MPI-INF/SWS Research Reports 1991-2021

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

Schwarz, Christian and Smid, Michiel

July 1991, 21 pages.

Status: available - back from printing

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.

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  AUTHOR = {Schwarz, Christian and Smid, Michiel},
  TITLE = {An O(n log n log log n) algorithm for the on-line closes pair problem},
  TYPE = {Research Report},
  INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},
  ADDRESS = {Im Stadtwald, D-66123 Saarbr{\"u}cken, Germany},
  NUMBER = {MPI-I-91-107},
  MONTH = {July},
  YEAR = {1991},
  ISSN = {0946-011X},