Optimal Separation and Strong Direct Sum for Randomized Query Complexity
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}}_\epsilon(f) = \Omega(...
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| Vydané v: | arXiv.org |
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| Hlavní autori: | , |
| Médium: | Paper |
| Jazyk: | English |
| Vydavateľské údaje: |
Ithaca
Cornell University Library, arXiv.org
02.08.2019
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| ISSN: | 2331-8422 |
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| Shrnutí: | 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 \(\overline{\mathrm{R}}_\epsilon(f) = \Omega( \mathrm{R}(f) \cdot \log\frac1\epsilon)\). * 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 \(\overline{\mathrm{R}}_\epsilon(f^k) = \Theta(k \cdot \overline{\mathrm{R}}_{\frac\epsilon k}(f))\). 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 \(\mathrm{R}^{\mathrm{cc}} (f^k) = \Theta( k \log k \cdot \mathrm{R}^{\mathrm{cc}}(f))\), answering a question of Feder, Kushilevitz, Naor, and Nisan (1995). |
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| Bibliografia: | SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50 |
| ISSN: | 2331-8422 |
| DOI: | 10.48550/arxiv.1908.01020 |