Efficient validation and construction of border arrays and validation of string matching automata
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 2, pp. 281-297.

We present an on-line linear time and space algorithm to check if an integer array $f$ is the border array of at least one string $w$ built on a bounded or unbounded size alphabet $\Sigma$. First of all, we show a bijection between the border array of a string $w$ and the skeleton of the DFA recognizing ${\Sigma }^{*}w$, called a string matching automaton (SMA). Different strings can have the same border array but the originality of the presented method is that the correspondence between a border array and a skeleton of SMA is independent from the underlying strings. This enables to design algorithms for validating and generating border arrays that outperform existing ones. The validating algorithm lowers the delay (maximal number of comparisons on one element of the array) from $O\left(|w|\right)$ to $1+min\left\{|\Sigma |,1+{log}_{2}|w|\right\}$ compared to existing algorithms. We then give results on the numbers of distinct border arrays depending on the alphabet size. We also present an algorithm that checks if a given directed unlabeled graph $G$ is the skeleton of a SMA on an alphabet of size $s$ in linear time. Along the process the algorithm can build one string $w$ for which $G$ is the SMA skeleton.

DOI : https://doi.org/10.1051/ita:2008030
Classification : 68R15,  68W05
Mots clés : combinatorics on words, period, border, string matching, string matching automata
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author = {Duval, Jean-Pierre and Lecroq, Thierry and Lefebvre, Arnaud},
title = {Efficient validation and construction of border arrays and validation of string matching automata},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {281--297},
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Duval, Jean-Pierre; Lecroq, Thierry; Lefebvre, Arnaud. Efficient validation and construction of border arrays and validation of string matching automata. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 2, pp. 281-297. doi : 10.1051/ita:2008030. http://www.numdam.org/articles/10.1051/ita:2008030/

[1] A.V. Aho, J.E. Hopcroft and J.D. Ullman, The design and analysis of computer algorithms. Addison-Wesley (1974). | MR 413592 | Zbl 0326.68005

[2] M. Crochemore, C. Hancart and T. Lecroq, Algorithms on Strings. Cambridge University Press (2007). | MR 2355493 | Zbl 1137.68060

[3] J.-P. Duval, T. Lecroq and A. Lefebvre, Border array on bounded alphabet. J. Autom. Lang. Comb. 10 (2005) 51-60. | MR 2192584 | Zbl 1089.68080

[4] F. Franěk, S. Gao, W. Lu, P.J. Ryan, W.F. Smyth, Y. Sun and L. Yang, Verifying a border array in linear time. J. Combin. Math. Combin. Comput. 42 (2002) 223-236. | MR 1929012 | Zbl 1009.68106

[5] C. Hancart, Analyse exacte et en moyenne d'algorithmes de recherche d'un motif dans un texte. Ph.D. thesis. Université Paris 7, France (1993).

[6] D.E. Knuth, J.H. Morris and V.R. Pratt Jr, Fast pattern matching in strings. SIAM J. Comput. 6 (1977) 323-350. | MR 451916 | Zbl 0372.68005

[7] D. Moore, W.F. Smyth and D. Miller, Counting distinct strings. Algorithmica 23 (1999) 1-13. | MR 1661152 | Zbl 0913.68088

[8] J.H. Morris and V.R. Pratt Jr, A linear pattern-matching algorithm. Technical Report 40, University of California, Berkeley (1970).

[9] M. Naylor, Abacaba-dabacaba. http://www.ac.wwu.edu/~mnaylor/abacaba/abacaba.html.

[10] I. Simon, String matching algorithms and automata, in Proceedings of the First South American Workshop on String Processing, edited by R. Baeza-Yates and N. Ziviani, Belo Horizonte, Brazil (1993) 151-157 | MR 1286978

[11] W.F. Smyth, Computing Pattern in Strings. Addison Wesley Pearson (2003).

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