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Problem 14: Binary Space Partition Size
- Is it possible to construct a binary space partition (BSP)
for n disjoint line segments in the plane of size
Θ(n log n)?
- Paterson and Yao [PY90].
- Solved by Csaba Tóth [Tót09].
- Partial and Related Results
- The upper bound of
O(n log n)
was established by Paterson and Yao [PY90].
Tóth [T01] improved the
lower bound to
Ω(n log n/loglog n).
Then in 2009 he established a matching upper bound [Tót09].
His proof is constructive and leads to a deterministic algorithm
that runs in
O(n polylog n) time.
As his algorithm produces BSP trees whose height might be
linear in n, it remains open whether his complexity bound
can be achieved while achieving O(log n) height.
- data structures; combinatorial geometry
- Entry Revision History
- J. O'Rourke, 2 Aug. 2001. Nina Amenta & JOR, 6 Jan. 2011.
J. S. B. Mitchell and Joseph O'Rourke.
Computational geometry column 42.
Internat. J. Comput. Geom. Appl., 11(5):573-582, 2001.
Also in SIGACT News 32(3):63-72 (2001), Issue 120.
M. S. Paterson and F. F. Yao.
Efficient binary space partitions for hidden-surface removal and
Discrete Comput. Geom., 5:485-503, 1990.
Binary plane partitions for disjoint line segments.
In Proc. 25th Sympos. on Comput. Geom., pages 71-79, 2009.
Discrete Comput. Geom., to appear.
Csaba David Tóth.
A note on binary plane partitions.
In Proc. 17th Annu. ACM Sympos. Comput. Geom., pages 151-156,
The Open Problems Project - December 08, 2012