Testing periodicity

A string α ∈ Σⁿ is called p-periodic, if for every i, j ∈ {1,..., n}, such that i ≡ j mod p, α_i = α_j, where α_i is the i-th place of α. A string a ∈ Σⁿ is said to be period(≤ g), if there exists p ∈ {1,..., g} such that α is p-periodic. An ∈-property tester for period(≤ g) is a randomized algorithm, that for an input α distinguishes between the case that a is in period(≤ g) and the case that one needs to change at least ∈-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 ∈-testers for period(≤ g) when g varies in the range 1,..., 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 ≥ (log n)^1+δ, the number of queries required and sufficient for testing period(≤ g) is polynomial in g. On the other hand, for each g ≤ logn/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 e-testing of periodicity (period(≤ n/2)) is Θ(√n log n) queries.