathematics and Computer Science.
T. Kociumaka—Supported by Polish budget funds for science in 2013-2017 as a research project under the ‘Diamond Grant’ program.
W. Rytter—This work was supported by the Polish National Science Center, grant no NCN2014/13/B/ST6/00770.
T. Waleń—Supported by the Polish Ministry of Science and Higher Education under the ‘Iuventus Plus’ program in 2015-2016 grant no 0392/IP3/2015/73.
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References
Anisiu, M., Anisiu, V., Kása, Z.: Total palindrome complexity of finite words. Discrete Mathematics 310(1), 109–114 (2010)
Brlek, S., Hamel, S., Nivat, M., Reutenauer, C.: On the palindromic complexity of infinite words. Int. J. Found. Comput. Sci. 15(2), 293–306 (2004)
Brlek, S., Lafrenière, N., Provençal, X.: Palindromic complexity of trees. In: Potapov, I. (ed.) DLT 2015. LNCS, vol. 9168, pp. 155–166. Springer, Heidelberg (2015). arXiv:1505.02695
Crochemore, M., Iliopoulos, C.S., Kociumaka, T., Kubica, M., Radoszewski, J., Rytter, W., Tyczyński, W., Waleń, T.: The maximum number of squares in a tree. In: Kärkkäinen, J., Stoye, J. (eds.) CPM 2012. LNCS, vol. 7354, pp. 27–40. Springer, Heidelberg (2012)
Droubay, X., Justin, J., Pirillo, G.: Episturmian words and some constructions of de Luca and Rauzy. Theor. Comput. Sci. 255(1–2), 539–553 (2001)
Glen, A., Justin, J., Widmer, S., Zamboni, L.Q.: Palindromic richness. Eur. J. Comb. 30(2), 510–531 (2009)
Groult, R., Prieur, É., Richomme, G.: Counting distinct palindromes in a word in linear time. Inf. Process. Lett. 110(20), 908–912 (2010)
Kociumaka, T., Radoszewski, J., Rytter, W., Waleń, T.: String powers in trees. In: Cicalese, F., Porat, E., Vaccaro, U. (eds.) CPM 2015. LNCS, vol. 9133, pp. 284–294. Springer, Heidelberg (2015)
Simpson, J.: Palindromes in circular words. Theor. Comput. Sci. 550, 66–78 (2014)
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Gawrychowski, P., Kociumaka, T., Rytter, W., Waleń, T. (2015). Tight Bound for the Number of Distinct Palindromes in a Tree. In: Iliopoulos, C., Puglisi, S., Yilmaz, E. (eds) String Processing and Information Retrieval. SPIRE 2015. Lecture Notes in Computer Science(), vol 9309. Springer, Cham. https://doi.org/10.1007/978-3-319-23826-5_26
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DOI: https://doi.org/10.1007/978-3-319-23826-5_26
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