### Abstract

The longest common extension (LCE) problem is to preprocess a given string ω of length n so that the length of the longest common prefix between suffixes of ω that start at any two given positions is answered quickly. In this paper, we present a data structure of O(z^{2} + n/t ) words of space which answers LCE queries in O(1) time and can be built in O(n log δ) time, where 1 ≤ T ≤ √n is a parameter, z is the size of the Lempel-Ziv 77 factorization of ω and φ is the alphabet size. The proposed LCE data structure does not access the input string ω when answering queries, and thus w can be deleted after preprocessing. On top of this main result, we obtain further results using (variants of) our LCE data structure, which include the following: For highly repetitive strings where the z^{2} term is dominated by n/x, we obtain a constant-time and sub-linear space LCE query data structure. Even when the input string is not well compressible via Lempel-Ziv 77 factorization, we still can obtain a constant-time and sub-linear space LCE data structure for suitable and for φ ≤ 2^{o(log n)}. The time-space trade-off lower bounds for the LCE problem by Bille et al. [J. Discrete Algorithms, 25:42-50, 2014] and by Kosolobov [CoRR, abs/1611.02891, 2016] do not apply in some cases with our LCE data structure.

Original language | English |
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Title of host publication | 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017 |

Editors | Kim G. Larsen, Jean-Francois Raskin, Hans L. Bodlaender |

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

ISBN (Electronic) | 9783959770460 |

DOIs | |

Publication status | Published - Nov 1 2017 |

Event | 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017 - Aalborg, Denmark Duration: Aug 21 2017 → Aug 25 2017 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 83 |

ISSN (Print) | 1868-8969 |

### Other

Other | 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017 |
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Country | Denmark |

City | Aalborg |

Period | 8/21/17 → 8/25/17 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software

### Cite this

*42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017*(Leibniz International Proceedings in Informatics, LIPIcs; Vol. 83). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.MFCS.2017.10

**Small-space LCE data structure with constant-time queries.** / Tanimura, Yuka; Nishimoto, Takaaki; Bannai, Hideo; Inenaga, Shunsuke; Takeda, Masayuki.

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

*42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017.*Leibniz International Proceedings in Informatics, LIPIcs, vol. 83, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017, Aalborg, Denmark, 8/21/17. https://doi.org/10.4230/LIPIcs.MFCS.2017.10

}

TY - GEN

T1 - Small-space LCE data structure with constant-time queries

AU - Tanimura, Yuka

AU - Nishimoto, Takaaki

AU - Bannai, Hideo

AU - Inenaga, Shunsuke

AU - Takeda, Masayuki

PY - 2017/11/1

Y1 - 2017/11/1

N2 - The longest common extension (LCE) problem is to preprocess a given string ω of length n so that the length of the longest common prefix between suffixes of ω that start at any two given positions is answered quickly. In this paper, we present a data structure of O(z2 + n/t ) words of space which answers LCE queries in O(1) time and can be built in O(n log δ) time, where 1 ≤ T ≤ √n is a parameter, z is the size of the Lempel-Ziv 77 factorization of ω and φ is the alphabet size. The proposed LCE data structure does not access the input string ω when answering queries, and thus w can be deleted after preprocessing. On top of this main result, we obtain further results using (variants of) our LCE data structure, which include the following: For highly repetitive strings where the z2 term is dominated by n/x, we obtain a constant-time and sub-linear space LCE query data structure. Even when the input string is not well compressible via Lempel-Ziv 77 factorization, we still can obtain a constant-time and sub-linear space LCE data structure for suitable and for φ ≤ 2o(log n). The time-space trade-off lower bounds for the LCE problem by Bille et al. [J. Discrete Algorithms, 25:42-50, 2014] and by Kosolobov [CoRR, abs/1611.02891, 2016] do not apply in some cases with our LCE data structure.

AB - The longest common extension (LCE) problem is to preprocess a given string ω of length n so that the length of the longest common prefix between suffixes of ω that start at any two given positions is answered quickly. In this paper, we present a data structure of O(z2 + n/t ) words of space which answers LCE queries in O(1) time and can be built in O(n log δ) time, where 1 ≤ T ≤ √n is a parameter, z is the size of the Lempel-Ziv 77 factorization of ω and φ is the alphabet size. The proposed LCE data structure does not access the input string ω when answering queries, and thus w can be deleted after preprocessing. On top of this main result, we obtain further results using (variants of) our LCE data structure, which include the following: For highly repetitive strings where the z2 term is dominated by n/x, we obtain a constant-time and sub-linear space LCE query data structure. Even when the input string is not well compressible via Lempel-Ziv 77 factorization, we still can obtain a constant-time and sub-linear space LCE data structure for suitable and for φ ≤ 2o(log n). The time-space trade-off lower bounds for the LCE problem by Bille et al. [J. Discrete Algorithms, 25:42-50, 2014] and by Kosolobov [CoRR, abs/1611.02891, 2016] do not apply in some cases with our LCE data structure.

UR - http://www.scopus.com/inward/record.url?scp=85038430729&partnerID=8YFLogxK

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U2 - 10.4230/LIPIcs.MFCS.2017.10

DO - 10.4230/LIPIcs.MFCS.2017.10

M3 - Conference contribution

AN - SCOPUS:85038430729

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017

A2 - Larsen, Kim G.

A2 - Raskin, Jean-Francois

A2 - Bodlaender, Hans L.

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

ER -