# Testing periodicity

Lachish, Oded and Newman, I.
(2005)
Testing periodicity.
In:
Chekuri, C. and Jansen, K. and Rolim, J.D.P. and Trevisan, L. (eds.)
*Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques.*
Lecture Notes in Computer Science.
Springer, pp. 366-377.
ISBN 9783540282396.

## Abstract

A string α∈Σ n is called p-periodic, if for every i,j ∈ {1,...,n}, such that i≡jmodp , α i = α j , where α i is the i-th place of α. A string α∈Σ n is said to be period(≤ g), if there exists p∈ {1,...,g} such that α is p-periodic. An ε-property tester for period(≤ g) is a randomized algorithm, that for an input α distinguishes between the case that α is in period(≤ g) and the case that one needs to change at least ε-fraction of the letters of α, so that it will become period(≤ g). The complexity of the tester is the number of letter-queries it makes to the input. We study here the complexity of ε-testers for period(≤ g) when g varies in the range 1,…,n2 . We show that there exists a surprising exponential phase transition in the query complexity around g=log n. That is, for every δ > 0 and for each g, such that g≥ (logn)1 + δ, the number of queries required and sufficient for testing period(≤ g) is polynomial in g. On the other hand, for each g≤logn4 , the number of queries required and sufficient for testing period(≤ g) is only poly-logarithmic in g. We also prove an exact asymptotic bound for testing general periodicity. Namely, that 1-sided error, non adaptive ε-testing of periodicity ( period(≤n2) ) is Θ(nlogn−−−−−√) queries.

## Metadata

Item Type: | Book Section |
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School: | Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences |

Depositing User: | Sarah Hall |

Date Deposited: | 25 May 2021 13:59 |

Last Modified: | 09 Aug 2023 12:50 |

URI: | https://eprints.bbk.ac.uk/id/eprint/44414 |

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