Discrete & Computational Geometry

, Volume 10, Issue 2, pp 143–155 | Cite as

How hard is half-space range searching?

  • Hervé Brönnimann
  • Bernard Chazelle
  • János Pach


We investigate the complexity ofhalf-space range searching: givenn points ind-space, build a data structure that allows us to determine efficiently how many points lie in a query half-space. We establish a tradeoff between the storagem and the worst-case query timet in the Fredman/Yao arithmetic model of computation. We show thatt must be at least on the order of
$$\frac{{(n/\log n)^{1 - (d - 1)/d(d + 1)} }}{{m^{1/d} }}$$

Although the bound is unlikely to be optimal, it falls reasonably close to the recent upper bound ofO(n/m 1/d ) established by Matoušek. We also show that it is possible to devise a sequence ofn inserts and half-space range queries that require a total time ofn 2-O(1/d) . Our results imply the first nontrivial lower bounds for spherical range searching in any fixed dimension. For example, they show that, with linear storage, circular range queries in the plane require Ω(n1/3) time (modulo a logarithmic factor).


Convex Body Discrete Comput Geom Range Query Query Time Storage Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Agarwal, P. K., Sharir, M. Applications of a new partitioning scheme,Discrete Comput. Geom. 9 (1993), 13–38.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barany, I., Larman, D. Convex bodies, economic cap coverings, random polytopes,Mathematika 35 (1988), 274–291.zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Burkhard, W. A., Fredman, M. L., Kleitman, D. J. Inherent complexity trade-offs for range query problems,Theoret. Comput. Sci. 16 (1981), 279–290.zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Chazelle, B. Lower bounds on the complexity of polytope range searching,J. Amer. Math. Soc. 2 (1989), 637–666.zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Chazelle, B. Lower bounds for orthogonal range searching: I. The reporting case,J. Assoc. Comput. Mach. 37 (1990), 200–212.zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Chazelle, B. Lower bounds for orthogonal range searching: II. The arithmetic model,J. Assoc. Comput. Mach. 37 (1990), 439–463.zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Chazelle, B., Rosenberg, B. The complexity of computing partial sums off-line,Internat. J. Comput. Geom. Applic. 1 (1991), 33–45.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Chazelle, B., Rosenberg, B. Lower bounds on the complexity of simplex range reporting on a pointer machine,Proc. 19th ICALP, LNCS 623, Springer-Verlag, Berlin, July 1992, pp. 439–449.Google Scholar
  9. 9.
    Chazelle, B., Sharir, M., Welzl, E. Quasi-optimal upper bounds for simplex range searching and new zone theorems,Algorithmica 8 (1992), 407–429.zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Chazelle, B., Welzl, E. Quasi-optimal range searching in spaces of finiteVC-dimension,Discrete Comput. Geom. 4 (1989), 467–489.zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Clarkson, K., Shor, P. Applications of random sampling to computational geometry, II,Discrete Comput. Geom. 4 (1989), 387–421.zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Cole, R., Yap, C. K. Geometric retrieval problems,Inform. Control 63 (1984), 39–57.zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Edelsbrunner, H., Welzl, E. Halfplanar range search in linear space andO(n0.695) query time,Inform. Process. Lett. 23 (1986), 289–293.zbMATHCrossRefGoogle Scholar
  14. 14.
    Ewald, G., Larman, D.G., Rogers, C.A. The directions of the line segments and of ther-dimensional balls on the boundary of a convex body in Euclidean space,Mathematika 17 (1970), 1–20.zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Fredman, M. L. A lower bound on the complexity of orthogonal range queries,J. Assoc. Comput. Mach. 28 (1981), 696–705.zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Fredman, M. L. Lower bounds on the complexity of some optimal data structures,SIAM J. Comput. 10 (1981), 1–10.zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Groemer, H. On the mean value of the volume of a random polytope in a convex set,Arch. Math. 25 (1974), 86–90.zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Grotschel, M., Lovasz, L., Schrijver, A.Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin, 1988.CrossRefGoogle Scholar
  19. 19.
    Haussler, D., Welzl, E. Epsilon-nets and simplex range queries,Discrete Comput. Geom. 2 (1987), 127–151.zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Macbeath, A. M. A theorem on non-homogeneous lattices,Ann. of Math. (2)56, (1952), 269–293.zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Matoušek, J. Reporting points in halfspaces,Proc. 32nd Ann. Symp. Found. Comput. Sci., 1991, pp. 207–215.Google Scholar
  22. 22.
    Matoušek, J. Range searching with effient hierarchical cuttings,Proc. 8th Ann. ACM Symp. Comput. Geom., 1992, pp. 276–285.Google Scholar
  23. 23.
    Mehlhorn, K.Data Structures and Algorithms 3: Multidimensional Searching and Computational Geometry, Springer-Verlag, Berlin, 1984.CrossRefGoogle Scholar
  24. 24.
    Santaló, L. A.Integral Geometry and Geometric Probability, Encyclopedia of Mathematics and Its Applications, Vol. 1 (Gian-Carlo Rota, ed.), Addison-Wesley, Reading, MA, 1976.Google Scholar
  25. 25.
    Spencer, J.Ten Lectures on the Probablistic Method, CBMS-NSF, SIAM, Philadelphia, PA, 1987.Google Scholar
  26. 26.
    Willard, D. E. Polygon retrieval,SIAM J. Comput. 11 (1982), 149–165.zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Yao, F. F. A 3-space partition and its applications,Proc. 15th Ann. ACM Symp. Theory Comput., 1983, pp. 258–263.Google Scholar
  28. 28.
    Yao, A. C. On the complexity of maintaining partial sums,SIAM J. Comput. 14 (1985), 277–288.zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Yao, A. C., Yao, F. F. A general approach tod-dimensional geometric queries,Proc. 17th Ann. ACM Symp. Theory Comput., 1985, pp. 163–168.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • Hervé Brönnimann
    • 1
  • Bernard Chazelle
    • 1
  • János Pach
    • 2
    • 3
  1. 1.Department of Computer SciencePrinceton UniversityPrincetonUSA
  2. 2.Department of Computer Science, City CollegeC.U.N.Y.New YorkUSA
  3. 3.Courant InstituteN.Y.U.New YorkUSA

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