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Settling the Complexity of Testing Grainedness of Distributions, and Application to Uniformity Testing in the Huge Object Model

Authors Clément L. Canonne , Sayantan Sen , Joy Qiping Yang

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  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Distribution testing
  • Uniformity testing
  • Huge Object Model
  • Lower bounds

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Abstract

In this work, we study the problem of testing m-grainedness of probability distributions over an n-element universe 𝒰, or, equivalently, of whether a probability distribution is induced by a multiset S ⊆ 𝒰 of size |S| = m. Recently, Goldreich and Ron (Computational Complexity, 2023) proved that Ω(n^c) samples are necessary for testing this property, for any c < 1 and m = Θ(n). They also conjectured that Ω(m/(log m)) samples are necessary for testing this property when m = Θ(n). In this work, we positively settle this conjecture.
Using a known connection to the Distribution over Huge objects (DoHo) model introduced by Goldreich and Ron (TheoretiCS, 2023), we leverage our results to provide improved bounds for uniformity testing in the DoHo model.

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Clément L. Canonne, Sayantan Sen, and Joy Qiping Yang. Settling the Complexity of Testing Grainedness of Distributions, and Application to Uniformity Testing in the Huge Object Model. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 26:1-26:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.26

Author Details

Clément L. Canonne
  • School of Computer Science, University of Sydney, Australia
Sayantan Sen
  • Centre for Quantum Technologies, National University of Singapore, Singapore
Joy Qiping Yang
  • School of Computer Science, University of Sydney, Australia

Funding

  • Canonne, Clément L.: Supported by an ARC DECRA (DE230101329).
  • Sen, Sayantan: Supported by the National Research Foundation, Singapore and A^{*}STAR under its Quantum Engineering Programme NRF2021-QEP2-02-P05.
  • Yang, Joy Qiping: Supported by a JD Technology Research Scholarship in Artificial intelligence.

Acknowledgements

We would like to thank the anonymous reviewers of ITCS 2025 for their suggestions which improved the presentation of the paper. SS would like to thank Clément Canonne and the Theory CS group at the University of Sydney for the warm hospitality during his academic visit, where this work was initiated.

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