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Nearly Tight Bounds on the Block Number of Boolean Functions in Terms of Sensitivity

Authors Sourav Chakraborty , Anna Gál

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LIPIcs.MFCS.2026.61.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Boolean functions
  • sensitivity
  • block sensitivity
  • block number

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Abstract

This paper explores the previously studied measure called block number of Boolean functions, that counts the maximum possible number of minimal sensitive blocks for any input. We present close to tight upper bounds on the block number in terms of the function’s sensitivity and the allowed block size, improving previous bounds by a quadratic factor. Moreover, our bound on the block number yields sharper upper bounds on DNF size and decision tree size. For some functions, our upper bounds on decision tree size and DNF size are exponentially smaller than those obtained by previous methods. We obtain these results by introducing and estimating a novel measure called brick number, which not only upper bounds the block number but also leads to a new characterization of block sensitivity.

Cite As Get BibTex

Sourav Chakraborty and Anna Gál. Nearly Tight Bounds on the Block Number of Boolean Functions in Terms of Sensitivity. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 61:1-61:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.61

Author Details

Sourav Chakraborty
  • Indian Statistical Institute, Kolkata, India
Anna Gál
  • University of Texas at Austin, Austin, TX, USA

Funding

  • Chakraborty, Sourav: The first author thanks the DST-NSF project (TPN-104427) for funding part of this work.

Acknowledgements

We thank the anonymous referees for helpful comments. We also thank Schloss Dagstuhl for its support and hospitality. Many of the discussions that led to the results in this work took place during various Dagstuhl Seminars.

References

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