License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2021.29
URN: urn:nbn:de:0030-drops-140988
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14098/
Björklund, Andreas ;
Kaski, Petteri
Counting Short Vector Pairs by Inner Product and Relations to the Permanent
Abstract
Given as input two n-element sets A, B ⊆ {0,1}^d with d = clog 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 n²/2^{Ω(√{log nlog log n/(clog² 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.
BibTeX - Entry
@InProceedings{bjorklund_et_al:LIPIcs.ICALP.2021.29,
author = {Bj\"{o}rklund, Andreas and Kaski, Petteri},
title = {{Counting Short Vector Pairs by Inner Product and Relations to the Permanent}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {29:1--29:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-195-5},
ISSN = {1868-8969},
year = {2021},
volume = {198},
editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14098},
URN = {urn:nbn:de:0030-drops-140988},
doi = {10.4230/LIPIcs.ICALP.2021.29},
annote = {Keywords: additive reconstruction, Chinese Remainder Theorem, counting, inner product, modular tomography, orthogonal vectors, permanent}
}
Keywords: |
|
additive reconstruction, Chinese Remainder Theorem, counting, inner product, modular tomography, orthogonal vectors, permanent |
Collection: |
|
48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) |
Issue Date: |
|
2021 |
Date of publication: |
|
02.07.2021 |