### Abstract

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.

Original language | English |
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Title of host publication | 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017 |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Volume | 78 |

ISBN (Electronic) | 9783959770392 |

DOIs | |

Publication status | Published - Jul 1 2017 |

Event | 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017 - Warsaw, Poland Duration: Jul 4 2017 → Jul 6 2017 |

### Other

Other | 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017 |
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Country | Poland |

City | Warsaw |

Period | 7/4/17 → 7/6/17 |

### All Science Journal Classification (ASJC) codes

- Software

### Cite this

*28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017*(Vol. 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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017.*vol. 78, 24, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017, Warsaw, Poland, 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

AN - SCOPUS:85027271339

VL - 78

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

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

ER -