Abstract
A string α∈Σn is called p-periodic, if for every i,j ∈ {1,...,n}, such that \(i\equiv j \bmod p\), α i = α j , where α i 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 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 \(1,\dots,\frac{n}{2}\). 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 ε-testing of periodicity (\(period(\leq \frac{n}{2})\)) is \(\Theta(\sqrt{n\log{n}})\) queries.
This research was supported by THE ISRAEL SCIENCE FOUNDATION (grant number 55/03).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Goldwasser, S., Goldreich, O., Ron, D.: Property testing and its connection to learning and approximation. Journal of the ACM 45, 653–750 (1998)
Rubinfeld, R., Sudan, M.: Robust characterization of polynomials with applications to program testing. SIAM Journal of Computing 25, 252–271 (1996)
Ergun, F., Muthukrishnan, S., Sahinalp, C.: Sub-linear methods for detecting periodic trends in data streams. In: Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 16–28. Springer, Heidelberg (2004)
Indyk, P., Koudas, N., Muthukrishnan, S.: Identifying representative trends in massive time series data sets using sketches. In: VLDB 2000, Proceedings of 26th International Conference on Very Large Data Bases, Cairo, Egypt, September 10-14, pp. 363–372. Morgan Kaufmann, San Francisco (2000)
Krauthgamer, R., Sasson, O.: Property testing of data dimensionality. In: Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms. Society for Industrial and Applied Mathematics, pp. 18–27 (2003)
Gilbert, C., Guha, S., Indyk, P., Muthukrishnan, S., Strauss, M.: Near-optimal sparse fourier representations via sampling. In: STOC 2002, Proceedings of the thirty-fourth annual ACM symposium on Theory of computing, pp. 152–161 (2002)
Samorodnitsky, A., Trevisan, L.: A PCP characterization of NP with optimal amortized query complexity. In: Proc. of the 32 ACM STOC, pp. 191–199 (2000)
Hästad, J., Wigderson, A.: Simple analysis of graph tests for linearity and pcp. Random Struct. Algorithms 22(2), 139–160 (2003)
Fischer, E.: The art of uninformed decisions: A primer to property testing. The computational complexity column of The Bulletin of the European Association for Theoretical Computer Science 75, 97–126 (2001)
Ron, D.: Property testing (a tutorial). In: Handbook of Randomized computing, pp. 597–649. Kluwer Press, Dordrecht (2001)
Hadamard, J.: Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques. Bull. Soc. Math. France 24, 199–220 (1896)
Poussin, V.: Recherces analytiques sur la théorie des nombres premiers. Ann. Soc. Sci. Bruxelles (1897)
Newman, D.J.: Simple analytic proof of the prime number theorem. Amer. Math. Monthly 87, 693–696 (1980)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lachish, O., Newman, I. (2005). Testing Periodicity. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2005 2005. Lecture Notes in Computer Science, vol 3624. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538462_31
Download citation
DOI: https://doi.org/10.1007/11538462_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28239-6
Online ISBN: 978-3-540-31874-3
eBook Packages: Computer ScienceComputer Science (R0)