Exponentially improved algorithms and lower bounds for testing signed majorities

Dana Ron*, Rocco A. Servedio

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


A signed majority function is a linear threshold function f : {+1, -1} n → {+1, -1} of the form f(x) = sign(∑i=1 n σix¡) where each σi ∈ {+1, -1}. Signed majority functions are a highly symmetrical subclass of the class of all linear threshold functions, which are functions of the form sign(∑i=1n WiX i - θ) for arbitrary real wi, θ. We study the query complexity of testing whether an unknown f : {+1, -1}n → {+1, -1} is a signed majority function versus ε-far from every signed majority function. While it is known [26] that the broader class of all linear threshold functions is testable with poly(1/ε) queries (independent of n), prior to our work the best upper bound for signed majority functions was O(√n)·poly(1/ε) queries (via a non-adaptive algorithm), and the best lower bound was Ω(log n) queries for non-adaptive algorithms [27]. As our main results we exponentially improve both these prior bounds for testing signed majority functions: • (Upper bound) We give a poly(log n, 1/ε)-query adaptive algorithm (which is computationally efficient) for this testing problem; • (Lower bound) We show that any non-adaptive algorithm for testing the class of signed majorities to constant accuracy must make n Ω(1) queries. This directly implies a lower bound of Ω(log n) queries for any adaptive algorithm. Our testing algorithm performs a sequence of restrictions together with consistency checks to ensure that each successive restriction is "compatible" with the function prior to restriction. This approach is used to transform the original n-variable testing problem into a testing problem over poly(log n, 1/ε) variables where a simple direct method can be applied. Analysis of the degree-1 Fourier coefficients plays an important role in our proofs.

Original languageEnglish
Title of host publicationProceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013
PublisherAssociation for Computing Machinery
Number of pages18
ISBN (Print)9781611972511
StatePublished - 2013
Event24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 - New Orleans, LA, United States
Duration: 6 Jan 20138 Jan 2013

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013
Country/TerritoryUnited States
CityNew Orleans, LA


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