Fine and Wilf's Theorem for Partial Words with Arbitrarily Many Weak Periods

Fine and Wilf's well-known theorem states that any word having periods p,q and length at least p+q-gcd(p,q) also has gcd(p,q) as a period. Moreover, the length p+q-gcd(p,q) is critical since counterexamples can be provided for shorter words. This result has since been extended to partial words, or finite sequences that may contain some "holes." More precisely, any partial word u with H holes having weak periods p,q and length at least the so-denoted lH(p,q) also has strong period gcd(p,q) provided u is not (H,(p,q))-special. This extension was done for one hole by Berstel and Boasson (where the class of (1,(p,q))-special partial words is empty), and for an arbitrary number of holes by Blanchet-Sadri. In this paper, we further extend these results, allowing an arbitrary number of weak periods. In addition to speciality, the concepts of intractable period sets and interference between periods play a role.

[1]  Lucian Ilie,et al.  Multiple spaced seeds for homology search , 2007, Bioinform..

[2]  A. M. Shur,et al.  Partial words and the interaction property of periods , 2004 .

[3]  Jean Berstel,et al.  Partial Words and a Theorem of Fine and Wilf , 1999, Theor. Comput. Sci..

[4]  Bin Ma,et al.  PatternHunter: faster and more sensitive homology search , 2002, Bioinform..

[5]  Shu Wang,et al.  A new approach to the periodicity lemma on strings with holes , 2009, Theoretical Computer Science.

[6]  Francine Blanchet-Sadri.  Algorithmic Combinatorics on Partial Words , 2012, Int. J. Found. Comput. Sci..

[7]  Tero Harju,et al.  Interaction Properties of Relational Periods , 2008, Discret. Math. Theor. Comput. Sci..

[8]  Francine Blanchet-Sadri,et al.  Graph connectivity, partial words, and a theorem of Fine and Wilf , 2008, Inf. Comput..

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

[10]  François Nicolas,et al.  Hardness of optimal spaced seed design , 2008, J. Comput. Syst. Sci..

[11]  Gad M. Landau,et al.  Optimal spaced seeds for faster approximate string matching , 2007, J. Comput. Syst. Sci..

[12]  Jeremy Buhler,et al.  Designing seeds for similarity search in genomic DNA , 2003, RECOMB '03.

[13]  Gregory Kucherov,et al.  Improved hit criteria for DNA local alignment , 2004, BMC Bioinformatics.

[14]  Lucian Ilie,et al.  Generalised fine and Wilf's theorem for arbitrary number of periods , 2005, Theor. Comput. Sci..

[15]  Bin Ma,et al.  On spaced seeds for similarity search , 2004, Discret. Appl. Math..