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: https://drops.dagstuhl.de/opus/volltexte/2023/17872/
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Cardinal, Jean ; Sharir, Micha

Improved Algebraic Degeneracy Testing

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LIPIcs-SoCG-2023-22.pdf (0.8 MB)


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}
}

Keywords: Degeneracy testing, k-SUM problem, incidence bounds, Hocroft’s problem, polynomial method, algebraic decision trees
Collection: 39th International Symposium on Computational Geometry (SoCG 2023)
Issue Date: 2023
Date of publication: 09.06.2023


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