A new approach to the periodicity lemma on strings with holes

We first give an elementary proof of the periodicity lemma for strings containing one hole (variously called a "wild card", a "don't-care" or an "indeterminate letter" in the literature). The proof is modelled on Euclid's algorithm for the greatest common divisor and is simpler than the original proof given in [J. Berstel, L. Boasson, Partial words and a theorem of Fine and Wilf, Theoret. Comput. Sci. 218 (1999) 135-141]. We then study the two-hole case, where our result agrees with the one given in [F. Blanchet-Sadri, Robert A. Hegstrom, Partial words and a theorem of Fine and Wilf revisited, Theoret. Comput. Sci. 270 (1-2) (2002) 401-419] but is more easily proved and enables us to identify a maximum-length prefix or suffix of the string to which the periodicity lemma does apply. Finally, we extend our result to three or more holes using elementary methods, and state a version of the periodicity lemma that applies to all strings with or without holes. We describe an algorithm that, given the locations of the holes in a string, computes maximum-length substrings to which the periodicity lemma applies, in time proportional to the number of holes. Our approach is quite different from that used by Blanchet-Sadri and Hegstrom, and also simpler

A New Approach to the Periodicity Lemma on Strings with Holes

Abstract We first give an elementary proof of the periodicity lemma for strings containing one hole (variously called a "wild card" or a "don't-care" or an "indeterminate letter" in the literature). The proof is modelled on Euclid's algorithm for the greatest common divisor and is simpler than the original proof given in [BB99]. We then study the two hole case, where our result agrees with the one given in [BSH02] but is more easily proved and enables us to identify a maximum-length prefix or suffix of the string to which the periodicity lemma does apply. Finally we extend our result to three or more holes using elementary methods and state a version of the periodicity lemma that applies to all strings with or without holes. We describe an algorithm that, given the locations of the holes in a string, computes maximum length substrings to which the periodicity lemma applies, in time proportional to the number of holes. Our approach is quite different from the one in [BSH02, BS04] and also simpler

Computing Covers Using Prefix Tables

An \emph{indeterminate string} $x = x[1..n]$ on an alphabet $\Sigma$ is a sequence of nonempty subsets of $\Sigma$; $x$ is said to be \emph{regular} if every subset is of size one. A proper substring $u$ of regular $x$ is said to be a \emph{cover} of $x$ iff for every $i \in 1..n$, an occurrence of $u$ in $x$ includes $x[i]$. The \emph{cover array} $\gamma = \gamma[1..n]$ of $x$ is an integer array such that $\gamma[i]$ is the longest cover of $x[1..i]$. Fifteen years ago a complex, though nevertheless linear-time, algorithm was proposed to compute the cover array of regular $x$ based on prior computation of the border array of $x$. In this paper we first describe a linear-time algorithm to compute the cover array of regular string $x$ based on the prefix table of $x$. We then extend this result to indeterminate strings.Comment: 14 pages, 1 figur

Periods in Partial Words: An Algorithm

Partial words are finite sequences over a finite alphabet that may contain some holes. A variant of the celebrated Fine–Wilf theorem shows the existence of a bound L=L(h,p,q) such that if a partial word of length at least L with h holes has periods p and q, then it also has period gcd(p,q). In this paper, we associate a graph with each p - and q -periodic word, and study two types of vertex connectivity on such a graph: modified degree connectivity and r -set connectivity where r = q mod p. As a result, we give an algorithm for computing L(h,p,q) in the general case and show how to use it to derive the closed formulas