### Abstract

In standard property testing, the task is to distinguish between objects that have a property P and those that are ε-far from P, for some ε > 0. In this setting, it is perfectly acceptable for the tester to provide a negative answer for every input object that does not satisfy P. This implies that property testing in and of itself cannot be expected to yield any information whatsoever about the distance from the object to the property. We address this problem in this paper, restricting our attention to monotonicity testing. A function f : {1, . . . , n] R is at distance ε_{f} from being monotone if it can (and must) be modified at ε_{f}n places to become monotone. For any fixed δ > 0, we compute, with probability at least 2/3, an interval [(1/2 - 8)ε, ε] that encloses ε_{f}. The running time of our algorithm is O(ε _{f}
^{-1} log log ε_{f}
^{-1} log n), which is optimal within a factor of log log ε_{f}
^{-1} and represents a substantial improvement over previous work. We give a second algorithm with an expected running time of O(ε_{f}
^{-1} log n log log log n). Finally, we extend our results to multivariate functions.

Original language | English (US) |
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Pages (from-to) | 371-383 |

Number of pages | 13 |

Journal | Random Structures and Algorithms |

Volume | 31 |

Issue number | 3 |

DOIs | |

State | Published - Oct 1 2007 |

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics

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## Cite this

*Random Structures and Algorithms*,

*31*(3), 371-383. https://doi.org/10.1002/rsa.20167