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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2020.9

The Power of Many Samples in Query Complexity

Authors: Andrew Bassilakis, Andrew Drucker, Mika Göös, Lunjia Hu, Weiyun Ma, and Li-Yang Tan

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract

The randomized query complexity 𝖱(f) of a boolean function f: {0,1}ⁿ → {0,1} is famously characterized (via Yao’s minimax) by the least number of queries needed to distinguish a distribution 𝒟₀ over 0-inputs from a distribution 𝒟₁ over 1-inputs, maximized over all pairs (𝒟₀,𝒟₁). We ask: Does this task become easier if we allow query access to infinitely many samples from either 𝒟₀ or 𝒟₁? We show the answer is no: There exists a hard pair (𝒟₀,𝒟₁) such that distinguishing 𝒟₀^∞ from 𝒟₁^∞ requires Θ(𝖱(f)) many queries. As an application, we show that for any composed function f∘g we have 𝖱(f∘g) ≥ Ω(fbs(f)𝖱(g)) where fbs denotes fractional block sensitivity.

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Andrew Bassilakis, Andrew Drucker, Mika Göös, Lunjia Hu, Weiyun Ma, and Li-Yang Tan. The Power of Many Samples in Query Complexity. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bassilakis_et_al:LIPIcs.ICALP.2020.9,
  author =	{Bassilakis, Andrew and Drucker, Andrew and G\"{o}\"{o}s, Mika and Hu, Lunjia and Ma, Weiyun and Tan, Li-Yang},
  title =	{{The Power of Many Samples in Query Complexity}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{9:1--9:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.9},
  URN =		{urn:nbn:de:0030-drops-124163},
  doi =		{10.4230/LIPIcs.ICALP.2020.9},
  annote =	{Keywords: Query complexity, Composition theorems}
}

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The Power of Many Samples in Query Complexity

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