Testing periodicity

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