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.
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Min, K., Kao, MY., Zhu, H. (2009). The Closest Pair Problem under the Hamming Metric. In: Ngo, H.Q. (eds) Computing and Combinatorics. COCOON 2009. Lecture Notes in Computer Science, vol 5609. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02882-3_21
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DOI: https://doi.org/10.1007/978-3-642-02882-3_21
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