A Composition Theorem via Conflict Complexity

Swagato Sanyal

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arXiv · 1801.03285

A Composition Theorem via Conflict Complexity

Swagato Sanyal

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Abstract

Let $\R(\cdot)$ stand for the bounded-error randomized query complexity. We show that for any relation $f \subseteq \{0,1\}^n \times \mathcal{S}$ and partial Boolean function $g \subseteq \{0,1\}^n \times \{0,1\}$, $\R_{1/3}(f \circ g^n) = \Omega(\R_{4/9}(f) \cdot \sqrt{\R_{1/3}(g)})$. Independently of us, Gavinsky, Lee and Santha \cite{newcomp} proved this result. By an example demonstrated in their work, this bound is optimal. We prove our result by introducing a novel complexity measure called the \emph{conflict complexity} of a partial Boolean function $g$, denoted by $\chi(g)$, which may be of independent interest. We show that $\chi(g) = \Omega(\sqrt{\R(g)})$ and $\R(f \circ g^n) = \Omega(\R(f) \cdot \chi(g))$.

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Swagato Sanyal. 2018-01-10. A Composition Theorem via Conflict Complexity. https://arxiv.org/abs/1801.03285

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