Optimal Separation and Strong Direct Sum for Randomized Query Complexity
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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 →{0,1} whose ϵ-error randomized query complexity satisfies R_ϵ(f) = Ω( R(f) ·log1/ϵ). * Strong direct sum theorem. For every function f and every k > 2, the randomized query complexity of computing k instances of f simultaneously satisfies R_ϵ(f^k) = Θ(k ·R_ϵ/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 R(f^k) = Θ( k log k ·R(f)). This answers an open question of Drucker (2012). Combining this result with the query-to-communication complexity lifting theorem of Göös, Pitassi, and Watson (2017), this also shows that there is a total function whose public-coin randomized communication complexity satisfies R^cc (f^k) = Θ( k log k ·R^cc(f)), answering a question of Feder, Kushilevitz, Naor, and Nisan (1995).
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