Abstract
We give a lower bound on the following problem, known as simplex range reporting: Given a collection P of n points in d-space and an arbitrary simplex q, find all the points in P ∩ q. It is understood that P is fixed and can be preprocessed ahead of time, while q is a query that must be answered on-line. We consider data structures for this problem that can be modeled on a pointer machine and whose query time is bounded by O(n δ+r), where r is the number of points to be reported and δ is an arbitrary fixed real. We prove that any such data structure of that form must occupy storage Ω(n d(1-δ)-ε), for any fixed ε > 0. This lower bound is tight within a factor of 2 ε.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
B. Chazelle. Filtering search: A new approach to query-answering. SIAM Journal on Computing, 15(3):703–724, 1986.
B. Chazelle. Lower bounds on the complexity of polytope range searching. Journal of the American Mathematical Society, 2(4):637–666, 1989.
B. Chazelle. Lower bounds for orthogonal range searching: I. The reporting case. Journal of the ACM, 37(2):200–212, April 1990.
B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds of simplex range searching and new zone theorems. In The Proceedings of the Sixth Annual Symposium on Computational Geometry, pages 23–33, 1990.
B. Chazelle and E. Welzl. Quasi-optimal range seaching in spaces of finite Vapnik Chervonenkis dimension. Discrete and Computational Geometry, 4(5):467–489, 1988.
H. Chernoff. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. The Annals of Mathematical Statistics, 23:137–142, 1952.
H. Edelsbrunner, D. G. Kirkpatrick, and H. A. Maurer. Polygonal intersection searching. Information Procesing Letters, 14(2):74–79, 1982.
H. Edelsbrunner and E. Welzl. Halfplanar range seach in linear space and O(n 0.695) query time. Information Processing Letters, 23:289–293, 1986.
P. Erdös and J. Spencer. Probabilistic Methods in Combinatorics. Academic Press, New York, 1974.
D. Haussler and E. Welzl. ε-nets and simplex range queries. Discrete & Computational Geometry, 2(3):237–256, 1987.
J. Komlós, E. Szemerédi, and J. Pintz. On Heilbronn's triangle problem. Journal of the London Mathematical Society, 24(2):385–396, 1981.
J. Komlós, E. Szemerédi, and J. Pintz. A lower bound for Heilbronn's problem. Journal of the London Mathematical Society, 25(2): 13–24, 1982.
J. Matoušek, Efficient partition trees. In Proceedings of the Seventh Annual Symposium on Computational Geometry, pages 1–9, 1991.
J. Matoušek. Range searching with efficient hierarchical cuttings. manuscript, 1991.
W. O. J. Moser. Problems on extremal properties of a finite set of points. Discrete Geometry and Convexity, 440:52–64, 1985.
M. Paterson and F. F. Yao. Point retrieval for polygons. Journal of Algorithms, 7:441–447, 1986.
R. E. Tarjan. A class of algorithms which require nonlinear time to maintain disjoint sets. Journal of Computing System Science, 18:110–127, 1979.
E. Welzl. Partition trees for triangle counting and other range searching problems. In Proceedings Fourth Annual Symposium on Computational Geometry, pages 23–33, 1988.
D. E. Willard. Polygon retrieval. SIAM Journal on Computing, 11:149–165, 1982.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1992 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chazelle, B., Rosenberg, B. (1992). Lower bounds on the complexity of simplex range reporting on a pointer machine. In: Kuich, W. (eds) Automata, Languages and Programming. ICALP 1992. Lecture Notes in Computer Science, vol 623. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55719-9_95
Download citation
DOI: https://doi.org/10.1007/3-540-55719-9_95
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-55719-7
Online ISBN: 978-3-540-47278-0
eBook Packages: Springer Book Archive