A Composition Theorem for Randomized Query Complexity

@inproceedings{Anshu2017ACT,
  title={A Composition Theorem for Randomized Query Complexity},
  author={Anurag Anshu and Dmitry Gavinsky and Rahul Jain and Srijita Kundu and Troy Lee and Priyanka Mukhopadhyay and Miklos Santha and Swagato Sanyal},
  booktitle={Electronic Colloquium on Computational Complexity},
  year={2017}
}
Let the randomized query complexity of a relation for error probability $\epsilon$ be denoted by $R_\epsilon(\cdot)$. We prove that for any relation $f \subseteq \{0,1\}^n \times \mathcal{R}$ and Boolean function $g:\{0,1\}^m \rightarrow \{0,1\}$, $R_{1/3}(f\circ g^n) = \Omega(R_{4/9}(f)\cdot R_{1/2-1/n^4}(g))$, where $f \circ g^n$ is the relation obtained by composing $f$ and $g$. We also show that $R_{1/3}\left(f \circ \left(g^\oplus_{O(\log n)}\right)^n\right)=\Omega(\log n \cdot R_{4/9}(f… CONTINUE READING
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