Binary Space Partitions for Fat Rectangles

We consider the practical problem of constructing binary space partitions (BSPs) for a set S of n orthogonal, nonintersecting, two-dimensional rectangles in ${\Bbb R}^3$ such that the aspect ratio of each rectangle in $S$ is at most $\alpha$, for some constant $\alpha \geq 1$. We present an $n2^{O(\...

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Bibliographic Details
Published in:SIAM journal on computing Vol. 29; no. 5; pp. 1422 - 1448
Main Authors: Agarwal, Pankaj K., Grove, Edward F., Murali, T. M., Vitter, Jeffrey Scott
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2000
Subjects:
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Artificial intelligence
Computer graphics
Computer science
Computer science; control theory; systems
Data processing. List processing. Character string processing
Exact sciences and technology
Mathematics
Memory organisation. Data processing
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis. Scientific computation
Pattern recognition. Digital image processing. Computational geometry
Sciences and techniques of general use
Software
Theoretical computing
Computational geometry
Partition
Solid modelling
Binary tree
Binary system
Computer graphics
Binary space partitioning
Aspect ratio
Decision tree
Hidden surface removal
Complexity
ISSN:0097-5397, 1095-7111
Online Access:Get full text
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Summary:We consider the practical problem of constructing binary space partitions (BSPs) for a set S of n orthogonal, nonintersecting, two-dimensional rectangles in ${\Bbb R}^3$ such that the aspect ratio of each rectangle in $S$ is at most $\alpha$, for some constant $\alpha \geq 1$. We present an $n2^{O(\sqrt{\log n})}$-time algorithm to build a binary space partition of size $n2^{O(\sqrt{\log n})}$ for $S$. We also show that if $m$ of the $n$ rectangles in $S$ have aspect ratios greater than $\alpha$, we can construct a BSP of size $n\sqrt{m}2^{O(\sqrt{\log n})}$ for $S$ in $n\sqrt{m}2^{O(\sqrt{\log n})}$ time. The constants of proportionality in the big-oh terms are linear in $\log \alpha$. We extend these results to cases in which the input contains nonorthogonal or intersecting objects.
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ISSN:0097-5397
1095-7111
DOI:10.1137/S0097539797320578