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

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 A is an element of C, the number of intersection points between the segments of A and those of B that lie in Delta. We present solutions in the algebraic decision-tree model whose cost is O (n60/31+8),for any e > 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.(c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under theCC BY license (http://creativecommons.org/licenses/by/4.0/).