Closest Pair of Points Problem

The closest pair of points problem or closest pair problem is a problem of computational geometry: given n points in metric space, find a pair of points with the smallest distance between them. The closest pair problem for points in the Euclidean plane was among the first geometric problems which were treated at the origins of the systematic study of the computational complexity of geometric algorithms.

A naive algorithm of finding distances between all pairs of points and selecting the minimum requires O(dn2) time. It turns out that the problem may be solved in O(n log n) time in a Euclidean space or Lp space of fixed dimension d. In the algebraic decision tree model of computation, the O(n log n) algorithm is optimal. The optimality follows from the observation that the element uniqueness problem (with the lower bound of Ω(n log n) for time complexity) is reducible to the closest pair problem: checking whether the minimal distance is 0 after the solving of the closest pair problem answers the question whether there are two coinciding points.

In the computational model which assumes that the floor function is computable in constant time the problem can be solved in O(n log log n) time. If we allow randomization to be used together with the floor function, the problem can be solved in O(n) time.