On the Difference Between Closest, Furthest, and Orthogonal Pairs: Nearly-Linear vs Barely-Subquadratic Complexity in Computational Geometry

TL;DR

This paper explores the computational complexity differences between closest and furthest point problems in high-dimensional spaces, establishing new lower bounds and reductions that suggest certain problems cannot be solved faster than near-quadratic time under common complexity assumptions.

Contribution

It introduces a novel self-reduction from the Orthogonal Vectors problem to higher dimensions, proving lower bounds for several geometric problems based on the Orthogonal Vectors Conjecture and ETH.

Findings

Barely-subquadratic problems like Euclidean diameter are hard in high dimensions.

Nearly-linear algorithms exist for closest pair in fixed dimensions.

Furthest pair problems require near-quadratic time in high dimensions.

Abstract

Point location problems for $n$ points in $d$-dimensional Euclidean space (and $\ell_p$ spaces more generally) have typically had two kinds of running-time solutions: * (Nearly-Linear) less than $d^{poly(d)} \cdot n \log^{O(d)} n$ time, or * (Barely-Subquadratic) $f(d) \cdot n^{2-1/\Theta(d)}$ time, for various $f$. For small $d$ and large $n$, "nearly-linear" running times are generally feasible, while "barely-subquadratic" times are generally infeasible. For example, in the Euclidean metric, finding a Closest Pair among $n$ points in ${\mathbb R}^d$ is nearly-linear, solvable in $2^{O(d)} \cdot n \log^{O(1)} n$ time, while known algorithms for Furthest Pair (the diameter of the point set) are only barely-subquadratic, requiring $\Omega(n^{2-1/\Theta(d)})$ time. Why do these proximity problems have such different time complexities? Is there a barrier to obtaining nearly-linear algorithms for problems which are currently only barely-subquadratic? We give a novel exact and deterministic self-reduction for the Orthogonal Vectors problem on $n$ vectors in $\{0,1\}^d$ to $n$ vectors in ${\mathbb Z}^{\omega(\log d)}$ that runs in $2^{o(d)}$ time. As a consequence, barely-subquadratic problems such as Euclidean diameter, Euclidean bichromatic closest pair, ray shooting, and incidence detection do not have $O(n^{2-\epsilon})$ time algorithms (in Turing models of computation) for dimensionality $d = \omega(\log \log n)^2$, unless the popular Orthogonal Vectors Conjecture and the Strong Exponential Time Hypothesis are false. That is, while poly-log-log-dimensional Closest Pair is in $n^{1+o(1)}$ time, the analogous case of Furthest Pair can encode larger-dimensional problems conjectured to require $n^{2-o(1)}$ time. We also show that the All-Nearest Neighbors problem in $\omega(\log n)$ dimensions requires $n^{2-o(1)}$ time to solve, assuming either of the above conjectures.