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.
@InProceedings{holmgren_et_al:LIPIcs.CCC.2024.11, author = {Holmgren, Justin and Rothblum, Ron}, title = {{Linear-Size Boolean Circuits for Multiselection}}, booktitle = {39th Computational Complexity Conference (CCC 2024)}, pages = {11:1--11:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-331-7}, ISSN = {1868-8969}, year = {2024}, volume = {300}, editor = {Santhanam, Rahul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.11}, URN = {urn:nbn:de:0030-drops-204070}, doi = {10.4230/LIPIcs.CCC.2024.11}, annote = {Keywords: Private Information Retrieval, Batch Selection, Boolean Circuits} }
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