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.


Testing Period Query Complexity Property Testing Input String Randomized Algorithm 
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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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