Optimal Separation and Strong Direct Sum for Randomized Query Complexity

Abstract: We establish two results regarding the query complexity of bounded-error randomized algorithms. * Bounded-error separation theorem. There exists a total function $f : {0,1}ⁿ \to {0,1}$ whose $\epsilon$-error randomized query complexity satisfies $$\overline{\mathrm{R}}<em>\epsilon(f) = \Omega( \mathrm{R}(f) \cdot \log\frac1\epsilon)$$. * Strong direct sum theorem. For every function $f$ and every $k \ge 2$, the randomized query complexity of computing $k$ instances of $f$ simultaneously satisfies $$\overline{\mathrm{R}}</em>\epsilon(f^k) = \Theta(k \cdot \overline{\mathrm{R}}_{\frac\epsilon k}(f))$$. As a consequence of our two main results, we obtain an optimal superlinear direct-sum-type theorem for randomized query complexity: there exists a function $f$ for which $$\mathrm{R}(f^k) = \Theta( k \log k \cdot \mathrm{R}(f))$$. This answers an open question of Drucker (2012). Combining this result with the query-to-communication complexity lifting theorem of G\"o\"os, Pitassi, and Watson (2017), this also shows that there is a total function whose public-coin randomized communication complexity satisfies $$\mathrm{R}^{{\mathrm{cc}}} (f^k) = \Theta( k \log k \cdot \mathrm{R}<sup>{\mathrm{cc}}(f))$$, answering a question of Feder, Kushilevitz, Naor, and Nisan (1995).