Structural properties of the string statistics problem
Journal of Computer and System Sciences
The Boyer Moore Galil string searching strategies revisited
SIAM Journal on Computing
Alphabet dependence in parameterized matching
Information Processing Letters
Parameterized pattern matching: algorithms and applications
Journal of Computer and System Sciences
Algorithms on strings, trees, and sequences: computer science and computational biology
Algorithms on strings, trees, and sequences: computer science and computational biology
Parameterized Duplication in Strings: Algorithms and an Application to Software Maintenance
SIAM Journal on Computing
Pattern matching algorithms
Two-Dimensional Periodicity in Rectangular Arrays
SIAM Journal on Computing
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
On improving the worst case running time of the Boyer-Moore string matching algorithm
Communications of the ACM
Generalizations of the Periodicity Theorem of Fine and Wilf
CAAP '94 Proceedings of the 19th International Colloquium on Trees in Algebra and Programming
Function matching: algorithms, applications, and a lower bound
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
δ γ --- Parameterized Matching
SPIRE '08 Proceedings of the 15th International Symposium on String Processing and Information Retrieval
Counting Parameterized Border Arrays for a Binary Alphabet
LATA '09 Proceedings of the 3rd International Conference on Language and Automata Theory and Applications
Cycle detection and correction
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Verifying and enumerating parameterized border arrays
Theoretical Computer Science
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Cycle detection and correction
ACM Transactions on Algorithms (TALG)
Approximate period detection and correction
SPIRE'12 Proceedings of the 19th international conference on String Processing and Information Retrieval
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One of the most beautiful and useful notions in the Mathematical Theory of Strings is that of a Period, i.e., an initial piece of a given string that can generate that string by repeating itself at regular intervals. Periods have an elegant mathematical structure and a wealth of applications [F. Mignosi and A. Restivo, Periodicity, Algebraic Combinatorics on Words, in: M. Lothaire (Ed.), Cambridge University Press, Cambridge, pp. 237-274, 2002]. At the hearth of their theory, there are two Periodicity Lemmas: one due to Lyndon and Schutzenberger [The equation a^M=b^Nc^P in a free group, Michigan Math. J. 9 (1962) 289-298], referred to as the Weak Version, and the other due to Fine and Wilf [Uniqueness theorems for periodic functions, Proc. Amer. Math. Soc. 16 (1965) 109-114]. In this paper, we investigate the notion of periodicity and the closely related one of repetition in connection with parameterized strings as introduced by Baker [Parameterized pattern matching: algorithms and applications, J. Comput. System Sci. 52(1) (1996) 28-42; Parameterized duplication in strings: algorithms and an application to software maintenance, SIAM J. Comput. 26(5) (1997) 1343-1362]. In such strings, the notion of pairwise match or ''equivalence'' of symbols is more relaxed than the usual one, in that it rests on some mapping, rather than identity, of symbols. It seems natural to try and extend notions of periods and periodicities to encompass parameterized strings. However, we know of no previous attempt in this direction. Our preliminary investigation yields results as follows. For periodicity, we get (a) a generalization of the Weak Version of the Periodicity Lemma for parameterized strings, showing that it is essential that the two mappings inducing the periodicity must commute; (b) a proof that an analogous of the Lemma by Fine and Wilf [Uniqueness theorems for periodic functions, Proc. Amer. Math. Soc. 16 (1965) 109-114] cannot hold for parameterized strings, even if the mappings inducing the periodicity ''commute'', in a sense to be specified below; (c) a proof that parameterized strings over an alphabet of at least three letters may have a set of periods which differ from those of any binary string of the same length-whereby the parameterized analog of a classic result by Guibas and Odlyzko [String overlaps, pattern matching, and nontransitive games, J. Combin. Theory Ser. A 30 (1981) 183-208] cannot hold. We also derive necessary and sufficient conditions characterizing parameterized repetitions, which are patterns of length at least twice that of the period, and show how the notion of root differs from the standard case, and highlight some of the implications on extending algorithmic criteria previously adopted for string searching, detection of repetitions and the likes. Finally, as a corollary of our main results, we also show that binary parameterized strings behave much in the same way as non-parameterized ones with respect to periodicity and repetitions, while there is a substantial difference for strings over alphabets of at least three symbols.