Settling the query complexity of non-adaptive junta testing

Xi Chen; Rocco A. Servedio; Li-Yang Tan; Erik Waingarten; Jinyu Xie

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arXiv · 1704.06314

Settling the query complexity of non-adaptive junta testing

Xi Chen·Rocco A. Servedio·Li-Yang Tan·Erik Waingarten·Jinyu Xie

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Abstract

We prove that any non-adaptive algorithm that tests whether an unknown Boolean function $f: \{0, 1\}^n\to \{0, 1\}$ is a $k$-junta or $\epsilon$-far from every $k$-junta must make $\widetilde{\Omega}(k^{3/2} / \epsilon)$ many queries for a wide range of parameters $k$ and $\epsilon$. Our result dramatically improves previous lower bounds from [BGSMdW13, STW15], and is essentially optimal given Blais's non-adaptive junta tester from [Blais08], which makes $\widetilde{O}(k^{3/2})/\epsilon$ queries. Combined with the adaptive tester of [Blais09] which makes $O(k\log k + k /\epsilon)$ queries, our result shows that adaptivity enables polynomial savings in query complexity for junta testing.

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BibTeX RIS

Xi Chen, Rocco A. Servedio, Li-Yang Tan, Erik Waingarten, Jinyu Xie. 2017-04-20. Settling the query complexity of non-adaptive junta testing. https://arxiv.org/abs/1704.06314

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