Abstract
We propose the first subquadratic-time algorithms to a number of natural problems in abelian pattern matching (also called jumbled pattern matching) for strings over a constant-sized alphabet. Two strings are considered equivalent in this model if the numbers of occurrences of respective symbols in both of them, specified by their so-called Parikh vectors, are the same. We propose the following algorithms for a string of length n:
-
Counting and finding longest/shortest abelian squares in \(O(n^2/\log ^2n)\) time. Abelian squares were first considered by Erdös (1961); Cummings and Smyth (1997) proposed an \(O(n^2)\)-time algorithm for computing them.
-
Computing all shortest (general) abelian periods in \(O(n^2/\sqrt{\log n})\) time. Abelian periods were introduced by Constantinescu and Ilie (2006) and the previous, quadratic-time algorithms for their computation were given by Fici et al. (2011) for a constant-sized alphabet and by Crochemore et al. (2012) for a general alphabet.
-
Finding all abelian covers in \(O(n^2/\log n)\) time. Abelian covers were defined by Matsuda et al. (2014).
-
Computing abelian border array in \(O(n^2/\log ^2n)\) time.
This work can be viewed as a continuation of a recent very active line of research on subquadratic space and time jumbled indexing for binary and constant-sized alphabets (e.g., Moosa and Rahman, 2012). All our algorithms work in linear space.
T. Kociumaka—Supported by Polish budget funds for science in 2013–2017 as a research project under the ‘Diamond Grant’ program.
J. Radoszewski—Supported by the Polish Ministry of Science and Higher Education under the ‘Iuventus Plus’ program in 2015–2016 grant no 0392/IP3/2015/73.
J. Radoszewski—Newton International Fellow at King’s College London.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Amir, A., Chan, T.M., Lewenstein, M., Lewenstein, N.: On hardness of jumbled indexing. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 114–125. Springer, Heidelberg (2014)
Burcsi, P., Cicalese, F., Fici, G., Lipták, Z.: On table arrangements, scrabble freaks, and jumbled pattern matching. In: Boldi, P., Gargano, L. (eds.) FUN 2010. LNCS, vol. 6099, pp. 89–101. Springer, Heidelberg (2010)
Chan, T.M., Lewenstein, M.: Clustered integer 3SUM via additive combinatorics. In: Servedio, R.A., Rubinfeld, R. (eds.) Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, 14–17 June 2015, pp. 31–40. ACM (2015)
Cicalese, F., Fici, G., Lipták, Z.: Searching for jumbled patterns in strings. In: Holub, J., Zdárek, J. (eds.) Proceedings of the Prague Stringology Conference 2009, Prague, Czech Republic, 31 August - 2 September 2009, pp. 105–117. Prague Stringology Club, Department of Computer Science and Engineering, Faculty of Electrical Engineering, Czech Technical University in Prague (2009)
Constantinescu, S., Ilie, L.: Fine and Wilf’s theorem for abelian periods. Bull. EATCS 89, 167–170 (2006)
Crochemore, M., Iliopoulos, C.S., Kociumaka, T., Kubica, M., Pachocki, J., Radoszewski, J., Rytter, W., Tyczyński, W., Waleń, T.: A note on efficient computation of all abelian periods in a string. Inf. Process. Lett. 113(3), 74–77 (2013)
Cummings, L.J., Smyth, W.F.: Weak repetitions in strings. J. Comb. Math. Comb. Comput. 24, 33–48 (1997)
Erdös, P.: Some unsolved problems. Hung. Acad. Sci. Mat. Kutató Intézet Közl 6, 221–254 (1961)
Fici, G., Lecroq, T., Lefebvre, A., Prieur-Gaston, É.: Computing abelian periods in words. In: Holub, J., Žd’árek, J. (eds.) Proceedings of the Prague Stringology Conference 2011, pp. 184–196. Czech Technical University in Prague, Czech Republic (2011)
Fici, G., Lecroq, T., Lefebvre, A., Prieur-Gaston, É., Smyth, W.: Quasi-linear time computation of the abelian periods of a word. In: Holub, J., Žd’árek, J. (eds.) Proceedings of the Prague Stringology Conference 2012, pp. 103–110. Czech Technical University in Prague, Czech Republic (2012)
Hermelin, D., Landau, G.M., Rabinovich, Y., Weimann, O.: Binary jumbled pattern matching via all-pairs shortest paths. CoRR, abs/1401.2065 (2014)
Kociumaka, T., Radoszewski, J., Rytter, W.: Efficient indexes for jumbled pattern matching with constant-sized alphabet. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 625–636. Springer, Heidelberg (2013)
Kociumaka, T., Radoszewski, J., Rytter, W.: Fast algorithms for abelian periods in words and greatest common divisor queries. In: Portier, N., Wilke, T. (eds.) 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, 27 February - 2 March 2013, Kiel, Germany, vol. 20 of LIPIcs, pp. 245–256. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)
Matsuda, S., Inenaga, S., Bannai, H., Takeda, M.: Computing abelian covers and abelian runs. In: Holub, J., Zdárek, J. (eds.) Proceedings of the Prague Stringology Conference 2014. Prague, Czech Republic, 1–3 September 2014, pp. 43–51. Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague (2014)
Moosa, T.M., Rahman, M.S.: Indexing permutations for binary strings. Inf. Process. Lett. 110(18–19), 795–798 (2010)
Moosa, T.M., Rahman, M.S.: Sub-quadratic time and linear space data structures for permutation matching in binary strings. J. Discrete Algorithms 10, 5–9 (2012)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Kociumaka, T., Radoszewski, J., Wiśniewski, B. (2016). Subquadratic-Time Algorithms for Abelian Stringology Problems. In: Kotsireas, I., Rump, S., Yap, C. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2015. Lecture Notes in Computer Science(), vol 9582. Springer, Cham. https://doi.org/10.1007/978-3-319-32859-1_27
Download citation
DOI: https://doi.org/10.1007/978-3-319-32859-1_27
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-32858-4
Online ISBN: 978-3-319-32859-1
eBook Packages: Computer ScienceComputer Science (R0)