Subquadratic algorithms for some 3SUM-hard geometric problems in the algebraic decision-tree model

Boris Aronov; Mark de Berg; Jean Cardinal; Esther Ezra; John Iacono; Micha Sharir

doi:10.1016/j.comgeo.2022.101945

Subquadratic algorithms for some 3SUM-hard geometric problems in the algebraic decision-tree model

Boris Aronov
, Mark de Berg
, Jean Cardinal
, Esther Ezra
, John Iacono
, Micha Sharir

Department of Computer Science and Artificial Intelligence

Polytechnic University
Eindhoven University of Technology
Université libre de Bruxelles
Tel Aviv University

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

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. 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 et al. (2021) [3]. 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.

Original language	English
Article number	101945
Journal	Computational Geometry: Theory and Applications
Volume	109
DOIs	https://doi.org/10.1016/j.comgeo.2022.101945
State	Published - Feb 2023

Bibliographical note

Publisher Copyright:
© 2022 The Author(s)

Funding

Work by B.A. was partially supported by NSF grants CCF-15-40656 and CCF-20-08551 , and by grant 2014/170 from the U.S-Israel Binational Science Foundation . Work by M.d.B. was partially supported by the Dutch Research Council (NWO) through Gravitation Grant NETWORKS (project no. 024.002.003 ). Work by J.C. was partially supported by the F.R.S.-FNRS ( Fonds National de la Recherche Scientifique ) under CDR Grant J.0146.18 . Work by E.E. was partially supported by NSF CAREER under grant CCF:AF-1553354 and by grant 824/17 from the Israel Science Foundation . Work by J.I. was partially supported by Fonds de la Recherche Scientifique FNRS under grant no. MISU F 6001 1 . Work by M.S. was partially supported by ISF grant 260/18 , by grant 1367/2016 from the German-Israeli Science Foundation ( GIF ), and by Blavatnik Research Fund in Computer Science at Tel Aviv University.

Funders	Funder number
Blavatnik Research Fund in Computer Science
German–Israeli Science Foundation
National Science Foundation	CCF-20-08551, 2014/170, CCF-15-40656
German-Israeli Foundation for Scientific Research and Development
United States-Israel Binational Science Foundation
Fonds De La Recherche Scientifique - FNRS	J.0146.18, 824/17, AF-1553354
Nederlandse Organisatie voor Wetenschappelijk Onderzoek	024.002.003
Israel Science Foundation	1367/2016, 260/18
Tel Aviv University

Keywords

3SUM-hard problems
Algebraic decision-tree model
Order type
Point location
Polynomial partitions

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10.1016/j.comgeo.2022.101945

Cite this

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title = "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. We present solutions in the algebraic decision-tree model whose cost is O(n60/31+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 et al. (2021) [3]. 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.",

keywords = "3SUM-hard problems, Algebraic decision-tree model, Order type, Point location, Polynomial partitions",

author = "Boris Aronov and \{de Berg\}, Mark and Jean Cardinal and Esther Ezra and John Iacono and Micha Sharir",

note = "Publisher Copyright: {\textcopyright} 2022 The Author(s)",

year = "2023",

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doi = "10.1016/j.comgeo.2022.101945",

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AU - Aronov, Boris

AU - de Berg, Mark

AU - Cardinal, Jean

AU - Ezra, Esther

AU - Iacono, John

AU - Sharir, Micha

PY - 2023/2

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N2 - 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. We present solutions in the algebraic decision-tree model whose cost is O(n60/31+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 et al. (2021) [3]. 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.

AB - 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. We present solutions in the algebraic decision-tree model whose cost is O(n60/31+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 et al. (2021) [3]. 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.

KW - 3SUM-hard problems

KW - Algebraic decision-tree model

KW - Order type

KW - Point location

KW - Polynomial partitions

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Subquadratic algorithms for some 3SUM-hard geometric problems in the algebraic decision-tree model

Abstract

Bibliographical note

Funding

Keywords

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