License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2019.29
URN: urn:nbn:de:0030-drops-108511
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10851/
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Blais, Eric ; Brody, Joshua

Optimal Separation and Strong Direct Sum for Randomized Query Complexity

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LIPIcs-CCC-2019-29.pdf (0.5 MB)

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 -> {0,1} whose epsilon-error randomized query complexity satisfies overline{R}_epsilon(f) = Omega(R(f) * log 1/epsilon).

Strong direct sum theorem. For every function f and every k >= 2, the randomized query complexity of computing k instances of f simultaneously satisfies overline{R}_epsilon(f^k) = Theta(k * overline{R}_{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 R(f^k) = Theta(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) = Theta(k log k * R^{cc}(f)), answering a question of Feder, Kushilevitz, Naor, and Nisan (1995).

BibTeX - Entry

@InProceedings{blais_et_al:LIPIcs:2019:10851,
  author =	{Eric Blais and Joshua Brody},
  title =	{{Optimal Separation and Strong Direct Sum for Randomized Query Complexity}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{29:1--29:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Amir Shpilka},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10851},
  URN =		{urn:nbn:de:0030-drops-108511},
  doi =		{10.4230/LIPIcs.CCC.2019.29},
  annote =	{Keywords: Decision trees, query complexity, communication complexity}
}

Keywords: Decision trees, query complexity, communication complexity
Collection: 34th Computational Complexity Conference (CCC 2019)
Issue Date: 2019
Date of publication: 16.07.2019

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