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 - 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

UR - http://www.scopus.com/inward/record.url?scp=85138450009&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2022.101945

DO - 10.1016/j.comgeo.2022.101945

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AN - SCOPUS:85138450009

SN - 0925-7721

VL - 109

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

M1 - 101945

ER -