Abstract
In this paper we develop some new data structures for storing sets of disks such that different types of queries can be answered efficiently. In particular we study intersection queries with lines and line segments and shooting queries. In the case of non-intersecting disks we obtain structures that use linear storage and have a query time of O(n β+ε+k) for intersection queries, where n is the number of disks, β=log2(1+\(\sqrt 5\)) − 1 ≈ 0.695, ε is an arbitrarily small positive constant, and k is the number of disks reported. For sets of intersecting disks we obtain a structure that uses O(n log n) storage and answers a query in time O(n 2/3log2 n). For ray shooting we obtain a structure that uses linear storage and has O(n β+ε) query time, for any ε>0.
The research of the first two authors was supported by the ESPRIT II Basic Research Actions Program of the EC under contract No. 3075 (project ALCOM). The work of the third author was supported by Dimacs (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center - NSF-STC88-09648.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Agarwal, P.K., Ray shooting and other applications of spanning trees with low stabbing number, Proc. 5th ann. symp. on Comp. Geometry (1989), pp. 315–325.
Chazelle, B., and L.J. Guibas, Visibility and intersection problems in plane geometry, Proc. 1st ann. symp. on Comp. Geometry (1985), pp. 135–146.
Chazelle, B., and L.J. Guibas, Fractional cascading: I. A data structuring technique, Algorithmica 1 (1986), pp. 133–162.
Chazelle, B., and E. Welzl, Quasi-optimal range searching in spaces of finite VC-dimension, Discr. & Comp. Geometry 4 (1989), pp. 467–489.
Cole, R., Searching and storing similar lists, J. of Algorithms 7 (1986), pp. 202–220.
Dobkin, D.P., and H. Edelsbrunner, Space searching for intersecting objects, J. of Algorithms 8 (1987), pp. 348–361.
Edelsbrunner, H., Algorithms in Combinatorial Geometry, Springer-Verlag, Berlin, 1987.
Edelsbrunner, H., L.J. Guibas, and J. Stolfi, Optimal point location in a monotone subdivision, SIAM J. Comput. 15 (1986), pp. 317–340.
Edelsbrunner, H., and E. Welzl, Halfplanar range search in linear space and O(n 0.695) query time, Inf. Proc. Lett. 23 (1986), pp. 289–293.
Guibas, L., M. Overmars, and M. Sharir, Ray shooting, implicit point location, and related queries in arrangements of segments, Techn. Rep. No. 433, New York University, 1989.
Haussler, D., and E. Welzl, ε-nets and simplex range queries, Discr. & Comp. Geometry 2 (1987), pp. 127–151.
Kedem, K., R. Livne, J. Pach, and M. Sharir, On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles, Discr. & Comp. Geometry 1 (1986), 59–71.
Overmars, M.H., H. Schipper, and M. Sharir, Storing line segments in partition trees, BIT, to appear.
Sharir, M., Efficient algorithm for reporting disc stabbing by points in the plane, preliminary note, 1989.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
van Kreveld, M., Overmars, M., Agarwal, P.K. (1990). Intersection queries in sets of disks. In: Gilbert, J.R., Karlsson, R. (eds) SWAT 90. SWAT 1990. Lecture Notes in Computer Science, vol 447. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-52846-6_107
Download citation
DOI: https://doi.org/10.1007/3-540-52846-6_107
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-52846-3
Online ISBN: 978-3-540-47164-6
eBook Packages: Springer Book Archive