Generalized pattern matching and periodicity under substring consistent equivalence relations

Let ź be a substring consistent equivalence relation (SCER) on strings such that for any two strings x , y , x ź y implies that (1) | x | = | y | and (2) x i . . j ź y i . . j for all 0 ź i ź j < | x | . Examples of SCER are parameterized pattern matching and order-preserving pattern matching. We present a generalized and efficient algorithm for pattern matching with SCER ź. Also, we show analogues of Fine and Wilf's periodicity lemma hold for SCER.

[1]  Peter Weiner,et al.  Linear Pattern Matching Algorithms , 1973, SWAT.

[2]  Brenda S. Baker.  Parameterized Pattern Matching: Algorithms and Applications , 1996, J. Comput. Syst. Sci..

[3]  H. Wilf,et al.  Uniqueness theorems for periodic functions , 1965 .

[4]  Shunsuke Inenaga,et al.  Palindrome pattern matching , 2013, Theor. Comput. Sci..

[5]  M. Schützenberger,et al.  The equation $a^M=b^Nc^P$ in a free group. , 1962 .

[6]  Wojciech Rytter,et al.  A linear time algorithm for consecutive permutation pattern matching , 2013, Inf. Process. Lett..

[7]  Joong Chae Na,et al.  A fast algorithm for order-preserving pattern matching , 2015, Inf. Process. Lett..

[8]  M. Lothaire,et al.  Applied Combinatorics on Words , 2005 .

[9]  Rudolf Fleischer,et al.  Order Preserving Matching , 2013, Theor. Comput. Sci..

[10]  Antonio Restivo,et al.  Fine and Wilf's Theorem for Three Periods and a Generalization of Sturmian Words , 1999, Theor. Comput. Sci..

[11]  S. Muthukrishnan,et al.  Alphabet Dependence in Parameterized Matching , 1994, Inf. Process. Lett..

[12]  Raffaele Giancarlo,et al.  Periodicity and repetitions in parameterized strings , 2008, Discret. Appl. Math..

[13]  Wojciech Rytter,et al.  Text Algorithms , 1994 .

[14]  Donald E. Knuth,et al.  Fast Pattern Matching in Strings , 1977, SIAM J. Comput..

[15]  Jacques Justin,et al.  On a paper by Castelli, Mignosi, Restivo , 2000, RAIRO Theor. Informatics Appl..

[16]  M. Lothaire.  Applied Combinatorics on Words (Encyclopedia of Mathematics and its Applications) , 2005 .