Testing Periodicity

Abstract

A string α∈Σn is called p-periodic, if for every i,j ∈ {1,...,n}, such that $i\equiv j \bmod p$, αi = αj, where αi is the i-th place of α. A string α∈Σn is said to be period(≤ g), if there exists p∈ {1,...,g} such that α is p-periodic. An e-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 e-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 e-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 e-testing of periodicity ($period(\leq \frac{n}{2})$) is $\Theta(\sqrt{n\log{n}})$ queries.