Counting short vector pairs by inner product and relations to the permanent

Andreas Björklund, Petteri Kaski

Research output: Chapter in Book/Report/Conference proceedingConference contributionScientificpeer-review

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Given as input two n-element sets A, B ⊆ {0, 1}d with d = c log n ≤ (log n)2/(log log n)4 and a target t ∈ {0, 1,⋯, d}, we show how to count the number of pairs (x, y) ∈ A × B with integer inner product 〉x, y〈 = t deterministically, in n2/2Ω(√log n log log n/(c log2 c)) time. This demonstrates that one can solve this problem in deterministic subquadratic time almost up to log2 n dimensions, nearly matching the dimension bound of a subquadratic randomized detection algorithm of Alman and Williams [FOCS 2015]. We also show how to modify their randomized algorithm to count the pairs w.h.p., to obtain a fast randomized algorithm. Our deterministic algorithm builds on a novel technique of reconstructing a function from sum-aggregates by prime residues, or modular tomography, which can be seen as an additive analog of the Chinese Remainder Theorem. As our second contribution, we relate the fine-grained complexity of the task of counting of vector pairs by inner product to the task of computing a zero-one matrix permanent over the integers.

Original languageEnglish
Title of host publication48th International Colloquium on Automata, Languages, and Programming, ICALP 2021
EditorsNikhil Bansal, Emanuela Merelli, James Worrell
PublisherSchloss Dagstuhl-Leibniz-Zentrum für Informatik
Number of pages21
ISBN (Electronic)9783959771955
Publication statusPublished - 1 Jul 2021
MoE publication typeA4 Article in a conference publication
EventInternational Colloquium on Automata, Languages and Programming - Virtual, Online, Glasgow, United Kingdom
Duration: 12 Jul 202116 Jul 2021
Conference number: 48

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
PublisherDagstuhl Publishing
ISSN (Print)1868-8969


ConferenceInternational Colloquium on Automata, Languages and Programming
Abbreviated titleICALP
Country/TerritoryUnited Kingdom


  • Additive reconstruction
  • Chinese Remainder Theorem
  • Counting
  • Inner product
  • Modular tomography
  • Orthogonal vectors
  • Permanent


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