Discrete & Computational Geometry

, Volume 6, Issue 4, pp 593–613 | Cite as

Onk-sets in arrangements of curves and surfaces

  • Micha Sharir


We extend the notion ofk-sets and (≤k)-sets (see [3], [12], and [19]) to arrangements of curves and surfaces. In the case of curves in the plane, we assume that each curve is simple and separates the plane. Ak-point is an intersection point of a pair of the curves which is covered by exactlyk interiors of (or half-planes bounded by) other curves; thek-set is the set of allk-points in such an arrangement, and the (≤k)-set is the union of allj-sets, forjk. Adapting the probabilistic analysis technique of Clarkson and Shor [13], we obtain bounds that relate the maximum size of the (≤k)-set to the maximum size of a 0-set of a sample of the curves. Using known bounds on the size of such 0-sets we obtain asympotically tight bounds for the maximum size of the (≤k)-set in the following special cases: (i) If each pair of curves intersect at most twice, the maximum size is Θ(nkα(nk)). (ii) If the curves are unbounded arcs and each pair of them intersect at most three times, then the maximum size is Θ(nkα(n/k)). (iii) If the curves arex-monotone arcs and each pair of them intersect in at most some fixed numbers of points, then the maximum size of the (≤k)-set is Θ(k2λ s (nk)), where λ s (m) is the maximum length of (m,s)-Davenport-Schinzel sequences. We also obtain generalizations of these results to certain classes of surfaces in three and higher dimensions. Finally, we present various applications of these results to arrangements of segments and curves, high-order Voronoi diagrams, partial stabbing of disjoint convex sets in the plane, and more. An interesting application yields andO(n logn) bound on the expected number of vertically visible features in an arrangement ofn horizontal discs when they are stacked on top of each other in random order. This in turn leads to an efficient randomized preprocessing ofn discs in the plane so as to allow fast stabbing queries, in which we want to report all discs containing a query point.


Intersection Point Voronoi Diagram Computational Geometry Query Point Expected 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. [1]
    P. K. Agarwal, Ray shooting and other applications of spanning trees with low stabbing number,Proc. 5th Symp. on Computational Geometry, 1989, pp. 315–325.Google Scholar
  2. [2]
    P. Agarwal, M. Sharir, and P. Shor, Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences,J. Combin. Theory Ser. A 52 (1989), 228–274.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    N. Alon and E. Györi, The number of small semispaces of a finite set of points in the plane,J. Combin. Theory Ser. A 41 (1986), 154–157.zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    H. Alt, R. Fleischer, M. Kaufmann, K. Mehlhorn, S. Näher, S. Schirra, and C. Uhrig, Approximate motion planning and the complexity of the boundary of the union of simple geometric figures,Proc. 6th Symp. on Computational Geometry, 1990, pp. 281–289.Google Scholar
  5. [5]
    M. Atallah, Some dynamic computational geometry problems,Comp. Math. Appl. 11 (1985), 1171–1181.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    F. Aurenhammer, Power diagrams: Properties, algorithms, and applicationsSIAM J. Comput. 16 (1987), 78–96.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    B. Chazelle, Lower bounds on the complexity of polytope range searching,Trans. Amer. Math. Soc. 2 (1989), 637–666.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    B. Chazelle and L. Guibas, Fractional cascading: II. Applications,Algorithmica 1 (1986), 163–191.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    B. Chazelle, L. Guibas, and D. T. Lee, The power of geometric duality,BIT 25 (1985), 76–90.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    B. Chazelle and D. T. Lee, On a circle placement problem,Computing 36 (1986), 1–16.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    B. Chazelle and F. P. Preparata, Halfspace range search: An algorithmic application ofk-sets,Discrete Comput. Geom. 1 (1986), 83–93.zbMATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    K. Clarkson, Applications of random sampling in computational geometry, II,Proc. 4th Symp. on Computational Geometry, 1988, pp. 1–11.Google Scholar
  13. [13]
    K. Clarkson and P. Short Applications of random sampling in computational geometry, II,Discrete Comput. Geom. 4 (1989), 387–421.zbMATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    H. Edelsbrunner,Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.zbMATHCrossRefGoogle Scholar
  15. [15]
    H. Edelsbrunner, L. Guibas, J. Hershberger, J. Pach, R. Pollack, R. Seidel, M. Sharir, and J. Snoeyink, On arrangements of Jordan arcs with three intersections per pair,Discrete Comput. Geom. 4 (1989), 523–539.zbMATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    H. Edelsbrunner, L. Guibas, J. Pach, R. Pollack, R. Seidel, and M. Sharir, Arrangements of curves in the plane: Topology, combinatorics and algorithms,Proc 15th Int. Colloq. on Automata, Programming and Languages, 1988, pp. 219–229.Google Scholar
  17. [17]
    H. Edelsbrunner, L. Guibas, and M. Sharir, The upper envelope of piecewise linear functions: Algorithms and applications,Discrete Comput. Geom. 4 (1989), 311–336.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    H. Edelsbrunner and M. Sharir, The maximum number of ways to stabn convex nonintersecting objects in the plane is 2n−2,Discrete Comput. Geom. 5 (1990), 35–42.zbMATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    J. Goodman and R. Pollack, On the number ofk-subsets of a set ofn points in the plane,J. Combin. Theory Ser. A 36 (1984), 101–104.zbMATHMathSciNetCrossRefGoogle Scholar
  20. [20]
    S. Hart and M. Sharir, Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes,Combinatorica 6 (1986), 151–177.zbMATHMathSciNetCrossRefGoogle Scholar
  21. [21]
    M. Katchalski, T. Lewis, and J. Zaks, Geometric permutations for convex sets,Discrete Math. 54 (1985), 271–284.zbMATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    K. Kedem, R. Livne, J. Pach, and M. Sharir, On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles,Discrete Comput. Geom. 1 (1986), 59–71.zbMATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    D. Leven and M. Sharir, Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams,Discrete Comput. Geom. 2 (1987), 9–31.zbMATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    J. Pach and M. Sharir, The upper envelope of piecewise linear functions and the boundary of a region enclosed by convex plates: Combinatorial analysis,Discrete Comput. Geom. 4 (1989), 291–309.zbMATHMathSciNetCrossRefGoogle Scholar
  25. [25]
    J. Pach, W. Steiger, and E. Szemerédi, An upper bound on the number of planark-sets,Proc. 30th IEEE Symp. on Foundations of Computer Science, 1989 pp. 72–79.Google Scholar
  26. [26]
    F. P. Preparata and M. I. Shamos,Computational Geometry: An Introduction, Springer-Verlag, Heidelberg, 1985.CrossRefGoogle Scholar
  27. [27]
    N. Sarnak and R. Tarjan, Planar point location using persistent search trees,Comm. ACM 29 (1986), 669–679.MathSciNetCrossRefGoogle Scholar
  28. [28]
    J. Schwartz and M. Sharir, On the 2-dimensional Davenport-Schinzel problem,J. Symbolic Comput. 10 (1990), 371–393.zbMATHMathSciNetCrossRefGoogle Scholar
  29. [29]
    R. Wenger, Upper bounds on geometric permutations for convex sets,Discrete Comput. Geom. 5 (1990), 27–33.zbMATHMathSciNetCrossRefGoogle Scholar
  30. [30]
    A. Wiernik and M. Sharir, Planar realization of nonlinear Davenport-Schinzel sequences by segments,Discrete Comput. Geom. 3 (1988), 15–47.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc 1991

Authors and Affiliations

  • Micha Sharir
    • 1
    • 2
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.School of Mathematical Sciences, Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations