Abstract
LetP be a set ofn points in ℝd (whered is a small fixed positive integer), and let Γ be a collection of subsets of ℝd, each of which is defined by a constant number of bounded degree polynomial inequalities. We consider the following Γ-range searching problem: GivenP, build a data structure for efficient answering of queries of the form, “Given a γ∈Γ, count (or report) the points ofP lying in γ.” Generalizing the simplex range searching techniques, we give a solution with nearly linear space and preprocessing time and withO(n 1−1/b+δ) query time, whered≤b≤2d−3 and δ>0 is an arbitrarily small constant. The acutal value ofb is related to the problem of partitioning arrangements of algebraic surfaces into cells with a constant description complexity. We present some of the applications of Γ-range searching problem, including improved ray shooting among triangles in ℝ3.
Article PDF
Similar content being viewed by others
References
P. K. Agarwal, Ray shooting and other applications of spanning trees with low stabbing number.SIAM J. Comput. 21 (1992), 540–570.
P. K. Agarwal, L. Guibas, M. Pellegrini, and M. Sharir. Ray shooting among spheres. Manuscript, 1993.
P. K. Agarwal, M. van Kreveld, and M. Overmars. Intersection queries for curved objects.J. Algorithms 15 (1993), 229–266.
P. K. Agarwal and J. Matoušek. Ray shooting and parametric search.SIAM J. Comput. 22 (1993), 794–806.
P. K. Agarwal and M. Sharir. Applications of a new space partitioning scheme.Discrete Comput. Geom. 9 (1993), 11–38.
A. Aggarwal, M. Hansen, and T. Leighton. Solving query-retrieval problems by compacting Voronoi diagrams.Proc. 21st ACM Symposium on Theory of Computing, 1990, pp. 331–340. Also to appear inJ. Assoc. Comput. Mach.
N. Alon, D. Haussler, E. Welzl, and G. Wöginger. Partitioning and geometric embedding of range spaces of finite Vapnik-Chervonenkis dimension.Proc. 3rd ACM Symposium on Computational Geometry, 1987, pp. 331–340.
B. Aronov, M. Pellegrini, and M. Sharir. On the zone of a surface in a hyperplane arrangement.Discrete Comput. Geom. 9 (1993), 177–186.
M. de Berg, D. Halperin, M. Overmars, J. Snoeyink, and M. van Kreveld. Efficient ray shooting and hidden surface removal.Proc. 7th Symposium on Computational Geometry, 1991, pp. 51–60.
B. Chazelle. Lower bounds on the complexity of polytope range searching.J. Amer. Math. Soc. 2 (1989), 637–666.
B. Chazelle. Cutting hyperplanes for divide-and-conquer.Discrete Comput. Geom. 9 (1993), 145–158.
B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir. A singly-exponential stratification scheme for real semi-algebraic varieties and its applications.Theoret. Comput. Sci. 84 (1991), 77–105. (Also inProc. 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.Proc. 21st ACM Symposium on Theory of Computing, 1989, pp. 382–393. Also to appear inAlgorithmica.
B. Chazelle and J. Friedman. A deterministic view of random sampling and its use in geometry.Combinatorica 10 (1990), 229–249.
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.Algorithmica 8 (1992), 407–430.
B. Chazelle and E. Welzl. Quasi-optimal range searching in spaces of finite VC-dimension.Discrete Comput. Geom. 4 (1989), 467–490.
K. L. Clarkson. A randomized algorithm for closest-point queries.SIAM J. Comput. 17 (1988), 830–847.
K. L. Clarkson and P. Shor. New applications of random sampling in computational geometry, II.Discrete Comput. Geom. 4 (1989), 387–421.
D. Dobkin and H. Edelsbrunner. Space searching for intersecting objects.J. Algorithms 8 (1987), 348–361.
H. Edelsbrunner,Algorithms in Combinatorial Geometry. Springer-Verlag, Berlin, 1987.
D. Haussler and E. Welzl. ε-nets and simplex range queries.Discrete Comput. Geom. 2 (1987), 127–151.
J. Hershberger and S. Suri. A pedestrian approach to ray shooting: shoot a ray, take a walk.Proc. 4th ACM-SIAM Symposium on Discrete Algorithms, 1993, pp. 54–63.
J. Komlós, J. Pach, and G. Wöginger. Almost tight bounds for epsilon-nets.Discrete Comput. Geom. 7 (1992), 163–173.
J. Matoušek. Approximations and optimal geometric divide-and-conquer.Proc. 23rd ACM Symposium on Theory of Computing, 1991, pp. 506–511.
J. Matoušek. Cutting hyperplane arrangements.Discrete Comput. Geom. 6 (1991), 385–406.
J. Matoušek. Efficient partition trees.Discrete Comput. Geom. 8 (1992), 315–334.
J. Matoušek. Reporting points in half- spaces,Comput. Geom. Theory Appl. 2 (1992), 169–186.
J. Matoušek. Range searching with efficient hierarchical cuttings.Discrete Comput. Geom. 10 (1993), 183–196.
M. Pellegrini. Ray shooting in 3-dimensional spaces.Algorithmica 9 (1993), 471–494.
F. Preparata and M. I. Shamos.Computational Geometry—An Introduction. Springer-Verlag, New York, 1985.
D. Sommerville,Analytical Geometry in Three Dimensions. Cambridge University Press, Cambridge, 1951.
R. Wenocur and R. Dudley. Some Vapnik-Chervonenkis classes.Discrete Math. 33 (1981), 313–318.
D. E. Willard. Polygon retrieval.SIAM J. Comput. 11 (1982), 149–165.
F. F. Yao and A. C. Yao. A general approach to geometric queries.Proc. 17th ACM Symposium on Theory of Computing, 1985, pp. 163–168.
Author information
Authors and Affiliations
Additional information
Part of the work by P. Agarwal was supported by National Science Foundation Grant CCR-91-06514. Part of the work by J. Matoušek was supported by a Humboldt Research Fellowship. A preliminary version of this paper appeared inProc. 17th Symp. on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, Vol. 629, Springer-Verlag, Berlin, 1992, pp. 1–13.
Rights and permissions
About this article
Cite this article
Agarwal, P.K., Matousek, J. On range searching with semialgebraic sets. Discrete Comput Geom 11, 393–418 (1994). https://doi.org/10.1007/BF02574015
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02574015