Abstract
We present improved algorithms for testing monotonicity of functions. Namely, given the ability to query an unknown function f: Σn ↦ Ξ, where Σ and Ξ are finite ordered sets, the test always accepts a monotone f, and rejects f with high probability if it is ε-far from being monotone (i.e., every monotone function differs from f on more than an ε fraction of the domain). For any ε > 0, the query complexity of the test is O((n/ε) · log ∣Σ ∣ · log ∣Ξ∣). The previous best known bound was \(\tilde{O}((n^2/\epsilon) \cdot \vert\Sigma\vert^2 \cdot \vert\Xi\vert)\).
We also present an alternative test for the boolean range Ξ = {0,1} whose query complexity O(n 2/ε 2 ) is independent of alphabet size ∣Σ∣.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and intractability of approximation problems. JACM 45(3), 501–555 (1998)
Arora, S., Sudan, S.: Improved low degree testing and its applications. In: Proceedings of STOC 1997, pp. 485–495 (1997)
Bellare, M., Coppersmith, D., Håstad, J., Kiwi, M., Sudan, M.: Linearity testing in characteristic two. In: Proceedings of FOCS 1995, pp. 432–441 (1995)
Bellare, M., Goldwasser, S., Lund, C., Russell, A.: Efficient probabilistically checkable proofs and applications to approximation. In: Proceedings of STOC 1993, pp. 294–304 (1993)
Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. JACM 47, 549–595 (1993)
Dodis, Y., Goldreich, O., Lehman, E., Raskhodnikova, S., Ron, D., Samorodnitsky, A.: Improved testing algorithms for monotonocity (1999), Available from ECCC http://www.eccc.uni-trier.de/eccc/
Ergun, F., Kannan, S., Kumar, S.R., Rubinfeld, R., Viswanathan, M.: Spotcheckers. In: Proceedings of STOC 1998, pp. 259–268 (1998)
Gemmell, P., Lipton, R., Rubinfeld, R., Sudan, M., Wigderson, A.: Selftesting/ correcting for polynomials and for approximate functions. In: Proceedings of STOC 1991, pp. 32–42 (1991)
Goldreich, O., Goldwasser, S., Lehman, E., Ron, D.: Testing monotonicity. In: Proceedings of FOCS 1998 (1998)
Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. JACM 45(4), 653–750 (1998); An extended abstract appeared in the proceedings of FOCS 1996
Goldreich, O., Ron, D.: Property testing in bounded degree graphs. In: Proceedings of STOC 1997, pp. 406–415 (1997)
Goldreich, O., Ron, D.: A sublinear bipartite tester for bounded degree graphs. In: Proceedings of STOC 1998, pp. 289–298 (1998); To appear in Combinatorica (1999)
Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a subconstant error-probability PCP characterization of NP. In: Proceedings of STOC 1997, pp. 475–484 (1997)
Rubinfeld, R., Sudan, M.: Robust characterization of polynomials with applications to program testing. SIAM Journal on Computing 25(2), 252–271 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dodis, Y., Goldreich, O., Lehman, E., Raskhodnikova, S., Ron, D., Samorodnitsky, A. (1999). Improved Testing Algorithms for Monotonicity. In: Hochbaum, D.S., Jansen, K., Rolim, J.D.P., Sinclair, A. (eds) Randomization, Approximation, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 1999 1999. Lecture Notes in Computer Science, vol 1671. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-48413-4_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-48413-4_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66329-4
Online ISBN: 978-3-540-48413-4
eBook Packages: Springer Book Archive