Abstract
Let P be a set of n points in ℝd (d 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 polynomials. The Γ-range searching problem is defined as: Preprocess P into a data structure, so that all points of P lying in a given γ Γ can be counted (or reported) efficiently. Generalizing the simplex range searching techniques, we construct a data structure for Γ-range searching with nearly linear space and preprocessing time, which can answer a query in time O(n 1−1/b+δ), where d≤b≤ 2d−3 and δ>0 is an arbitrarily small constant. The actual value of b is related to the problem of partitioning arrangements of algebraic surfaces into constant-complexity cells.
Part of the work by P.A. was supported by National Science Foundation Grant CCR-91-06514. Part of the work by J.M. was supported by Humboldt Research Fellowship.
Preview
Unable to display preview. Download preview PDF.
References
A. Aggarwal, M. Hansen, and T.Leighton. Solving query-retrieval problems by compacting Voronoi diagrams. Proc. 21st ACM Symposium on Theory of Computing, 1990, 331–340.
N. Alon, D. Haussler, E. Welzl, and G. Wöginger. Partitioning and geometric embedding of range spaces of finite Vapnik-Chervonenkis dimension. In Proc. 3. ACM Symposium on Computational Geometry, pages 331–340, 1987.
B. Aronov, M. Pellegrini, and M. Sharir, On the zone of a surface in a hyperplane arrangement. Discrete & Computational Geometry, to appear.
P. K. Agarwal and M. Sharir. Applications of a new space partitioning scheme. In Proc. 2. Workshop on Algorithms and Data Structures, 1991.
B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir. Point-location in real-algebraic varieties and its applications. In Proc. 16th International Colloquium on Automata, Languages and Programming, pages 179–192, 1989.
B. Chazelle and J. Friedman. A deterministic view of random sampling and its use in geometry. Combinatorica, 10(3):229–249, 1990.
B. Chazelle. Lower bounds on the complexity of polytope range searching. J. Amer. Math. Soc, 2(4):637–666, 1989.
B. Chazelle. Cutting hyperplanes for divide-and-conquer. Tech. report CS-TR-335-91, Princeton University, 1991. Preliminary version: Proc. 32. IEEE Symposium on Foundations of Computer Science, October 1991.
K. L. Clarkson and P. Shor. New applications of random sampling in computational geometry II. Discrete & Computational Geometry, 4:387–421, 1989.
B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. In Proc. 6. ACM Symposium on Computational Geometry, pages 23–33, 1990.
B. Chazelle and E. Welzl. Quasi-optimal range searching in spaces of finite VC-dimension. Discrete & Computational Geometry, 4:467–490, 1989.
D. Dobkin and H. Edelsbrunner. Space searching for intersecting objects. Journal of Algorithms, 8:348–361, 1987.
H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Verlag, Berlin-Heidelberg-New York, 1987.
D. Haussler and E. Welzl. ε-nets and simplex range queries. Discrete & Computational Geometry, 2:127–151, 1987.
J. Komlós, J. Pach, and G. Wöginger. Almost tight bounds for epsilon-nets. Discrete & Computational Geometry, 1992. To appear.
J. Matoušek. Approximations and optimal geometric divide-and-conquer. In Proc. 23. ACM Symposium on Theory of Computing, pages 506–511, 1991.
J. Matoušek. Cutting hyperplane arrangements. Discrete & Computational Geometry, 6(5):385–406, 1991.
J. Matoušek. Efficient partition trees. In Proc. 7. ACM Symposium on Computational Geometry, pages 1–9, 1991. Also to appear in Discrete & Computational Geometry.
J. Matoušek. Reporting points in halfspaces. Proc. 32nd IEEE Symposium on Foundations of Computer Science, 1991, pp. 207–215.
J. Matoušek. Range searching with efficient hierarchical cuttings. In Proc. 8. ACM Symposium on Computational Geometry, 1992. To appear.
F. Preparata and M. I. Shamos. Computational Geometry — An Introduction. Springer-Verlag, 1985.
D. E. Willard. Polygon retrieval. SIAM Journal on Computing, 11:149–165, 1982.
F. F. Yao and A. C. Yao. A general approach to geometric queries. In Proc. 17. ACM Symposium on Theory of Computing, pages 163–168, 1985.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1992 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Agarwal, P.K., Matoušek, J. (1992). On range searching with semialgebraic sets. In: Havel, I.M., Koubek, V. (eds) Mathematical Foundations of Computer Science 1992. MFCS 1992. Lecture Notes in Computer Science, vol 629. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-55808-X_1
Download citation
DOI: https://doi.org/10.1007/3-540-55808-X_1
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-55808-8
Online ISBN: 978-3-540-47291-9
eBook Packages: Springer Book Archive