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Abstract

A string α∈Σ n is called p-periodic, if for every i,j ∈ {1,...,n}, such that \(i\equiv j \bmod p\), 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 \(1,\dots,\frac{n}{2}\). 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 \(g\leq \frac{log{n}}{4}\), 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 (\(period(\leq \frac{n}{2})\)) is \(\Theta(\sqrt{n\log{n}})\) queries.

Keywords

Testing Period Query Complexity Property Testing Input String Randomized Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Oded Lachish
    • 1
  • Ilan Newman
    • 1
  1. 1.Haifa UniversityHaifaIsrael

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