## Abstract

The Arithmetic Progressions Tree (APT) is an encoding of a monotonic sequence ℒ in [1..n]. Previous work on APT coding focused on its theoretical and experimental compression guarantees. This paper is the first to consider computations over APT compressed data. In particular: (1) We show how to perform a search for any sub-sequence of the monotone sequence ℒ in time proportional to the query sub-sequence length multiplied by the size of the APT compressed representation of ℒ. (2) We show how, given the APT compressed representation of the monotone sequence ℒ, we can find a minimum run-length of ℒ in constant time, a maximum run-length of ℒ in O(log n) time, and all runs of ℒ in constant time plus the output size. (3) Most importantly, we show how, given the APT compressed representation of the monotone sequence ℒ, we can answer whether a periodic pattern P appears in ℒ in O(log n) time and report its locations in the output size time. (4) In addition, we improve the APT construction algorithm time and space complexity.

Original language | English |
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Title of host publication | Proceedings - DCC 2024 |

Subtitle of host publication | 2024 Data Compression Conference |

Editors | Ali Bilgin, James E. Fowler, Joan Serra-Sagrista, Yan Ye, James A. Storer |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 153-162 |

Number of pages | 10 |

ISBN (Electronic) | 9798350385878 |

DOIs | |

State | Published - 2024 |

Externally published | Yes |

Event | 2024 Data Compression Conference, DCC 2024 - Snowbird, United States Duration: 19 Mar 2024 → 22 Mar 2024 |

### Publication series

Name | Data Compression Conference Proceedings |
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ISSN (Print) | 1068-0314 |

### Conference

Conference | 2024 Data Compression Conference, DCC 2024 |
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Country/Territory | United States |

City | Snowbird |

Period | 19/03/24 → 22/03/24 |

### Bibliographical note

Publisher Copyright:© 2024 IEEE.

## Keywords

- APT compression
- Arithmetic progressions
- Compact Data Structures
- Monotonic sequences
- Periodic patterns