## Abstract

In this work, we develop new insights into the fundamental problem of convexity testing of real-valued functions over the domain [n]. Specifically, we present a nonadaptive algorithm that, given inputs ε ∈ (0, 1), s ∈ N, and oracle access to a function, ε-tests convexity in O(log(s)/ε), where s is an upper bound on the number of distinct discrete derivatives of the function. We also show that this bound is tight. Since s ≤ n, our query complexity bound is at least as good as that of the optimal convexity tester (Ben Eliezer; ITCS 2019) with complexity O(^{log}_{ε}^{εn} ); our bound is strictly better when s = o(n). The main contribution of our work is to appropriately parameterize the complexity of convexity testing to circumvent the worst-case lower bound (Belovs et al.; SODA 2020) of Ω(log(_{ε}^{εn)} ) expressed in terms of the input size and obtain a more efficient algorithm.

Original language | English |
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Title of host publication | SIAM Symposium on Simplicity in Algorithms, SOSA 2022 |

Publisher | Society for Industrial and Applied Mathematics Publications |

Pages | 174-181 |

Number of pages | 8 |

ISBN (Electronic) | 9781713852087 |

State | Published - 2022 |

Event | 5th SIAM Symposium on Simplicity in Algorithms, SOSA 2022, co-located with SODA 2022 - Virtual, Online Duration: 10 Jan 2022 → 11 Jan 2022 |

### Publication series

Name | SIAM Symposium on Simplicity in Algorithms, SOSA 2022 |
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### Conference

Conference | 5th SIAM Symposium on Simplicity in Algorithms, SOSA 2022, co-located with SODA 2022 |
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City | Virtual, Online |

Period | 10/01/22 → 11/01/22 |

### Bibliographical note

Publisher Copyright:Copyright © 2022 by SIAM.

## ASJC Scopus subject areas

- Computational Theory and Mathematics
- Computational Mathematics
- Numerical Analysis
- Theoretical Computer Science