When Is Amplification Necessary for Composition in Randomized Query Complexity?

Authors Shalev Ben-David, Mika Göös, Robin Kothari, Thomas Watson

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Shalev Ben-David
  • University of Waterloo, Canada
Mika Göös
  • Stanford University, CA, USA
Robin Kothari
  • Microsoft Quantum and Microsoft Research, Redmond, WA, USA
Thomas Watson
  • University of Memphis, TN, USA


We thank Badih Ghazi for interesting discussions about this work, and we thank anonymous reviewers for their comments.

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Shalev Ben-David, Mika Göös, Robin Kothari, and Thomas Watson. When Is Amplification Necessary for Composition in Randomized Query Complexity?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Suppose we have randomized decision trees for an outer function f and an inner function g. The natural approach for obtaining a randomized decision tree for the composed function (f∘ gⁿ)(x¹,…,xⁿ) = f(g(x¹),…,g(xⁿ)) involves amplifying the success probability of the decision tree for g, so that a union bound can be used to bound the error probability over all the coordinates. The amplification introduces a logarithmic factor cost overhead. We study the question: When is this log factor necessary? We show that when the outer function is parity or majority, the log factor can be necessary, even for models that are more powerful than plain randomized decision trees. Our results are related to, but qualitatively strengthen in various ways, known results about decision trees with noisy inputs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Oracles and decision trees
  • Amplification
  • composition
  • query complexity


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