### 抄録

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.

元の言語 | 英語 |
---|---|

ページ（範囲） | 6959-6981 |

ページ数 | 23 |

ジャーナル | Theoretical Computer Science |

巻 | 412 |

発行部数 | 50 |

DOI | |

出版物ステータス | 出版済み - 11 25 2011 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### これを引用

**Verifying and enumerating parameterized border arrays.** / I, Tomohiro; Inenaga, Shunsuke; Bannai, Hideo; Takeda, Masayuki.

研究成果: ジャーナルへの寄稿 › 記事

*Theoretical Computer Science*, 巻. 412, 番号 50, pp. 6959-6981. https://doi.org/10.1016/j.tcs.2011.09.008

}

TY - JOUR

T1 - Verifying and enumerating parameterized border arrays

AU - I, Tomohiro

AU - Inenaga, Shunsuke

AU - Bannai, Hideo

AU - Takeda, Masayuki

PY - 2011/11/25

Y1 - 2011/11/25

N2 - 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 B2 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∞kn/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 nn2.5) time, where Bn denotes the number of p-border arrays of length n.

AB - 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 B2 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∞kn/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 nn2.5) time, where Bn denotes the number of p-border arrays of length n.

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U2 - 10.1016/j.tcs.2011.09.008

DO - 10.1016/j.tcs.2011.09.008

M3 - Article

VL - 412

SP - 6959

EP - 6981

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 50

ER -