Partition bound is quadratically tight for product distributions
(1512.01968)Abstract
Let $f : {0,1}n \times {0,1}n \rightarrow {0,1}$ be a 2-party function. For every product distribution $\mu$ on ${0,1}n \times {0,1}n$, we show that $$\mathsf{CC}\mu_{0.49}(f) = O\left(\left(\log \mathsf{prt}{1/8}(f) \cdot \log \log \mathsf{prt}{1/8}(f)\right)2\right),$$ where $\mathsf{CC}\mu_\varepsilon(f)$ is the distributional communication complexity of $f$ with error at most $\varepsilon$ under the distribution $\mu$ and $\mathsf{prt}{1/8}(f)$ is the {\em partition bound} of $f$, as defined by Jain and Klauck [{\em Proc. 25th CCC}, 2010]. We also prove a similar bound in terms of $\mathsf{IC}{1/8}(f)$, the {\em information complexity} of $f$, namely, $$\mathsf{CC}\mu_{0.49}(f) = O\left(\left(\mathsf{IC}{1/8}(f) \cdot \log \mathsf{IC}{1/8}(f)\right)2\right).$$ The latter bound was recently and independently established by Kol [{\em Proc. 48th STOC}, 2016] using a different technique. We show a similar result for query complexity under product distributions. Let $g : {0,1}n \rightarrow {0,1}$ be a function. For every bit-wise product distribution $\mu$ on ${0,1}n$, we show that $$\mathsf{QC}\mu_{0.49}(g) = O\left(\left( \log \mathsf{qprt}{1/8}(g) \cdot \log \log\mathsf{qprt}{1/8}(g) \right)2 \right),$$ where $\mathsf{QC}\mu_{\varepsilon}(g)$ is the distributional query complexity of $f$ with error at most $\varepsilon$ under the distribution $\mu$ and $\mathsf{qprt}_{1/8}(g))$ is the {\em query partition bound} of the function $g$. Partition bounds were introduced (in both communication complexity and query complexity models) to provide LP-based lower bounds for randomized communication complexity and randomized query complexity. Our results demonstrate that these lower bounds are polynomially tight for {\em product} distributions.
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