# Testing periodicity

Lachish, Oded and Newman, I.
(2011)
Testing periodicity.
*Algorithmica* 60
(2),
pp. 401-420.
ISSN 0178-4617.

## Abstract

We study the string-property of being periodic and having periodicity smaller than a given bound. Let Σ be a fixed alphabet and let p,n be integers such that p≤n2 . A length-n string over Σ, α=(α 1,…,α n ), has the property Period(p) if for every i,j∈{1,…,n}, α i =α j whenever i≡j (mod p). For an integer parameter g≤n2, the property Period(≤g) is the property of all strings that are in Period(p) for some p≤g. The property Period(≤n2) is also called Periodicity. An ε-test for a property P of length-n strings is a randomized algorithm that for an input α distinguishes between the case that α is in P and the case where one needs to change at least an ε-fraction of the letters of α to get a string in P. The query complexity of the ε-test is the number of letter queries it makes for the worst case input string of length n. We study the query complexity of ε-tests for Period(≤g) as a function of the parameter g, when g varies from 1 to n2 , while ignoring the exact dependence on the proximity parameter ε. We show that there exists an exponential phase transition in the query complexity around g=log n. That is, for every δ>0 and g≥(log n)1+δ , every two-sided error, adaptive ε-test for Period(≤g) has a query complexity that is polynomial in g. On the other hand, for g≤logn6 , there exists a one-sided error, non-adaptive ε-test for Period(≤g), whose query complexity is poly-logarithmic in g. We also prove that the asymptotic query complexity of one-sided error non-adaptive ε-tests for Periodicity is Θ(nlogn−−−−−√) , ignoring the dependence on ε.

## Metadata

Item Type: | Article |
---|---|

School: | Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences |

Depositing User: | Sarah Hall |

Date Deposited: | 30 May 2013 13:53 |

Last Modified: | 09 Aug 2023 12:33 |

URI: | https://eprints.bbk.ac.uk/id/eprint/7112 |

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