On Closest Pair in Euclidean Metric: Monochromatic is as Hard as Bichromatic
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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( n) was raised as an open question in recent works (Abboud-Rubinstein-Williams [FOCS'17], Williams [SODA'18], David-Karthik-Laekhanukit [SoCG'18]). 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=( 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 n dimensions in the ℓ_p-metric. 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 ( 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-Miccancio-Sudan [IEEE Trans. Inf. Theory'03].
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