Testing periodicity

Oded Lachish, Ilan Newman

Research output: Contribution to journalConference articlepeer-review


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 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.

Original languageEnglish
Pages (from-to)366-377
Number of pages12
JournalLecture Notes in Computer Science
StatePublished - 2005
Event8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2005 and 9th International Workshop on Randomization and Computation, RANDOM 2005 - Berkeley, CA, United States
Duration: 22 Aug 200524 Aug 2005

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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