Testing Periodicity

A string α ∈ Σn is called p-periodic, if for every i, j ∈ {1, . . . , n}, such that i ≡ j mod 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 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 -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 othe...

• Type: Conference
• Year: 2005
• Where: APPROX
• Authors: Oded Lachish, Ilan Newman
• Keywords: Algorithms, APPROX 2005, I-th Place, Mod P, String