Testing monotonicity

Oded Goldreich; Shafi Goldwasser; Eric Lehman; Dana Ron; Alex Samorodnitsky

doi:10.1007/s004930070011

Testing monotonicity

Oded Goldreich, Shafi Goldwasser, Eric Lehman, Dana Ron, Alex Samorodnitsky

School of Electrical Engineering

Research output: Contribution to journal › Article › peer-review

Abstract

We present a (randomized) test for monotonicity of Boolean functions. Namely, given the ability to query an unknown function f : (0,1)ⁿ → (0,1) at arguments of its choice, the test always accepts a monotone f, and rejects f with high probability if it is ∈-far from being monotone (i.e., every monotone function differs from f on more than an ∈ fraction of the domain). The complexity of the test is O(n/∈). The analysis of our algorithm relates two natural combinatorial quantities that can be measured with respect to a Boolean function; one being global to the function and the other being local to it. A key ingredient is the use of a switching (or sorting) operator on functions.

Original language	English
Pages (from-to)	301-337
Number of pages	37
Journal	Combinatorica
Volume	20
Issue number	3
DOIs	https://doi.org/10.1007/s004930070011
State	Published - 2000

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abstract = "We present a (randomized) test for monotonicity of Boolean functions. Namely, given the ability to query an unknown function f : (0,1)n → (0,1) at arguments of its choice, the test always accepts a monotone f, and rejects f with high probability if it is ∈-far from being monotone (i.e., every monotone function differs from f on more than an ∈ fraction of the domain). The complexity of the test is O(n/∈). The analysis of our algorithm relates two natural combinatorial quantities that can be measured with respect to a Boolean function; one being global to the function and the other being local to it. A key ingredient is the use of a switching (or sorting) operator on functions.",

author = "Oded Goldreich and Shafi Goldwasser and Eric Lehman and Dana Ron and Alex Samorodnitsky",

note = "Funding Information: * A preliminary (and weaker) version of th is work appeared in [25] † Work done wh ile visiting LCS, MIT. ‡Supported in part by DARPA grant DABT63-96-C-0018 an in part by a Guastella fellowsh ip. § Th is work was done wh ile visiting LCS, MIT, and was supported by an ONR Science Sch olar Fellowsh ip at th e Bunting Institute.",

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AU - Ron, Dana

AU - Samorodnitsky, Alex

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Testing monotonicity

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