Abstract
For a word equation E of length n in one variable x occurring #x times in E a resolution algorithm of O(n + #x log n) time complexity is presented here. This is the best result known and for the equations that feature #x < n/logn it yields time complexity of O(n) which is optimal. Additionally, we prove that the set of solutions of one-variable word equations is either of the form F where F is a set of O(logn) words or of the form F ∪ (uv)+ u where F is a set of O(logn) words and u, v are some words such that uv is a primitive word.
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References
Angluin D., Finding pattern common to a set of string, in Proc. STOC’79, 130–141, 1979.
Charatonik W., Pacholski L., Word equations in two variables, Proc. IWWERT’91, LNCS 677, 43–57, 1991.
Crochemore M., Rytter W., Text Algorithms, Oxford University Press, 1994.
Crochemore M., Rytter W., Periodic suffixes of strings, in: Acts of Sequences, 1991.
Diekert V., Makanin’s algorithm, a chapter in a book on combinatorics of words, 1998, personal comunication (Volker.Diekert@informatik.uni-stuttgart.de).
Jaffar J., Minimal and complete word unification, Journal of the A CM 37(1), 47–85, 1990.
Eyono Obono S., Goralcik P., Maksimienko M., Efficient Solving of the Word Equations in One Variable, Lecture Notes in Computer Science 841, Springer Verlag, Proc. of the 19th International Symposium MFCS 1994, Kosice, Slovakia, 336–341, Springer, 1994.
Gutierrez C., Satisfiability of word equations with constants is in exponential space, in: Proc. FOCS’98, IEEE Computer Society Press, Palo Alto, California.
Ilie L., Plandowski W., Two-variable word equations, RAIRO Theoretical Informatics and Applications 34, 467–501, 2000.
Koscielski A., Pacholski L., Complexity of Makanin’s Algorithm, Journal of the ACM 43(4), 670–684, 1996.
Makanin G. S., The problem of solvability of equations in a free semigroup, Mat. Sb., 103(2), 147–236. In Russian; English translation in: Math. USSR Sbornik, 32, 129-198, 1977.
Neraud J., Equations in words: an algorithmic contribution, Bull. Belg. Math. Soc. 1, 253–283, 1994.
Plandowski W., Rytter W., Application of Lempel-Ziv encodings to the solution of word equations, in: Proc. ICALP’98, LNCS 1443, 731–742, 1998.
Plandowski W., Satisfiability of word equations with constants is in NEXPTIME, Proc. STOC’99, 721–725, ACM Press, 1999.
Plandowski W., Satisfiability of word equations with constants is in PSPACE, Proc. FOCS’99, 495–500, IEEE Computer Society Press, 1999.
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Dabrowski, R., Plandowski, W. (2002). On Word Equations in One Variable. In: Diks, K., Rytter, W. (eds) Mathematical Foundations of Computer Science 2002. MFCS 2002. Lecture Notes in Computer Science, vol 2420. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45687-2_17
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DOI: https://doi.org/10.1007/3-540-45687-2_17
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