## Abstract

For a permutation π : [k] → [k], a function f : [n] → ℝ contains a π-appearance if there exists 1 ≤ i_{1} < i_{2} < · · · < i_{k} ≤ n such that for all s, t ∈ [k], f(is) < f(it) if and only if π(s) < π(t). The function is π-free if it has no π-appearances. In this paper, we investigate the problem of testing whether an input function f is π-free or whether f differs on at least εn values from every π-free function. This is a generalization of the well-studied monotonicity testing and was first studied by Newman, Rabinovich, Rajendraprasad and Sohler [28]. We show that for all constants k ∈ N, ε ∈ (0, 1), and permutation π : [k] → [k], there is a one-sided error ε-testing algorithm for π-freeness of functions f : [n] → R that makes Õ(n^{o(1)}) queries. We improve significantly upon the previous best upper bound O(n^{1−1/(k−1))} by Ben-Eliezer and Canonne [7]. Our algorithm is adaptive, while the earlier best upper bound is known to be tight for nonadaptive algorithms.

Original language | English |
---|---|

Title of host publication | 49th EATCS International Conference on Automata, Languages, and Programming, ICALP 2022 |

Editors | Mikolaj Bojanczyk, Emanuela Merelli, David P. Woodruff |

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

ISBN (Electronic) | 9783959772358 |

DOIs | |

State | Published - 1 Jul 2022 |

Event | 49th EATCS International Conference on Automata, Languages, and Programming, ICALP 2022 - Paris, France Duration: 4 Jul 2022 → 8 Jul 2022 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
---|---|

Volume | 229 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 49th EATCS International Conference on Automata, Languages, and Programming, ICALP 2022 |
---|---|

Country/Territory | France |

City | Paris |

Period | 4/07/22 → 8/07/22 |

### Bibliographical note

Funding Information:The author was supported by the Israel Science Foundation, grant number

Publisher Copyright:

© Ilan Newman and Nithin Varma; licensed under Creative Commons License CC-BY 4.0

## Keywords

- Pattern freeness
- Property testing
- Sublinear algorithms

## ASJC Scopus subject areas

- Software