The Closest Pair Problem under the Hamming Metric

  • Authors:

  • Kerui Min;Ming-Yang Kao;Hong Zhu

  • Affiliations:

  • School of Computer Science, Fudan University, Shanghai, China;Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, United States;Software School, East China Normal University, Shanghai, China

  • Venue:
  • COCOON '09 Proceedings of the 15th Annual International Conference on Computing and Combinatorics
  • Year:
  • 2009

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Abstract

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.