The Curse of Medium Dimension for Geometric Problems in Almost Every Norm.

Given a point-set, finding the Closest Pair of points in the set, determining its Diameter, and computing a Euclidean Minimum Spanning Tree are amongst the most fundamental problems in Computer Science and Computational Geometry. In this paper, we study the complexity of these three problems in medium dimension, i.e., dimension $d=Theta(log N)$, and for various $ell_p$-norms. We show a reduction from SAT to these geometric problems in medium dimension, thus showing that these problems have no subquadratic-time algorithm, i.e., no $O(N^{2-varepsilon}2^{o(d)})$-time algorithm for any $varepsilonu003e0$, under the Strong Exponential-Time Hypothesis (SETH). In particular, under SETH we prove the following results: $bullet$ For every $pgeq 0$, there is no subquadratic-time algorithm for the Diameter problem, in the $ell_p$-norm and dimension $d=Theta(log N)$. $bullet$ There is no subquadratic-time algorithm for the Euclidean Minimum Spanning Tree problem in dimension $d=Theta(log N)$. $bullet$ For every $pin mathbb{R}_{u003e2}cup {infty}$, there is no subquadratic-time algorithm for the Closest Pair problem in the $ell_p$-norm and dimension $d=Theta(log N)$. Additionally, we prove a subquadratic lower bound for a generalization of the Closest Pair problem, namely the Set Closest Pair problem for all norms $ell_p$, $pinmathbb{R}_{geq 1}cup{infty}$. Many of our proofs go through a reduction from the Bichromatic Closest Pair problem; we prove that for every $pinmathbb{R}^+cup {infty}$, there is no subquadratic-time algorithm for the Bichromatic Closest Pair problem in the $ell_p$-norm and dimension $d=Theta(log N)$, extending a result by Alman and Williams [FOCS 2015]. To show certain limitations of our techniques, we also prove a Point-Set Separation lemma for the $ell_2$-norm, which might be of independent interest.