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An O(n log n log log n) algorithm for the on-line closes pair problem

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Schwarz,  Christian
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Smid,  Michiel
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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MPI-I-91-107.pdf
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Citation

Schwarz, C., & Smid, M.(1991). An O(n log n log log n) algorithm for the on-line closes pair problem (MPI-I-91-107). Saarbrücken: Max-Planck-Institut für Informatik.


Cite as: https://hdl.handle.net/11858/00-001M-0000-0014-7B0B-9
Abstract
Let $V$ be a set of $n$ points in $k$-dimensional space. It is shown how the closest pair in $V$ can be maintained under insertions in $O(\log n \log\log n)$ amortized time, using $O(n)$ space. Distances are measured in the $L_{t}$-metric, where $1 \leq t \leq \infty$. This gives an $O(n \log n \log\log n)$ time on-line algorithm for computing the closest pair. The algorithm is based on Bentley's logarithmic method for decomposable searching problems. It uses a non-trivial extension of fractional cascading to $k$-dimensional space. It is also shown how to extend the method to maintain the closest pair during semi-online updates. Then, the update time becomes $O((\log n)^{2})$, even in the worst case.