Discrete & Computational Geometry

, Volume 4, Issue 5, pp 433–466 | Cite as

Implicitly representing arrangements of lines or segments

  • Herbert Edelsbrunner
  • Leonidas Guibas
  • John Hershberger
  • Raimund Seidel
  • Micha Sharir
  • Jack Snoeyink
  • Emo Welzl


Anarrangement ofn lines (or line segments) in the plane is the partition of the plane defined by these objects. Such an arrangement consists ofO(n2) regions, calledfaces. In this paper we study the problem of calculating and storing arrangementsimplicitly, using subquadratic space and preprocessing, so that, given any query pointp, we can calculate efficiently the face containingp. First, we consider the case of lines and show that with Λ(n) space1 and Λ(n3/2) preprocessing time, we can answer face queries in Λ(√n)+O(K) time, whereK is the output size. (The query time is achieved with high probability.) In the process, we solve three interesting subproblems: (1) given a set ofn points, find a straight-edge spanning tree of these points such that any line intersects only a few edges of the tree, (2) given a simple polygonal path Γ, form a data structure from which we can find the convex hull of any subpath of Γ quickly, and (3) given a set of points, organize them so that the convex hull of their subset lying above a query line can be found quickly. Second, using random sampling, we give a tradeoff between increasing space and decreasing query time. Third, we extend our structure to report faces in an arrangement of line segments in Λ(n1/3)+O(K) time, givenΛ(n4/3) space and Λ(n5/3) preprocessing time. Lastly, we note that our techniques allow us to computem faces in an arrangement ofn lines in time Λ(m2/3n2/3+n), which is nearly optimal.


