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 -