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Revision #1 to TR17-107 | 22nd June 2017 19:05

A Composition Theorem for Randomized Query complexity

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Revision #1
Authors: Anurag Anshu, Dmitry Gavinsky, Rahul Jain, Srijita Kundu, Troy Lee, Priyanka Mukhopadhyay, Miklos Santha, Swagato Sanyal
Accepted on: 22nd June 2017 19:05
Downloads: 54
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Abstract:

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 using an XOR lemma that $\R_{1/3}\left(f \circ \left(g^\oplus_{O(\log n)}\right)^n\right)=\Omega(\log n \cdot \R_{4/9}(f) \cdot \R_{1/3}(g))$, where $g^\oplus_{O(\log n)}$ is the function obtained by composing the XOR function on $O(\log n)$ bits and $g$.



Changes to previous version:

Corrected typos


Paper:

TR17-107 | 1st June 2017 17:39

A Composition Theorem for Randomized Query complexity


Abstract:

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 $\R_{1/3}\left(f \circ \left(g^\oplus_{O(\log n)}\right)^n\right)=\Omega(\log n \cdot \R_{4/9}(f) \cdot \R_{1/3}(g))$, where $g^\oplus_{O(\log n)}$ is the function obtained by composing the xor function on $O(\log n)$ bits and $g^t$.



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