Discrete & Computational Geometry

, Volume 5, Issue 5, pp 427–448

Construction of ɛ-nets

  • Jiří Matoušek


LetS be a set ofn points in the plane and letɛ be a real number, 0<ɛ<1. We give a deterministic algorithm, which in timeO(−2 log(1/ɛ)+ɛ−8) (resp.O(−2 log(1/ɛ)+ɛ−10) constructs anɛ-netNS of sizeO((1/ɛ) (log(1/ɛ))2) for intersections ofS with double wedges (resp. triangles); this means that any double wedge (resp. triangle) containing more thatɛn points ofS contains a point ofN. This givesO(n logn) deterministic preprocessing for the simplex range-counting algorithm of Haussler and Welzl [HW] (in the plane).

We also prove that given a setL ofn lines in the plane, we can cut the plane intoO(ɛ−2) triangles in such a way that no triangle is intersected by more thanɛn lines ofL. We give a deterministic algorithm for this with running timeO(−2 log(1/ɛ)). This has numerous applications in various computational geometry problems.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A1]
    P. K. Agarwal: A deterministic algorithm for partitioning arrangements of lines and its applications,Proc. 5th Ann. ACM Symposium on Computational Geometry (1989), pp. 11–21 (full version to apear inDiscrete Comput. Geom.).Google Scholar
  2. [A2]
    P. K. Agarwal: Ray shooting and other applications of spanning trees with low stabbing number,Proc. 5th Ann. ACM Symposium on Computational Geometry (1989), pp. 315–325 (full version to appear inDiscrete Comput. Geom.).Google Scholar
  3. [AKS]
    M. Ajtai, J. Komlós, E. Szemerédi: AnO(n logn) sorting network,Proc. 15th ACM Symp. on Theory on Computing (1983), pp. 1–9.Google Scholar
  4. [CF]
    B. Chazelle, J. Friedman: A deterministic view of random sampling and its use in geometry, Report CS-TR-181-88 (extended abstract inProc. 29th Ann. IEEE Symposium on Foundations of Computer Science (1988), pp. 539–549).Google Scholar
  5. [CW]
    B. Chazelle, E. Welzl: Quasi-optimal range searching in spaces of finite VC-dimension,Discrete Comput. Geom. 4 (1989), 467–490.MathSciNetCrossRefMATHGoogle Scholar
  6. [CH]
    V. Chvátal: A greedy heuristics for the set-covering problem,Math. Oper. Res. 4 (1979), 233–235.MathSciNetCrossRefMATHGoogle Scholar
  7. [CS]
    K. Clarkson, P. Shor: Applications of random sampling in computational geometry, II,Discrete Comput. Geom. 4 (1989), 387–421.MathSciNetCrossRefMATHGoogle Scholar
  8. [CEG*]
    K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, E. Welzl: Combinatorial complexity for arrangements of curves and surfaces,Proc. 29th Ann. IEEE Symposium on Foundations of Computer Science (1988), pp. 568–579.Google Scholar
  9. [Co]
    R. Cole: Slowing down sorting networks to obtain faster sorting algorithms,J. Assoc. Comput. Mach. 31 (1984), 200–208.Google Scholar
  10. [CSSS]
    R. Cole, J. Salowe, W. L. Steiger, E. Szemerédi: An optimal-time algorithm for slope selection,SIAM J. Comput. 18 (1989), 792–810.MathSciNetCrossRefMATHGoogle Scholar
  11. [E]
    H. Edelsbrunner:Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.CrossRefMATHGoogle Scholar
  12. [EGH*]
    H. Edelsbrunner, L. Guibas, J. Herschberger, R. Seidel, M. Sharir, J. Snoeyink, E. Welzl: Implicitly representing arrangements of lines or segments,Discrete Comput. Geom. 4 (1989), 433–466.MathSciNetCrossRefMATHGoogle Scholar
  13. [EW]
    H. Edelsbrunner, E. Welzl: Constructing belts in 2-dimensional arrangements,SIAM J. Comput. 15 (1986), 271–284.MathSciNetCrossRefMATHGoogle Scholar
  14. [GJ]
    M. R. Garey, D. S. Johnson;Computers and Intractability, Freeman, San Francisco, 1979.MATHGoogle Scholar
  15. [HW]
    D. Haussler, E. Welzl:ɛ-nets and simplex range queries,Discrete Comput. Geom. 2 (1987), 127–151.MathSciNetCrossRefMATHGoogle Scholar
  16. [L]
    L. Lovász: On the ratio of optimal integral and fractional cover,Discrete Math. 13 (1975), 383–390.MathSciNetCrossRefMATHGoogle Scholar
  17. [M1]
    J. Matoušek:Approximate Halfplanar Range Counting, KAM Series 59-87, Charles University, Prague, 1987.Google Scholar
  18. [M2]
    J. Matoušek: Cutting hyperplane arrangements, 6th ACM Symposium on Computational Geometry, 1990.Google Scholar
  19. [M3]
    J. Matoušek: Spanning trees with low crossing number, to appear inInform. Theoret. Applic. Google Scholar
  20. [Me]
    N. Megiddo: Applying parallel computation algorithm in the design of serial algorithms,J. Assoc. Comput. Mach. 30 (1983), 852–865.MathSciNetCrossRefMATHGoogle Scholar
  21. [PSS]
    J. Pach, W. Steiger, E. Szemerédi: An upper bound for the number of planark-sets,Proc. 30th Ann. IEEE Symposium on Foundations of Computer Science (1989), pp. 72–81.Google Scholar
  22. [S]
    S. Suri: A linear algorithm for minimum link paths inside a simple polygon,Comput. Vision Graphics Image Process. 35 (1986), 99–110.CrossRefMATHGoogle Scholar
  23. [VC]
    V. N. Vapnik, A. Ya. Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities,Theory Probab. Appl. 16 (1971), 264–280.CrossRefMATHGoogle Scholar
  24. [W1]
    E. Welzl: Partition trees for triangle counting and other range searching problems,Proc. 4th ACM Symposium on Computational Geometry (1988), pp. 23–33.Google Scholar
  25. [W2]
    E. Welzl: More onk-sets of finite sets in the plane,Discrete Comput. Geom. 1 (1986), 95–100.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Jiří Matoušek
    • 1
  1. 1.Department of Computer ScienceCharles UniversityPraha 1Czechoslovakia

Personalised recommendations