Discrete & Computational Geometry

, Volume 1, Issue 1, pp 83–93 | Cite as

Halfspace range search: An algorithmic application ofk-sets

  • B. Chazelle
  • F. P. Preparata
Article

Abstract

Given a fixed setS ofn points inE3 and a query planeπ, the halfspace range search problem asks for the retrieval of all points ofS on a chosen side ofπ. We prove that withO(n(logn)8 (loglogn)4) storage it is possible to solve this problem inO(k+logn) time, wherek is the number of points to be reported. This result rests crucially on a new combinatorial derivation. We show that the total number ofj-sets (j=1, ...,k) realized by a set ofn points inE3 isO(nk5); ak-set is any subset ofS of sizek which can be separated from the rest ofS by a plane.

References

  1. 1.
    J. L. Bentley and H. A. Maurer, A note on Euclidean near neighbor searching in the plane, Inform. Process. Lett. 8 (1979), 133–136.MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    B. Chazelle, Filtering search: A new approach to query-answering, Proc. 24th IEEE Annu. Symp. Found. Comput. Sci., (1983), 122–132 (also to appear in J. SIAM on Comput.)Google Scholar
  3. 3.
    B. Chazelle, Criticality considerations in the design of geometric algorithms, Proc. 22nd Annu. Allerton Conf. on Comm., Contr., and Comput. (1984).Google Scholar
  4. 4.
    B. Chazelle, R. Cole, F. P. Preparata, and C. K. Yap, New upper bounds for neighbor searching, Tech. Rept. CS-84-11 (1984), Brown Univ.Google Scholar
  5. 5.
    B. Chazelle, L. J. Guibas, and D. T. Lee, The power of geometric duality, Proc. 24th IEEE Annu. Symp. Found. Comput. Sci., (1983), 217–225.Google Scholar
  6. 6.
    R. Cole, M. Sharir, and C. K. Yap, Onk-hulls and related problems, Proc. 16th Annu. ACM Symp. on Theory of Computing, (1984), 154–166. To appear in SIAM J. on Comput.Google Scholar
  7. 7.
    R. Cole and C. K. Yap, Geometric retrieval problems, Proc. 24th IEEE Annu. Symp. Found. Comput. Sci., (1983), 112–121.Google Scholar
  8. 8.
    H. Edelsbrunner, Arrangements and geometric computations, to appear.Google Scholar
  9. 9.
    H. Edelsbrunner, J. O'Rourke, and R. Seidel, Constructing arrangements of lines and hyperplanes with applications, Proc. 24th IEEE Annu. Symp. Found. Comput. Sci., (1983), 83–91.Google Scholar
  10. 10.
    P. Erdős, L. Lovász, A. Simmons, and E. G. Straus, Dissection graphs of planar point sets, in A Survey of Combinatorial Theory, J. N. Srivastava et al., eds., North-Holland (1973), 139–149.Google Scholar
  11. 11.
    D. T. Lee, Onk-nearest neighbor Voronoi diagrams in the plane, IEEE Trans. Comp., Vol C-31, No. 6, June 1982, pp. 478–487.CrossRefMATHGoogle Scholar
  12. 12.
    F. P. Preparata and M. I. Shamos, Introduction to Computational Geometry, Springer-Verlag, to appear.Google Scholar
  13. 13.
    F. F. Yao, D. P. Dobkin, and H. Edelsbrunner, A partition of 3-space with applications, forthcoming report.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • B. Chazelle
    • 1
  • F. P. Preparata
    • 2
  1. 1.Department of Computer ScienceBrown UniversityProvidence
  2. 2.Coordinated Science LaboratoryUniversity of IllinoisUrbana-Champaign

Personalised recommendations