Subquadratic Algorithms for Algebraic Generalizations of 3SUM

TL;DR

This paper develops subquadratic algorithms for a polynomial generalization of the 3SUM problem, enabling faster solutions for related geometric problems and introducing new algebraic tools for computational geometry.

Contribution

It introduces subquadratic algorithms for the 3POL problem, a polynomial generalization of 3SUM, and extends key geometric tools to polynomial settings.

Findings

Existence of bounded-degree algebraic decision trees of depth O(n^{12/7+ε}) for 3POL.

3POL can be solved in O(n^2 (log log n)^{3/2} / (log n)^{1/2}) time in the real-RAM model.

Subquadratic algorithms for GPT when points lie on o((log n)^{1/6}/(log log n)^{1/2}) polynomial curves.

Abstract

The 3SUM problem asks if an input $n$-set of real numbers contains a triple whose sum is zero. We consider the 3POL problem, a natural generalization of 3SUM where we replace the sum function by a constant-degree polynomial in three variables. The motivations are threefold. Raz, Sharir, and de Zeeuw gave a $O(n^{11/6})$ upper bound on the number of solutions of trivariate polynomial equations when the solutions are taken from the cartesian product of three $n$-sets of real numbers. We give algorithms for the corresponding problem of counting such solutions. Gr\o nlund and Pettie recently designed subquadratic algorithms for 3SUM. We generalize their results to 3POL. Finally, we shed light on the General Position Testing (GPT) problem: "Given $n$ points in the plane, do three of them lie on a line?", a key problem in computational geometry. We prove that there exist bounded-degree algebraic decision trees of depth $O(n^{\frac{12}{7}+\varepsilon})$ that solve 3POL, and that 3POL can be solved in $O(n^2 {(\log \log n)}^\frac{3}{2} / {(\log n)}^\frac{1}{2})$ time in the real-RAM model. Among the possible applications of those results, we show how to solve GPT in subquadratic time when the input points lie on $o({(\log n)}^\frac{1}{6}/{(\log \log n)}^\frac{1}{2})$ constant-degree polynomial curves. This constitutes a first step towards closing the major open question of whether GPT can be solved in subquadratic time. To obtain these results, we generalize important tools --- such as batch range searching and dominance reporting --- to a polynomial setting. We expect these new tools to be useful in other applications.