Improved testing algorithms for monotonicity

Yevgeniy Dodis; Oded Goldreich; Eric Lehman; Sofya Raskhodnikova; Dana Ron; Alex Samorodnitsky

doi:10.1007/978-3-540-48413-4_10

Improved testing algorithms for monotonicity

Yevgeniy Dodis, Oded Goldreich, Eric Lehman, Sofya Raskhodnikova, Dana Ron, Alex Samorodnitsky

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

136 Scopus citations

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 _ 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 Õ ((n²/ϵ) ·|Σ|). We also present an alternative test for the boolean range Ξ = {0,1} whose query complexity O(n²/ϵ²) is independent of alphabet size |Σ|.

Original language	English
Title of host publication	Randomization, Approximation, and Combinatorial Optimization
Subtitle of host publication	Algorithms and Techniques - 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999, Proceedings
Editors	Jose D. P. Rolim, Alistair Sinclair, Dorit Hochbaum, Klaus Jansen
Publisher	Springer Verlag
Pages	97-108
Number of pages	12
ISBN (Print)	3540663290, 9783540663294
DOIs	https://doi.org/10.1007/978-3-540-48413-4_10
State	Published - 1999
Externally published	Yes
Event	3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999 - Berkeley, United States Duration: 8 Aug 1999 → 11 Aug 1999

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	1671
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999
Country/Territory	United States
City	Berkeley
Period	8/08/99 → 11/08/99

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1999.

Access to Document

10.1007/978-3-540-48413-4_10

Cite this

Dodis, Y., Goldreich, O., Lehman, E., Raskhodnikova, S., Ron, D., & Samorodnitsky, A. (1999). Improved testing algorithms for monotonicity. In J. D. P. Rolim, A. Sinclair, D. Hochbaum, & K. Jansen (Eds.), Randomization, Approximation, and Combinatorial Optimization: Algorithms and Techniques - 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999, Proceedings (pp. 97-108). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1671). Springer Verlag. https://doi.org/10.1007/978-3-540-48413-4_10

Dodis, Yevgeniy ; Goldreich, Oded ; Lehman, Eric et al. / Improved testing algorithms for monotonicity. Randomization, Approximation, and Combinatorial Optimization: Algorithms and Techniques - 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999, Proceedings. editor / Jose D. P. Rolim ; Alistair Sinclair ; Dorit Hochbaum ; Klaus Jansen. Springer Verlag, 1999. pp. 97-108 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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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 _ 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 {\~O} ((n2/ϵ) ·|Σ|). We also present an alternative test for the boolean range Ξ = {0,1} whose query complexity O(n2/ϵ2) is independent of alphabet size |Σ|.",

author = "Yevgeniy Dodis and Oded Goldreich and Eric Lehman and Sofya Raskhodnikova and Dana Ron and Alex Samorodnitsky",

note = "Publisher Copyright: {\textcopyright} Springer-Verlag Berlin Heidelberg 1999.; 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999 ; Conference date: 08-08-1999 Through 11-08-1999",

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doi = "10.1007/978-3-540-48413-4_10",

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isbn = "3540663290",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

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Dodis, Y, Goldreich, O, Lehman, E, Raskhodnikova, S, Ron, D & Samorodnitsky, A 1999, Improved testing algorithms for monotonicity. in JDP Rolim, A Sinclair, D Hochbaum & K Jansen (eds), Randomization, Approximation, and Combinatorial Optimization: Algorithms and Techniques - 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1671, Springer Verlag, pp. 97-108, 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999, Berkeley, United States, 8/08/99. https://doi.org/10.1007/978-3-540-48413-4_10

Improved testing algorithms for monotonicity. / Dodis, Yevgeniy; Goldreich, Oded; Lehman, Eric et al.
Randomization, Approximation, and Combinatorial Optimization: Algorithms and Techniques - 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999, Proceedings. ed. / Jose D. P. Rolim; Alistair Sinclair; Dorit Hochbaum; Klaus Jansen. Springer Verlag, 1999. p. 97-108 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1671).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Goldreich, Oded

AU - Lehman, Eric

AU - Raskhodnikova, Sofya

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N2 - 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 _ 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 Õ ((n2/ϵ) ·|Σ|). We also present an alternative test for the boolean range Ξ = {0,1} whose query complexity O(n2/ϵ2) is independent of alphabet size |Σ|.

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Dodis Y, Goldreich O, Lehman E, Raskhodnikova S, Ron D, Samorodnitsky A. Improved testing algorithms for monotonicity. In Rolim JDP, Sinclair A, Hochbaum D, Jansen K, editors, Randomization, Approximation, and Combinatorial Optimization: Algorithms and Techniques - 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999, Proceedings. Springer Verlag. 1999. p. 97-108. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-540-48413-4_10

Improved testing algorithms for monotonicity

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