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A Composition Theorem for Randomized Query Complexity

Authors Anurag Anshu, Dmitry Gavinsky, Rahul Jain, Srijita Kundu, Troy Lee, Priyanka Mukhopadhyay, Miklos Santha, Swagato Sanyal

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Keywords
  • Query algorithms and complexity
  • Decision trees
  • Composition theorem
  • XOR lemma
  • Hardness amplification

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Abstract

Let the randomized query complexity of a relation for error probability epsilon be denoted by R_epsilon(). We prove that for any relation f contained in {0,1}^n times R and Boolean function g:{0,1}^m -> {0,1},  R_{1/3}(f o g^n) = Omega(R_{4/9}(f).R_{1/2-1/n^4}(g)), where f o g^n is the relation obtained by composing f and g. We also show using an XOR lemma that R_{1/3}(f o (g^{xor}_{O(log n)})^n) = Omega(log n . R_{4/9}(f) . R_{1/3}(g))$, where g^{xor}_{O(log n)} is the function obtained by composing the XOR function on O(log n) bits and g.

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Anurag Anshu, Dmitry Gavinsky, Rahul Jain, Srijita Kundu, Troy Lee, Priyanka Mukhopadhyay, Miklos Santha, and Swagato Sanyal. A Composition Theorem for Randomized Query Complexity. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 10:1-10:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.FSTTCS.2017.10

Author Details

Anurag Anshu
Dmitry Gavinsky
Rahul Jain
Srijita Kundu
Troy Lee
Priyanka Mukhopadhyay
Miklos Santha
Swagato Sanyal

References

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