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Linear-Size Boolean Circuits for Multiselection

Authors Justin Holmgren , Ron Rothblum

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Author Details

Justin Holmgren
  • NTT Research, Sunnyvale, CA, USA
Ron Rothblum
  • Technion, Haifa, Israel

Funding

  • Rothblum, Ron: Ron Rothblum is funded by the European Union (ERC, FASTPROOF, 101041208). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

Acknowledgements

We thank Yuval Ishai for useful discussions and his encouragement and an anonymous reviewer for useful comments.

Cite As Get BibTex

Justin Holmgren and Ron Rothblum. Linear-Size Boolean Circuits for Multiselection. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.CCC.2024.11

Abstract

We study the circuit complexity of the multiselection problem: given an input string x ∈ {0,1}ⁿ along with indices i_1,… ,i_q ∈ [n], output (x_{i_1},… ,x_{i_q}). A trivial lower bound for the circuit size is the input length n + q⋅log(n), but the straightforward construction has size Θ(q⋅n).
Our main result is an O(n+q⋅log³(n))-size and O(log(n+q))-depth circuit for multiselection. In particular, for any q ≤ n/log³(n) the circuit has linear size and logarithmic depth. Prior to our work no linear-size circuit for multiselection was known for any q = ω(1) and regardless of depth.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Private Information Retrieval
  • Batch Selection
  • Boolean Circuits

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