Abstract
The exponent of a string is the quotient of the string’s length over the string’s smallest period. The exponent and the period of a string can be computed in time proportional to the string’s length. We design an algorithm to compute the maximal exponent of factors of an overlap-free string. Our algorithm runs in linear-time on a fixed-size alphabet, while a naive solution of the question would run in cubic time. The solution for non overlap-free strings derives from algorithms to compute all maximal repetitions, also called runs, occurring in the string. We show there is a linear number of maximal-exponent repeats in an overlap-free string. The algorithm can locate all of them in linear time.
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
Bell, T.C., Cleary, J.G., Witten, I.H.: Text compression. Prentice-Hall, Englewood Cliffs (1990)
Berkman, O., Iliopoulos, C.S., Park, K.: The subtree max gap problem with application to parallel string covering. Information and Computation 123(1), 127–137 (1995)
Brodal, G.S., Lyngsø, R.B., Pedersen, C.N.S., Stoye, J.: Finding Maximal Pairs with Bounded Gap. In: Crochemore, M., Paterson, M. (eds.) CPM 1999. LNCS, vol. 1645, pp. 134–149. Springer, Heidelberg (1999)
Brodal, G.S., Pedersen, C.N.S.: Finding Maximal Quasiperiodicities in Strings. In: Giancarlo, R., Sankoff, D. (eds.) CPM 2000. LNCS, vol. 1848, pp. 397–411. Springer, Heidelberg (2000)
Christou, M., Crochemore, M., Iliopoulos, C.S., Kubica, M., Pissis, S.P., Radoszewski, J., Rytter, W., Szreder, B., Waleń, T.: Efficient Seeds Computation Revisited. In: Giancarlo, R., Manzini, G. (eds.) CPM 2011. LNCS, vol. 6661, pp. 350–363. Springer, Heidelberg (2011)
Crochemore, M., Hancart, C., Lecroq, T.: Algorithms on Strings, 392 pages. Cambridge University Press (2007)
Crochemore, M., Ilie, L.: Maximal repetitions in strings. Journal of Computer and System Sciences 74, 796–807 (2008), doi:10.1016/j.jcss.2007.09.003
Crochemore, M., Ilie, L., Tinta, L.: The “runs” conjecture. Theoretical Computer Science 412(27), 2931–2941 (2011)
Crochemore, M., Tischler, G.: Computing longest previous non-overlapping factors. Information Processing Letters 111, 291–295 (2011)
Currie, J.D., Rampersad, N.: A proof of Dejean’s conjecture. Mathematics of Computation 80(274), 1063–1070 (2011)
Dejean, F.: Sur un théorème de Thue. Journal of Combinatorial Theory, Series A 13(1), 90–99 (1972)
Gusfield, D.: Algorithms on strings, trees and sequences: computer science and computational biology. Cambridge University Press, Cambridge (1997)
Iliopoulos, C.S., Moore, D.W.G., Park, K.: Covering a string. Algorithmica 16(3), 288–297 (1996)
Kolpakov, R., Kucherov, G.: On maximal repetitions in words. Journal of Discrete Algorithms 1(1), 159–186 (2000)
Kolpakov, R., Kucherov, G., Ochem, P.: On maximal repetitions of arbitrary exponent. Information Processing Letters 110(7), 252–256 (2010)
Rao, M.: Last cases of Dejean’s conjecture. Theoretical Computer Science 412(27), 3010–3018 (2011)
Rytter, W.: The number of runs in a string. Information and Computation 205(9), 1459–1469 (2007)
Thue, A.: Über unendliche Zeichenreihen. Norske Vid. Selsk. Skr. I Math-Nat. Kl. 7, 1–22 (1906)
Ziv, J., Lempel, A.: A universal algorithm for sequential data compression 23, 337–343 (1977)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Badkobeh, G., Crochemore, M., Toopsuwan, C. (2012). Computing the Maximal-Exponent Repeats of an Overlap-Free String in Linear Time. In: Calderón-Benavides, L., González-Caro, C., Chávez, E., Ziviani, N. (eds) String Processing and Information Retrieval. SPIRE 2012. Lecture Notes in Computer Science, vol 7608. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34109-0_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-34109-0_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34108-3
Online ISBN: 978-3-642-34109-0
eBook Packages: Computer ScienceComputer Science (R0)