On Closest Pair in Euclidean Metric: Monochromatic is as Hard as Bichromatic

Abstract

Given a set of n points in ℝ^d, the (monochromatic) Closest Pair problem asks to find a pair of distinct points in the set that are closest in the ℓ_p-metric. Closest Pair is a fundamental problem in Computational Geometry and understanding its fine-grained complexity in the Euclidean metric when d = ω(log n) was raised as an open question in recent works (Abboud-Rubinstein-Williams [6], Williams [48], David-Karthik-Laekhanukit [17]).

In this paper, we show that for every p ∈ ℝ_≥1 ∪ {0}, under the Strong Exponential Time Hypothesis (SETH), for every ε > 0, the following holds:

• No algorithm running in time O(n^2−ε) can solve the Closest Pair problem in $d = {\left({\log n} \right)^{{\Omega _\varepsilon}\left(1 \right)}}$ dimensions in the ℓ_p-metric.

• There exists δ = δ(ε) > 0 and c = c(ε) ≥ 1 such that no algorithm running in time O(n^1.5−ε) can approximate Closest Pair problem to a factor of (1 + δ) in d ≥ c log n dimensions in the ℓ_p-metric.

In particular, our first result is shown by establishing the computational equivalence of the bichromatic Closest Pair problem and the (monochromatic) Closest Pair problem (up to n^ε factor in the running time) for $d = {\left({\log n} \right)^{{\Omega _\varepsilon}\left(1 \right)}}$ dimensions.

Additionally, under SETH, we rule out nearly-polynomial factor approximation algorithms running in subquadratic time for the (monochromatic) Maximum Inner Product problem where we are given a set of n points in n^o(1)-dimensional Euclidean space and are required to find a pair of distinct points in the set that maximize the inner product.

At the heart of all our proofs is the construction of a dense bipartite graph with low contact dimension, i.e., we construct a balanced bipartite graph on n vertices with n^2−ε edges whose vertices can be realized as points in a ${\left({\log n} \right)^{{\Omega _\varepsilon}\left(1 \right)}}$-dimensional Euclidean space such that every pair of vertices which have an edge in the graph are at distance exactly 1 and every other pair of vertices are at distance greater than 1. This graph construction is inspired by the construction of locally dense codes introduced by Dumer-Micciancio-Sudan [18].