## Abstract

Given ɛ ∈ [0, 1), the ɛ-Relative Error Periodic Pattern Problem (REPP) is the following: INPUT: An n-long sequence S of numbers s_{i} ∈ N in increasing order. OUTPUT: The longest ɛ-relative error periodic pattern, i.e., the longest subsequence s_{i1},s_{i2},…,s_{ik} of S, for which there exists a number p such that the absolute difference between any two consecutive numbers in the subsequence is at least p and at most p(1 + ɛ). The best known algorithm for this problem has O(n^{3} ) time complexity. This bound is too high for large inputs in practice. In this paper we give a new algorithm for finding the longest ɛ-relative error periodic pattern (the REPP problem). Our method is based on a transformation of the input sequence into a different representation: the ɛ-active maximal intervals list L, defined in this paper. We show that the transformation of S to the list L can be done efficiently (quadratic in n and linear in the size of L) and prove that our algorithm is linear in the size of L. This enables us to prove that our algorithm works in sub-cubic time on inputs for which the best known algorithm works in O(n^{3} ) time. Moreover, though it may happen that our algorithm would still be cubic, it is never worse than the known O(n^{3} )-algorithm and in many situations its complexity is O(n^{2} )time.

Original language | English |
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Title of host publication | Design and Analysis of Algorithms - 1st Mediterranean Conference on Algorithms, MedAlg 2012, Proceedings |

Editors | Guy Even, Dror Rawitz |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 1-12 |

Number of pages | 12 |

ISBN (Print) | 9783642348617 |

DOIs | |

State | Published - 2012 |

Event | 1st Mediterranean Conference on Algorithms, MedAlg 2012 - Kibbutz Ein Gedi, Israel Duration: 3 Dec 2012 → 5 Dec 2012 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7659 LNNS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 1st Mediterranean Conference on Algorithms, MedAlg 2012 |
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Country/Territory | Israel |

City | Kibbutz Ein Gedi |

Period | 3/12/12 → 5/12/12 |

### Bibliographical note

Publisher Copyright:© Springer-Verlag Berlin Heidelberg 2012.

### Funding

The best known algorithm for this problem has O(n3) time complexity. This bound is too high for large inputs in practice. In this paper we give a new algorithm for finding the longest ϵ-relative error periodic pattern (the REPP problem). Our method is based on a transformation ★ Partly supported by NSF grant CCR-09-04581, ISF grant 347/09, and BSF grant 2008217. ★★ Partly supported by BSF grant 2008217. ★★★ Partly supported by the National Science Foundation Award 0904246, Israel Sci-ence Foundation grant 347/09, Yahoo, Grant No. 2008217 from the United States-Israel Binational Science Foundation (BSF) and DFG. † Partly supported by ISF grant 347/09. Partly supported by NSF grant CCR-09-04581, ISF grant 347/09, and BSF grant 2008217.?? Partly supported by BSF grant 2008217.Partly supported by the National Science Foundation Award 0904246, Israel Science Foundation grant 347/09, Yahoo, Grant No. 2008217 from the United StatesIsrael Binational Science Foundation (BSF) and DFG.Partly supported by ISF grant 347/09.

Funders | Funder number |
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Israel Sci-ence Foundation | |

United StatesIsrael Binational Science Foundation | |

National Science Foundation | CCR-09-04581, 0904246 |

Deutsche Forschungsgemeinschaft | |

United States-Israel Binational Science Foundation | 2008217 |

Israel Science Foundation | 347/09 |