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Revision #1 to TR15-200 | 8th December 2015 22:57
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#### Almost quadratic gap between partition complexity and query/communication complexity

**Abstract:**
We show nearly quadratic separations between two pairs of complexity measures:

1. We show that there is a Boolean function $f$ with $D(f)=\Omega((D^{sc}(f))^{2-o(1)})$ where $D(f)$ is the deterministic query complexity of $f$ and $D^{sc}$ is the subcube partition complexity of $f$;

2. As a consequence, we obtain that there is $f(x, y)$ such that $D^{cc}(f)=\Omega(\log^{2-o(1)}\chi(f))$ where $D^{cc}(f)$ is the deterministic 2-party communication complexity of $f$ (in the standard 2-party model of communication) and $\chi(f)$ is the partition number of $f$.

Both of those separations are nearly optimal: it is well known that $D(f)=O((D^{sc}(f))^{2})$ and $D^{cc}(f)=O(\log^2\chi(f))$.

**Changes to previous version:**
List of authors corrected.

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TR15-200 | 4th December 2015 16:27
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#### Almost quadratic gap between partition complexity and query/communication complexity

**Abstract:**
We show nearly quadratic separations between two pairs of complexity measures:

1. We show that there is a Boolean function $f$ with $D(f)=\Omega((D^{sc}(f))^{2-o(1)})$ where $D(f)$ is the deterministic query complexity of $f$ and $D^{sc}$ is the subcube partition complexity of $f$;

2. As a consequence, we obtain that there is $f(x, y)$ such that $D^{cc}(f)=\Omega(\log^{2-o(1)}\chi(f))$ where $D^{cc}(f)$ is the deterministic 2-party communication complexity of $f$ (in the standard 2-party model of communication) and $\chi(f)$ is the partition number of $f$.

Both of those separations are nearly optimal: it is well known that $D(f)=O((D^{sc}(f))^{2})$ and $D^{cc}(f)=O(\log^2\chi(f))$.