max planck institut
informatik

# MPI-I-91-107

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

### Schwarz, Christian and Smid, Michiel

MPI-I-91-107. July 1991, 21 pages. | Status: available - back from printing | Next --> Entry | Previous <-- Entry

Abstract in LaTeX format:
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.
Acknowledgement:
References to related material:

12035 KBytes
Please note: If you don't have a viewer for PostScript on your platform, try to install GhostScript and GhostView
URL to this document: http://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1991-107
BibTeX
@TECHREPORT{SchwarzSmid91,
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},