Question

Randomized algorithms to find closest pairs of points

Answer 1

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[1] was among the first geometric problems that 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 in a space of dimension d and selecting the minimum requires O(n2) 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.[2] In the algebraic decision treemodel of computation, the O(n log n)algorithm is optimal, by a reduction from the element uniqueness problem.



randomized algorithm is an algorithm that employs a degree of randomness as part of its logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random bits. Formally, the algorithm's performance will be a random variable determined by the random bits; thus either the running time, or the output (or both) are random variables.