Convex Hull Span Tree Steiner Tree Query Point Query Time 
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. [AS]
    B. Aronov and M. Sharir. Triangles in space, or building and analyzing castles in the air. InProceedings of the 4th ACM Symposium on Computational Geometry, pages 381–391, ACM, June 1988.Google Scholar
  2. [BO]
    J. L. Bentley and T. A. Ottman. Algorithms for reporting and counting geometric intersections.IEEE Transactions on Computers, 28(9):643–647, 1979.CrossRefMATHGoogle Scholar
  3. [B]
    K. Q. Brown. Geometric Transforms for Fast Geometric Algorithms. Ph.D. thesis, Carnegie-Mellon University, 1980.Google Scholar
  4. [CE]
    B. Chazelle and H. Edelsbrunner. An optimal algorithm for intersecting line segments in the plane. InProceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science, pages 590–600, IEEE, 1988.Google Scholar
  5. [CG1]
    B. Chazelle and L. Guibas. Visibility and intersection problems in plane geometry. InProceedings of the ACM Symposium on Computational Geometry, pages 135–146, ACM, 1985. Submitted toDiscrete and Computational Geometry.Google Scholar
  6. [CG2]
    B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications.Algorithmica, 1:163–191, 1986.MathSciNetCrossRefMATHGoogle Scholar
  7. [CGL]
    B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality.BIT, 25:76–90, 1985.MathSciNetCrossRefMATHGoogle Scholar
  8. [CW]
    B. Chazelle and E. Welzl. Range searching and VC-dimension: a characterization of efficiency. 1988. Manuscript.Google Scholar
  9. [C]
    K. Clarkson. New applications of random sampling in computational geometry.Discrete and Computational Geometry, 2:195–222, 1987.MathSciNetCrossRefMATHGoogle Scholar
  10. [CEG*]
    K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, and E. Welzl. Combinatorial complexity bounds for arrangements of curves and surfaces. InProceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science, pages 568–579, IEEE, 1988.Google Scholar
  11. [E]
    H. Edelsbrunner.Algorithms in Combinatorial Geometry. EATCS Monographs on Theoretical Computer Science, Volume 10, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar
  12. [EG]
    H. Edelsbrunner and L. J. Guibas. Topologically sweeping an arrangement. InProceedings of the 18th ACM Symposium on Theory of Computing, pages 389–403, ACM, May 1986.Google Scholar
  13. [EGP*]
    H. Edelsbrunner, L. Guibas, J. Pach, R. Pollack, R. Seidel, and M. Sharir. Arrangements of curves in the plane: Topology, combinatorics, and algorithms. InProceedings of the 15th ICALP, 1988.Google Scholar
  14. [EGS1]
    H. Edelsbrunner, L. J. Guibas, and M. Sharir. The complexity of many faces in arrangements of lines and of segments. InProceedings of the 4th ACM Symposium on Computational Geometry, pages 44–55, ACM, June 1988.Google Scholar
  15. [EGS2]
    H. Edelsbrunner, L. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision.SIAM Journal on Computing, 15:317–340, 1986.MathSciNetCrossRefMATHGoogle Scholar
  16. [EOS]
    H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of lines and hyperplanes with applications.SIAM Journal on Computing, 15:341–363, 1986.MathSciNetCrossRefMATHGoogle Scholar
  17. [EW]
    H. Edelsbrunner and E. Welzl. Halfplanar range search in linear space andO(n 0 695) query time.Information Processing Letters, 23:289–293, 1986.CrossRefMATHGoogle Scholar
  18. [GH]
    L. Guibas and J. Hershberger. Optimal shortest path queries in a simple polygon. InProceedings of the 3rd ACM Symposium on Computational Geometry, pages 50–63, ACM, June 1987.Google Scholar
  19. [GHS]
    L. Guibas, J. Hershberger, and J. Snoeyink. Bridge trees: a data structure for convex hulls. 1989. In preparation.Google Scholar
  20. [GOS]
    L. Guibas, M. Overmars, and M. Sharir. Interesting line segments, ray shooting, and other applications of geometric partitioning techniques. InProceedings of the First Scandinavian Workshop on Algorithm Theory, pages 64–73. Lecture Notes in Computer Science, Volume 318, Springer-Verlag, Berlin, 1988.Google Scholar
  21. [HW]
    D. Haussler and E. Welzl. Epsilon-nets and simplex range queries.Discrete and Computational Geometry, 2:127–151, 1987.MathSciNetCrossRefMATHGoogle Scholar
  22. [Ki]
    D. Kirkpatrick. Optimal search in planar subdivisions.SIAM Journal on Computing, 12:28–35, 1983.MathSciNetCrossRefMATHGoogle Scholar
  23. [Kn]
    D. E. Knuth.Sorting and Searching. The Art of Computer Programming, Volume 3, Addison-Wesley, Reading, MA, 1973.Google Scholar
  24. [M]
    J. Matoušek. Spanning trees with low crossing number. 1988. Manuscript.Google Scholar
  25. [OL]
    M. Overmars and H. van Leeuwen. Maintenance of configurations in the plane.Journal of Computer and System Sciences, 23:166–204, 1981.MathSciNetCrossRefMATHGoogle Scholar
  26. [PS]
    F. P. Preparata and M. I. Shamos.Computational Geometry. Springer-Verlag, New York, 1985.CrossRefGoogle Scholar
  27. [ST]
    N. Sarnak and R. E. Tarjan. Planar point location using persistent search trees.Communications of the ACM, 29:669–679, 1986.MathSciNetCrossRefGoogle Scholar
  28. [SH]
    M. I. Shamos and D. Hoey. Geometric intersection problems. InProceedings of the 17th IEEE Symposium on Foundations of Computer Science, pages 208–215, IEEE, 1976.Google Scholar
  29. [We]
    E. Welzl. Partition trees for triangle counting and other range searching problems. InProceedings of the 4th ACM Symposium on Computational Geometry, pages 23–33, ACM, June 1988.Google Scholar
  30. [Wi]
    D. E. Willard. Polygon retrieval.SIAM Journal on Computing, 11:149–165, 1982.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • Herbert Edelsbrunner
    • 1
  • Leonidas Guibas
    • 2
    • 3
  • John Hershberger
    • 3
  • Raimund Seidel
    • 4
    • 5
  • Micha Sharir
    • 6
    • 7
  • Jack Snoeyink
    • 2
  • Emo Welzl
    • 8
  1. 1.University of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Stanford UniversityStanfordUSA
  3. 3.DEC Systems Research CenterPalo AltoUSA
  4. 4.IBM Almaden Research CenterSan JoseUSA
  5. 5.University of California at BerkeleyBerkeleyUSA
  6. 6.New York UniversityNew YorkUSA
  7. 7.Tel Aviv UniversityTel AvivIsrael
  8. 8.Freie Universität BerlinBerlin 33Federal Republic of Germany

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