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The rectangle enclosure and point-dominance problems revisited

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http://pubman.mpdl.mpg.de/cone/persons/resource/persons44551

Gupta,  Prosenjit
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons45509

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

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MPI-I-94-142.pdf
(Any fulltext), 204KB

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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: http://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