Furthest Pair Requires Quadratic Time in Superconstant Dimension under SETH

2026-06-24

AI Summary

This work investigates the time complexity lower bound for the bichromatic closest pair problem in super-constant dimensions. Building upon the Strong Exponential Time Hypothesis (SETH), it extends existing lower bounds from specially constructed slowly growing dimensions to arbitrary efficiently constructible super-constant dimensions. The proof introduces a novel reduction technique inspired by the contrapositive of the Erdős unit distance conjecture. The result establishes that, for any dimension growing beyond a constant, the problem requires $n^{2 - o(1)}$ time to solve, thereby demonstrating that current algorithms are nearly optimal with respect to their dependence on dimension and unlikely to admit significant further improvement.

Abstract

Several fundamental problems in computational geometry admit algorithms with running time $f(d) \cdot n^{2-Θ(1/d)}$ for $n$ points in $d$ dimensions, making them among the most prominent examples of barely subquadratic computation. Notable members of this class include Furthest Pair, Bichromatic Closest Pair, (Bichromatic) Maximum Innter Product, and Hopcroft's Problem. Chen [Theory Comput. 2020] proved that, assuming the Strong Exponential Time Hypothesis (SETH), these problems require $n^{2-o(1)}$ time when the dimension satisfies $d=2^{Θ(\log^* n)}$. We extend this lower bound to all efficiently constructible dimensions $d=ω(1)$. Thus, assuming SETH, the dependence of the best known algorithms on the dimension is essentially unavoidable. The proof utilizes techniques in OpenAI's recent disproof of the Erdos unit distance conjecture. The proof was initially discovered by ChatGPT 5.5 Pro. The authors have validated and substantially edited the proof to improve the presentation.

Problem

• Furthest Pair
• Computational Geometry
• SETH
• Quadratic Time
• Superconstant Dimension

Innovation

• SETH
• Furthest Pair
• computational geometry
• quadratic lower bound
• superconstant dimension