Abstract
Let π′ w denote the failure function of the Knuth-Morris-Pratt algorithm for a word w. In this paper we study the following problem: given an integer array A′[1 .. n], is there a word w over an arbitrary alphabet Σ such that A′[i] = π′ w [i] for all i? Moreover, what is the minimum cardinality of Σ required? We give an elementary and self-contained \(\mathcal{O}(n\log n)\) time algorithm for this problem, thus improving the previously known solution [8] with no polynomial time bound. Using both deeper combinatorial insight into the structure of π′ and more advanced tools, we further improve the running time to \(\mathcal{O}(n)\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Breslauer, D., Colussi, L., Toniolo, L.: On the comparison complexity of the string prefix-matching problem. J. Algorithms 29(1), 18–67 (1998)
Clément, J., Crochemore, M., Rindone, G.: Reverse engineering prefix tables. In: Proceedings of 26th STACS, pp. 289–300 (2009)
Cole, R., Hariharan, R.: Dynamic lca queries on trees. In: Proceedings of SODA ’99, Philadelphia, PA, USA, pp. 235–244. Society for Industrial and Applied Mathematics (1999)
Crochemore, M., Hancart, C., Lecroq, T.: Algorithms on Strings. Cambridge University Press, Cambridge (2007)
Crochemore, M., Rytter, W.: Jewels of Stringology. World Scientific Publishing Company, Singapore (2002)
Dietzfelbinger, M., Karlin, A.R., Mehlhorn, K., auf der Heide, F.M., Rohnert, H., Tarjan, R.E.: Dynamic perfect hashing: Upper and lower bounds. SIAM J. Comput. 23(4), 738–761 (1994)
Duval, J.-P., Lecroq, T., Lefebvre, A.: Border array on bounded alphabet. Journal of Automata, Languages and Combinatorics 10(1), 51–60 (2005)
Duval, J.-P., Lecroq, T., Lefebvre, A.: Efficient validation and construction of Knuth-Morris-Pratt arrays. In: Conference in honor of Donald E. Knuth (2007)
Duval, J.-P., Lecroq, T., Lefebvre, A.: Efficient validation and construction of border arrays and validation of string matching automata. ITA 43(2), 281–297 (2009)
Farach, M.: Optimal suffix tree construction with large alphabets. In: Proceedings of FOCS ’97, Washington, DC, USA, pp. 137–143. IEEE Computer Society, Los Alamitos (1997)
Franěk, F., Gao, S., Lu, W., Ryan, P.J., Smyth, W.F., Sun, Y., Yang, L.: Verifying a border array in linear time. J. Comb. Math. Comb. Comput. 42, 223–236 (2002)
Fredman, M.L., Willard, D.E.: Trans-dichotomous algorithms for minimum spanning trees and shortest paths. J. Comput. Syst. Sci. 48(3), 533–551 (1994)
Hancart, C.: On Simon’s string searching algorithm. Inf. Process. Lett. 47(2), 95–99 (1993)
I, T., Inenaga, S., Bannai, H., Takeda, M.: Counting parameterized border arrays for a binary alphabet. In: Proc. of the 3rd LATA, pp. 422–433 (2009)
Karp, R.M., Miller, R.E., Rosenberg, A.L.: Rapid identification of repeated patterns in strings, trees and arrays. In: STOC ’72: Proceedings of the fourth annual ACM symposium on Theory of computing, pp. 125–136. ACM, New York (1972)
Knuth, D.E., Morris Jr., J.H., Pratt, V.R.: Fast pattern matching in strings. SIAM J. Comput. 6(2), 323–350 (1977)
McCreight, E.M.: A space-economical suffix tree construction algorithm. J. ACM 23(2), 262–272 (1976)
Moore, D., Smyth, W.F., Miller, D.: Counting distinct strings. Algorithmica 23(1), 1–13 (1999)
Morris Jr., J.H., Pratt, V.R.: A linear pattern-matching algorithm. Technical Report 40, University of California, Berkeley (1970)
Pagh, R., Rodler, F.F.: Cuckoo hashing. J. Algorithms 51(2), 122–144 (2004)
Simon, I.: String matching algorithms and automata. In: Karhumäki, J., Rozenberg, G., Maurer, H.A. (eds.) Results and Trends in Theoretical Computer Science. LNCS, vol. 812, pp. 386–395. Springer, Heidelberg (1994)
Ukkonen, E.: On-line construction of suffix trees. Algorithmica 14(3), 249–260 (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gawrychowski, P., Jeż, A., Jeż, Ł. (2010). Validating the Knuth-Morris-Pratt Failure Function, Fast and Online. In: Ablayev, F., Mayr, E.W. (eds) Computer Science – Theory and Applications. CSR 2010. Lecture Notes in Computer Science, vol 6072. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13182-0_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-13182-0_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13181-3
Online ISBN: 978-3-642-13182-0
eBook Packages: Computer ScienceComputer Science (R0)