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

## Abstract

Given a set of n points in ℝd, the (monochromatic) Closest Pair problem asks to find a pair of distinct points in the set that are closest in the 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(n2−ε) can solve the Closest Pair problem in $$d = {\left({\log n} \right)^{{\Omega _\varepsilon}\left(1 \right)}}$$ dimensions in the p-metric.

• There exists δ = δ(ε) > 0 and c = c(ε) ≥ 1 such that no algorithm running in time O(n1.5−ε) can approximate Closest Pair problem to a factor of (1 + δ) in dc log n dimensions in the 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 nε 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 no(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 n2−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].

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## Acknowledgements

We are grateful to Madhu Sudan for extremely helpful and informative discussion about AG codes; in particular, Madhu pointed us to [46]. We thank Bundit Laekhanukit and Or Meir for general discussions, and the Simons Institute for their wonderful work-space. Finally, we would like to thank Lijie Chen and Orr Paradise for useful comments on the paper.

## Author information

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### Corresponding author

Correspondence to C. S. Karthik.

Full version of the paper available at http://arxiv.org/abs/1812.00901

Supported by Irit Dinur’s ERC-CoG grant 772839 and BSF grant 2014371.

Supported by NSF under Grants No. CCF 1655215 and CCF 1815434.

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Karthik, C.S., Manurangsi, P. On Closest Pair in Euclidean Metric: Monochromatic is as Hard as Bichromatic. Combinatorica 40, 539–573 (2020). https://doi.org/10.1007/s00493-019-4113-1