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 $\Delta \in C$, 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 $O\left(n^{60/31+\epsilon }\right)$, for any $\epsilon >0$.

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.