Oded Lachish

We study the string-property of being periodic and having periodicity smaller than a given bound. Let Σ be a fixed alphabet and let p, n be integers such that p ≤ n2 . A length-n string over Σ, α = (α1, . . . , αn), has the property Period(p) if for every i, j ∈ {1, . . . , n}, αi = αj whenever i ≡ j (mod p). For an integer function g = g(n) ≤ n2 , the property Period(≤ g) is the property of all strings that are in Period(p) for some p ≤ g. The property Period(≤ n2 ) is also called Periodicity. An -test for a property P of length-n strings is a randomized algorithm that for an input α distinguishes between the case that α is in P and the case where one needs to change at least an -fraction of the letters of α to get a string in P . The query complexity of the -test is the number of letter queries it makes for the worst case input string of length n. We study the query complexity of -tests for Period(≤ g) as a function of g, when g varies from 1 to n2 , while ignoring the exact dependence on the proximity parameter . We show that there exists an exponential phase transition in the query complexity around g = log n. That is, for every δ > 0 and g ≥ (log n) , every two-sided, adaptive -test for Period(≤ g) has a query complexity that is polynomial in g. On the other hand, for g ≤ logn 6 , there exists a one-sided error, non-adaptive -test for Period(≤ g), whose query complexity is poly-logarithmic in g. We also prove that the asymptotic query complexity of one-sided error non-adaptive -tests for Periodicity is Θ( √ n log n), ignoring the dependence on .