Abstract
We prove the existence of linear size binary space partitions for sets of objects in the plane under certain conditions that are often satisfied in practical situations. In particular, we construct linear size binary space partitions for sets of fat objects, for sets of line segments where the ratio between the lengths of the longest and shortest segment is bounded by a constant, and for homothetic objects. For all cases we also show how to turn the existence proofs into efficient algorithms.
Supported by the Dutch Organisation for Scientific Research (N.W.O.) and by ESPRIT Basic Research Action No. 7141 (project ALCOM II: Algorithms and Complexity)
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
C. Ballieux. Motion planning using binary space partition. Report inf/src/93-25, Utrecht University, 1993.
B. Chazelle. personal communication, 1993.
N. Chin and S. Feiner. Near real time shadow generation using bsp trees. In SIGGRAPH'90, pages 99–106, 1990.
F. d'Amore and P. G. Franciosa. On the optimal binary plane partition for sets of isothetic rectangles. In Proc. 4th Canad. Conf. Comput. Geom., pages 1–5, 1992.
M. de Berg. Efficient algorithms for ray shooting and hidden surface removal. Ph.D. dissertation, Dept. Comput. Sci., Univ. Utrecht, Utrecht, Netherlands, 1992.
H. Edelsbrunner. Computing the extreme distances between two convex polygons. J. Algorithms, 6:213–224, 1985.
H. Fuchs, Z. M. Kedem, and B. Naylor. On visible surface generation by a priori tree structures. Comput. Graph., 14(3):124–133, 1980.
B. Naylor, J. Amanatides, and W. Thibault. Merging bsp trees yields polyhedral set operations. In SIGGRAPH'90, pages 115–124, 1990.
M. H. Overmars. Range searching in a set of line segments. In Proc. 1st Annu. ACM Sympos. Comput. Geom., pages 177–185, 1985.
M. S. Paterson and F. F. Yao. Efficient binary space partitions for hidden-surface removal and solid modeling. Discrete Comput. Geom., 5:485–503, 1990.
M. S. Paterson and F. F. Yao. Optimal binary space partitions for orthogonal objects. J. Algorithms, 13:99–113, 1992.
F. P. Preparata. A new approach to planar point location. SIAM J. Comput., 10:473–482, 1981.
F. P. Preparata and M. I. Shamos. Computational Geometry: an Introduction. Springer-Verlag, New York, NY, 1985.
W. C. Thibault and B. F. Naylor. Set operations on polyhedra using binary space partitioning trees. In Proc. SIGGRAPH'87, pages 153–162, 1987.
A. van der Stappen, D. Halperin, and M. Overmars. The complexity of the free space for a robot moving amidst fat obstacles. Computational Geometry: Theory and Applications, 3:353–373, 1993.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
de Berg, M., de Groot, M., Overmars, M. (1994). New results on binary space partitions in the plane (extended abstract). In: Schmidt, E.M., Skyum, S. (eds) Algorithm Theory — SWAT '94. SWAT 1994. Lecture Notes in Computer Science, vol 824. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58218-5_6
Download citation
DOI: https://doi.org/10.1007/3-540-58218-5_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58218-2
Online ISBN: 978-3-540-48577-3
eBook Packages: Springer Book Archive