### 抄録

A substring Q of a string S is called a shortest unique substring (SUS) for interval [s, t] in S, if Q occurs exactly once in S, this occurrence of Q contains interval [s, t], and every substring of S which contains interval [s, t] and is shorter than Q occurs at least twice in S. The SUS problem is, given a string S, to preprocess S so that for any subsequent query interval [s, t] all the SUSs for interval [s, t] can be answered quickly. When s = t, we call the SUSs for [s, t] as point SUSs, and when s ≤ t, we call the SUSs for [s, t] as interval SUSs. There exist optimal O(n)-time preprocessing scheme which answers queries in optimal O(k) time for both point and interval SUSs, where n is the length of S and k is the number of outputs for a given query. In this paper, we reveal structural, combinatorial properties underlying the SUS problem: Namely, we show that the number of intervals in S that correspond to point SUSs for all query positions in S is less than 1.5n, and show that this is a matching upper and lower bound. Also, we consider the maximum number of intervals in S that correspond to interval SUSs for all query intervals in S.

元の言語 | 英語 |
---|---|

ホスト出版物のタイトル | 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017 |

出版者 | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

巻 | 78 |

ISBN（電子版） | 9783959770392 |

DOI | |

出版物ステータス | 出版済み - 7 1 2017 |

イベント | 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017 - Warsaw, ポーランド 継続期間: 7 4 2017 → 7 6 2017 |

### その他

その他 | 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017 |
---|---|

国 | ポーランド |

市 | Warsaw |

期間 | 7/4/17 → 7/6/17 |

### All Science Journal Classification (ASJC) codes

- Software

### これを引用

*28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017*(巻 78). [24] Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.CPM.2017.24

**Tight bounds on the maximum number of shortest unique substrings.** / Mieno, Takuya; Inenaga, Shunsuke; Bannai, Hideo; Takeda, Masayuki.

研究成果: 著書/レポートタイプへの貢献 › 会議での発言

*28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017.*巻. 78, 24, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017, Warsaw, ポーランド, 7/4/17. https://doi.org/10.4230/LIPIcs.CPM.2017.24

}

TY - GEN

T1 - Tight bounds on the maximum number of shortest unique substrings

AU - Mieno, Takuya

AU - Inenaga, Shunsuke

AU - Bannai, Hideo

AU - Takeda, Masayuki

PY - 2017/7/1

Y1 - 2017/7/1

N2 - A substring Q of a string S is called a shortest unique substring (SUS) for interval [s, t] in S, if Q occurs exactly once in S, this occurrence of Q contains interval [s, t], and every substring of S which contains interval [s, t] and is shorter than Q occurs at least twice in S. The SUS problem is, given a string S, to preprocess S so that for any subsequent query interval [s, t] all the SUSs for interval [s, t] can be answered quickly. When s = t, we call the SUSs for [s, t] as point SUSs, and when s ≤ t, we call the SUSs for [s, t] as interval SUSs. There exist optimal O(n)-time preprocessing scheme which answers queries in optimal O(k) time for both point and interval SUSs, where n is the length of S and k is the number of outputs for a given query. In this paper, we reveal structural, combinatorial properties underlying the SUS problem: Namely, we show that the number of intervals in S that correspond to point SUSs for all query positions in S is less than 1.5n, and show that this is a matching upper and lower bound. Also, we consider the maximum number of intervals in S that correspond to interval SUSs for all query intervals in S.

AB - A substring Q of a string S is called a shortest unique substring (SUS) for interval [s, t] in S, if Q occurs exactly once in S, this occurrence of Q contains interval [s, t], and every substring of S which contains interval [s, t] and is shorter than Q occurs at least twice in S. The SUS problem is, given a string S, to preprocess S so that for any subsequent query interval [s, t] all the SUSs for interval [s, t] can be answered quickly. When s = t, we call the SUSs for [s, t] as point SUSs, and when s ≤ t, we call the SUSs for [s, t] as interval SUSs. There exist optimal O(n)-time preprocessing scheme which answers queries in optimal O(k) time for both point and interval SUSs, where n is the length of S and k is the number of outputs for a given query. In this paper, we reveal structural, combinatorial properties underlying the SUS problem: Namely, we show that the number of intervals in S that correspond to point SUSs for all query positions in S is less than 1.5n, and show that this is a matching upper and lower bound. Also, we consider the maximum number of intervals in S that correspond to interval SUSs for all query intervals in S.

UR - http://www.scopus.com/inward/record.url?scp=85027271339&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85027271339&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.CPM.2017.24

DO - 10.4230/LIPIcs.CPM.2017.24

M3 - Conference contribution

VL - 78

BT - 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -