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.SoCG.2023.22
URN: urn:nbn:de:0030-drops-178723
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17872/
Cardinal, Jean ;
Sharir, Micha
Improved Algebraic Degeneracy Testing
Abstract
In the classical linear degeneracy testing problem, we are given n real numbers and a k-variate linear polynomial F, for some constant k, and have to determine whether there exist k numbers a_1,…,a_k from the set such that F(a_1,…,a_k) = 0. We consider a generalization of this problem in which F is an arbitrary constant-degree polynomial, we are given k sets of n real numbers, and have to determine whether there exists a k-tuple of numbers, one in each set, on which F vanishes. We give the first improvement over the naïve O^*(n^{k-1}) algorithm for this problem (where the O^*(⋅) notation omits subpolynomial factors).
We show that the problem can be solved in time O^*(n^{k - 2 + 4/(k+2)}) for even k and in time O^*(n^{k - 2 + (4k-8)/(k²-5)}) for odd k in the real RAM model of computation. We also prove that for k = 4, the problem can be solved in time O^*(n^2.625) in the algebraic decision tree model, and for k = 5 it can be solved in time O^*(n^3.56) in the same model, both improving on the above uniform bounds.
All our results rely on an algebraic generalization of the standard meet-in-the-middle algorithm for k-SUM, powered by recent algorithmic advances in the polynomial method for semi-algebraic range searching. In fact, our main technical result is much more broadly applicable, as it provides a general tool for detecting incidences and other interactions between points and algebraic surfaces in any dimension. In particular, it yields an efficient algorithm for a general, algebraic version of Hopcroft’s point-line incidence detection problem in any dimension.
BibTeX - Entry
@InProceedings{cardinal_et_al:LIPIcs.SoCG.2023.22,
author = {Cardinal, Jean and Sharir, Micha},
title = {{Improved Algebraic Degeneracy Testing}},
booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)},
pages = {22:1--22:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-273-0},
ISSN = {1868-8969},
year = {2023},
volume = {258},
editor = {Chambers, Erin W. and Gudmundsson, Joachim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17872},
URN = {urn:nbn:de:0030-drops-178723},
doi = {10.4230/LIPIcs.SoCG.2023.22},
annote = {Keywords: Degeneracy testing, k-SUM problem, incidence bounds, Hocroft’s problem, polynomial method, algebraic decision trees}
}
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Keywords: |
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Degeneracy testing, k-SUM problem, incidence bounds, Hocroft’s problem, polynomial method, algebraic decision trees |
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Collection: |
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39th International Symposium on Computational Geometry (SoCG 2023) |
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Issue Date: |
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2023 |
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Date of publication: |
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09.06.2023 |
