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#### The rectangle enclosure and point-dominance problems revisited

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MPI-I-94-142.pdf

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##### Citation

Gupta, P., Janardan, R., Smid, M., & Dasgupta, B.(1994). *The
rectangle enclosure and point-dominance problems revisited* (MPI-I-94-142). Saarbrücken: Max-Planck-Institut für
Informatik.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0014-B525-8

##### Abstract

We consider the problem of reporting the pairwise enclosures
among a set of $n$ axes-parallel rectangles in $\IR^2$,
which is equivalent to reporting dominance pairs in a set
of $n$ points in $\IR^4$. For more than ten years, it has been
an open problem whether these problems can be solved faster than
in $O(n \log^2 n +k)$ time, where $k$ denotes the number of
reported pairs. First, we give a divide-and-conquer algorithm
that matches the $O(n)$ space and $O(n \log^2 n +k)$ time
bounds of the algorithm of Lee and Preparata,
but is simpler.
Then we give another algorithm that uses $O(n)$ space and runs
in $O(n \log n \log\log n + k \log\log n)$ time. For the
special case where the rectangles have at most $\alpha$
different aspect ratios, we give an algorithm tha