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

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 $$d = {\left({\log n} \right)^{{\Omega _\varepsilon}\left(1 \right)}}$$ d = ( log n ) Ω ε ( 1 ) 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 $$d = {\left({\log n} \right)^{{\Omega _\varepsilon}\left(1 \right)}}$$ d = ( log n ) Ω ε ( 1 ) 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 $${\left({\log n} \right)^{{\Omega _\varepsilon}\left(1 \right)}}$$ ( log n ) Ω ε ( 1 ) -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].