# 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:

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**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},`

` ADDRESS = {Im Stadtwald, D-66123 Saarbr{\"u}cken, Germany},`

` NUMBER = {MPI-I-91-107},`

` MONTH = {July},`

` YEAR = {1991},`

` ISSN = {0946-011X},`

`}`