TY - JOUR

T1 - Subquadratic algorithms for some 3SUM-hard geometric problems in the algebraic decision-tree model

AU - Aronov, Boris

AU - de Berg, Mark

AU - Cardinal, Jean

AU - Ezra, Esther

AU - Iacono, John

AU - Sharir, Micha

N1 - Funding Information:
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.
Publisher Copyright:
© 2022 The Author(s)

PY - 2023/2

Y1 - 2023/2

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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U2 - 10.1016/j.comgeo.2022.101945

DO - 10.1016/j.comgeo.2022.101945

M3 - Article

AN - SCOPUS:85138450009

VL - 109

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

M1 - 101945

ER -