We present several applications of a recent space-partitioning technique of Chazelle, Sharir, and Welzl (Proceedings of the 6th Annual ACM Symposium on Computational Geometry, 1990, pp. 23–33). Our results include efficient algorithms for output-sensitive hidden surface removal, for ray shooting in two and three dimensions, and for constructing spanning trees with low stabbing number.
P. K. Agarwal, Ray shooting and other applications of spanning trees with low stabbing number,SIAM J. Comput. 21 (1992), in press.
P. K. Agarwal, Partitioning arrangements of lines, I: An efficient deterministic algorithm,Discrete Comput. Geom. 5 (1990), 449–483.
P. K. Agarwal, Partitioning arrangements of lines, II: Applications,Discrete Comput. Geom. 5 (1990), 533–573.
P. K. Agarwal, M. van Kreveld, and M. Overmars, Searching and storing curved objects,Proc. 7th ACM Symp. on Computational Geometry, 1991, pp. 41–50. (Also to appear inJ. Algorithms.)
P. K. Agarwal and J. Matoušek, Ray shooting and parametric search,Proc. 24th ACM Symp. on Theory of Computing, 1992.
P. K. Agarwal and J. Matoušek, Range searching with non-linear objects, in preparation, 1991.
B. Aronov and M. Sharir, On the zone of a surface in a hyperplane arrangement,Proc. 2nd Workshop on Algorithms and Data Structures, Lecture Notes in Computer Science, Vol. 519, Springer-Verlag, Berlin, 1991, pp. 13–19.
R. Bar Yehuda and S. Fogel, Good splitters with applications to ray shooting,Proc. 2nd Canadian Conf. on Computational Geometry, 1990, pp. 81–85. (Also to appear inAlgorithmica.)
J. Bentley and J. Saxe, Decomposable searching problems, I: Static-to-dynamic transformation,J. Algorithms,1 (1980), 301–358.
M. de Berg, D. Halperin, M. Overmars, J. Snoeyink, and M. van Kreveld, Efficient ray shooting and hidden surface removal,Proc. 7th Symp. on Computational Geometry, 1991, pp. 21–30.
B. Chazelle, An optimal computing convex hull algorithm and new results on cuttings,Proc. 32nd IEEE Symp. on Foundations of Computer Science, 1991, pp. 29–38.
B. Chazelle, H. Edelsbrunner, M. Grigni, L. Guibas, J. Hershberger, M. Sharir, and J. Snoeyink Ray shooting in polygons using geodesic triangulations,Proc. 18th International Colloquium on Automata, Languages, and Programming, 1991, pp. 661–673.
B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir, A singly-exponential stratification scheme for real semi-algebraic varieties and its applications,Proc. 16th International Colloquium on Automata, Languages and Programming, 1989, pp. 179–192.
B. Chazelle, H. Edelsbrunner, L. Guibas, M. Sharir, and J. Stolfi, Lines in space: Combinatorics and algorithms, Technical Report 491, Dept. of Computer Science, New York University, February 1990.
B. Chazelle and L. Guibas, Visibility and intersection problems in plane geometry,Discrete Comput. Geom. 4 (1989), 551–589.
B. Chazelle, M. Sharir, and E. Welzl, Quasi-optimal upper bounds for simplex range searching and new zone theorems,Proc. 6th ACM Symp. on Computational Geometry, 1990, pp. 23–33.
B. Chazelle and E. Welzl, Quasi-optimal range searching in spaces of finite Vapnik-Chervonenkis dimensions,Discrete Comput. Geom. 4 (1989), 467–489.
S. W. Cheng and R. Janardan, New results on dynamic planar point location,Proc. 31st IEEE Symp. on Foundations of Computer Science, 1990, pp. 96–105.
S. W. Cheng and R. Janardan, Space efficient ray shooting and intersection searching: Algorithms, dynamization, and applications,Proc. 2nd SIAM-ACM Symp. on Discrete Algorithms, 1991, pp. 7–16.
K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, and E. Welzl, Combinatorial complexity bounds for arrangements of curves and spheres,Discrete Comput. Geom. 5 (1990), 99–160.
