## Abstract

The parameterized pattern matching problem is to check if there exists a renaming bijection on the alphabet with which a given pattern can be transformed into a substring of a given text. A parameterized border array (p-border array) is a parameterized version of a standard border array, and we can efficiently solve the parameterized pattern matching problem using p-border arrays. In this paper, we present a linear time algorithm to verify if a given integer array is a valid p-border array for a binary alphabet. We also show a linear time algorithm to compute all binary parameterized strings sharing a given p-border array. In addition, we give an algorithm which computes all p-border arrays of length at most n, where n is a given threshold. This algorithm runs in O(B _{2} ^{n}) time, where B_{2} ^{n} is the number of all p-border arrays of length n for a binary parameter alphabet. The problems with a larger alphabet are much more difficult. Still, we present an O(n ^{1.5})time O(n)space algorithm to verify if a given integer array of length n is a valid p-border array for an unbounded alphabet. The best previously known solution to this task takes time proportional to the n-th Bell number 1/e∑_{k=0}∞k^{n}/k!, and hence our algorithm is much more efficient. Also, we show that it is possible to enumerate all p-border arrays of length at most n for an unbounded alphabet in O(B ^{n}n^{2.5}) time, where B^{n} denotes the number of p-border arrays of length n.

Original language | English |
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Pages (from-to) | 6959-6981 |

Number of pages | 23 |

Journal | Theoretical Computer Science |

Volume | 412 |

Issue number | 50 |

DOIs | |

Publication status | Published - Nov 25 2011 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- General Computer Science