The Closest Pair Problem under the Hamming Metric

Finding the closest pair among a given set of points under Hamming Metric is a fundamental problem with many applications. Let n be the number of points and D the dimensionality of all points. We show that for 0 < D ≤ n 0.294, the problem, with the binary alphabet set, can be solved within time complexity $O\left(n^{2+o(1)}\right)$, whereas for n 0.294 < D ≤ n , it can be solved within time complexity $O\left(n^{1.843} D^{0.533}\right)$. We also provide an alternative approach not involving algebraic matrix multiplication, which has the time complexity $O\left(n^2D/\log^2 D\right)$ with small constant, and is effective for practical use. Moreover, for arbitrary large alphabet set, an algorithm with the time complexity $O\left(n^2\sqrt{D}\right)$ is obtained for 0 < D ≤ n 0.294, whereas the time complexity is $O\left(n^{1.921} D^{0.767}\right)$ for n 0.294 < D ≤ n . In addition, the algorithms propose in this paper provides a solution to the open problem stated by Kao et al.