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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2021.3
URN: urn:nbn:de:0030-drops-154363
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/15436/
Aronov, Boris ;
de Berg, Mark ;
Cardinal, Jean ;
Ezra, Esther ;
Iacono, John ;
Sharir, Micha
Subquadratic Algorithms for Some 3Sum-Hard Geometric Problems in the Algebraic Decision Tree Model
Abstract
We present subquadratic algorithms in the algebraic decision-tree model for several 3Sum-hard geometric problems, all of which can be reduced to the following question: Given two sets A, B, each consisting of n pairwise disjoint segments in the plane, and a set C of n triangles in the plane, we want to count, for each triangle Δ ∈ C, the number of intersection points between the segments of A and those of B that lie in Δ. The problems considered in this paper have been studied by Chan (2020), who gave algorithms that solve them, in the standard real-RAM model, in O((n²/log²n) log^O(1) log n) time. We present solutions in the algebraic decision-tree model whose cost is O(n^{60/31+ε}), for any ε > 0.
Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal, Aronov, Ezra, and Zahl (2020).
A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the order type of the lines, a "handicap" that turns out to be beneficial for speeding up our algorithm.
BibTeX - Entry
@InProceedings{aronov_et_al:LIPIcs.ISAAC.2021.3,
author = {Aronov, Boris and de Berg, Mark and Cardinal, Jean and Ezra, Esther and Iacono, John and Sharir, Micha},
title = {{Subquadratic Algorithms for Some 3Sum-Hard Geometric Problems in the Algebraic Decision Tree Model}},
booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
pages = {3:1--3:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-214-3},
ISSN = {1868-8969},
year = {2021},
volume = {212},
editor = {Ahn, Hee-Kap and Sadakane, Kunihiko},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/15436},
URN = {urn:nbn:de:0030-drops-154363},
doi = {10.4230/LIPIcs.ISAAC.2021.3},
annote = {Keywords: Computational geometry, Algebraic decision-tree model, Polynomial partitioning, Primal-dual range searching, Order types, Point location, Hierarchical partitions}
}
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Keywords: |
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Computational geometry, Algebraic decision-tree model, Polynomial partitioning, Primal-dual range searching, Order types, Point location, Hierarchical partitions |
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Collection: |
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32nd International Symposium on Algorithms and Computation (ISAAC 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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30.11.2021 |