R. Cole and M. Sharir, Visibility problems for polyhedral terrains,J. Symbolic Comput. 7 (1989), 11–30.
D. Dobkin and H. Edelsbrunner, Space searching for intersecting objects,J. Algorithms 8 (1987), 348–361.
D. Dobkin and D. Kirkpatrick, A linear algorithm for determining the separation of convex polyhedra,J. Algorithms 6 (1985), 381–392.
D. Dobkin and D. Kirkpatrick, Determining the separation of preprocessed polyhedra: a unified approach,Proc. 17th International Colloquium on Automata, Languages, and Programming, 1990, pp. 400–413.
H. Edelsbrunner, L. Guibas, J. Hershberger, R. Seidel, M. Sharir, J. Snoeyink, and E. Welzl, Implicitly representing arrangements of lines and of segments,Discrete Comput. Geom. 4 (1989), 433–466.
L. Guibas, M. Overmars, and M. Sharir, Ray shooting, implicit point location, and related queries in arrangements of segments, Technical Report 433, Courant Institute, New York University, 1989.
J. Matoušek, More on cutting hyperplanes and spanning trees with low crossing number, Technical Report B-90-2, Freie Universität Berlin, 1990.
J. Matoušek, Spanning trees with low crossing numbers,Inform. Théoret. Applic. 25 (1991), 103–123.
J. Matoušek, Approximations and optimal geometric divide-and-conquer,Proc. 23rd ACM Symp. on Theory of Computing, 1991, 506–511.
J. Matoušek, Efficient partition trees,Proc. 7th ACM Symp. on Computational Geometry, 1991, pp. 1–9.
J. Matoušek, Range searching with efficient hierarchical cuttings,Proc. 8th ACM Symp. on Computational Geometry, 1992, to appear.
K. Mehlhorn,Data Structures and Algorithms, 3: Multi-Dimensional Searching and Computational Geometry, Springer-Verlag, Berlin, 1984.
M. Overmars and J. van Leeuwen, Maintenance of configurations in the plane,J. Comput. System Sci. 23 (1981), 166–204.
M. Overmars and J. van Leeuwen, Worst-case optimal insertion and deletion methods for decomposable searching,Inform. Process. Lett. 12 (1981), 168–173.
M. Overmars, H. Schipper, and M. Sharir, Storing line segments in partition trees,BIT 30 (1990), 385–403.
M. Overmars and M. Sharir, Output-sensitive hidden surface removal,Proc. 30th IEEE Symp. on Foundations of Computer Science, 1989, pp. 598–603.
M. Overmars and M. Sharir, An improved technique for output-sensitive hidden surface removal, Technical Report RUU-CS-89-32, Computer Science Department, University of Utrecht, December 1989. (To appear inAlgorithmica.)
M. Overmars and M. Sharir, Merging visibility maps,Comput. Geom. Theory Applic. 1 (1991), 35–50.
M. Pellegrini, Combinatorial and algorithmic analysis of stabbing and visibility problems in 3-dimensional space, Ph.D. Thesis, New York University, 1991.
A. Schmitt, H. Müller, and W. Leister, Ray tracing algorithms—theory and algorithms, inTheoretical Foundations of Computer Graphics and CAD (ed. R. Earnshaw), NATO Series, Springer-Verlag, Berlin, 1988, pp. 997–1030.
M. Sharir and M. Overmars, A simple output-sensitive hidden surface removal algorithm,ACM Trans. Graphics 11 (1992), 1–11.
D. M. H. Sommerville,Analytical Geometry in Three Dimensions, Cambridge University Press, Cambridge, 1951.
J. Stolfi, Primitives for computational geometry, Ph.D. Thesis, Stanford University, 1989.
E. Welzl, Partition trees for triangle counting and other range searching problems,Proc. 4th ACM Symp. on Computational Geometry, 1988, pp. 23–33.
Work on this paper has been supported by DIMACS, an NSF Science and Technology Center, under Grant STC-88-09684. The second author has been supported by Office of Naval Research Grants N00014-89-J-3042 and N00014-90-J-1284, by National Science Foundation Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.
About this article
Cite this article
Agarwal, P.K., Sharir, M. Applications of a new space-partitioning technique. Discrete Comput Geom 9, 11–38 (1993). https://doi.org/10.1007/BF02189304
- Span Tree
- Range Query
- Query Time
- Partition Tree
- Double Wedge