# On*k*-sets in arrangements of curves and surfaces

- 135 Downloads
- 52 Citations

## Abstract

We extend the notion of*k*-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. A*k*-point is an intersection point of a pair of the curves which is covered by exactly*k* interiors of (or half-planes bounded by) other curves; the*k*-set is the set of all*k*-points in such an arrangement, and the (≤*k*)-set is the union of all*j*-sets, for*j*≤*k*. 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(*nk*α(*nk*)). (ii) If the curves are unbounded arcs and each pair of them intersect at most three times, then the maximum size is Θ(*nk*α(*n/k*)). (iii) If the curves are*x*-monotone arcs and each pair of them intersect in at most some fixed number*s* of points, then the maximum size of the (≤*k*)-set is Θ(*k*^{2}λ_{ 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 and*O*(*n* log*n*) bound on the expected number of vertically visible features in an arrangement of*n* horizontal discs when they are stacked on top of each other in random order. This in turn leads to an efficient randomized preprocessing of*n* discs in the plane so as to allow fast stabbing queries, in which we want to report all discs containing a query point.

## Keywords

Intersection Point Voronoi Diagram Computational Geometry Query Point Expected Time## References

- [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]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]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]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]M. Atallah, Some dynamic computational geometry problems,
*Comp. Math. Appl.***11**(1985), 1171–1181.zbMATHMathSciNetCrossRefGoogle Scholar - [6]F. Aurenhammer, Power diagrams: Properties, algorithms, and applications
*SIAM J. Comput.***16**(1987), 78–96.zbMATHMathSciNetCrossRefGoogle Scholar - [7]B. Chazelle, Lower bounds on the complexity of polytope range searching,
*Trans. Amer. Math. Soc.***2**(1989), 637–666.zbMATHMathSciNetCrossRefGoogle Scholar - [8]B. Chazelle and L. Guibas, Fractional cascading: II. Applications,
*Algorithmica***1**(1986), 163–191.zbMATHMathSciNetCrossRefGoogle Scholar - [9]B. Chazelle, L. Guibas, and D. T. Lee, The power of geometric duality,
*BIT***25**(1985), 76–90.zbMATHMathSciNetCrossRefGoogle Scholar - [10]B. Chazelle and D. T. Lee, On a circle placement problem,
*Computing***36**(1986), 1–16.zbMATHMathSciNetCrossRefGoogle Scholar - [11]B. Chazelle and F. P. Preparata, Halfspace range search: An algorithmic application of
*k*-sets,*Discrete Comput. Geom.***1**(1986), 83–93.zbMATHMathSciNetCrossRefGoogle Scholar - [12]K. Clarkson, Applications of random sampling in computational geometry, II,
*Proc. 4th Symp. on Computational Geometry*, 1988, pp. 1–11.Google Scholar - [13]K. Clarkson and P. Short Applications of random sampling in computational geometry, II,
*Discrete Comput. Geom.***4**(1989), 387–421.zbMATHMathSciNetCrossRefGoogle Scholar - [14]H. Edelsbrunner,
*Algorithms in Combinatorial Geometry*, Springer-Verlag, Heidelberg, 1987.zbMATHCrossRefGoogle Scholar - [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]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]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]H. Edelsbrunner and M. Sharir, The maximum number of ways to stab
*n*convex nonintersecting objects in the plane is 2*n*−2,*Discrete Comput. Geom.***5**(1990), 35–42.zbMATHMathSciNetCrossRefGoogle Scholar - [19]J. Goodman and R. Pollack, On the number of
*k*-subsets of a set of*n*points in the plane,*J. Combin. Theory Ser. A***36**(1984), 101–104.zbMATHMathSciNetCrossRefGoogle Scholar - [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]M. Katchalski, T. Lewis, and J. Zaks, Geometric permutations for convex sets,
*Discrete Math.***54**(1985), 271–284.zbMATHMathSciNetCrossRefGoogle Scholar - [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]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]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]J. Pach, W. Steiger, and E. Szemerédi, An upper bound on the number of planar
*k*-sets,*Proc. 30th IEEE Symp. on Foundations of Computer Science*, 1989 pp. 72–79.Google Scholar - [26]F. P. Preparata and M. I. Shamos,
*Computational Geometry: An Introduction*, Springer-Verlag, Heidelberg, 1985.CrossRefGoogle Scholar - [27]N. Sarnak and R. Tarjan, Planar point location using persistent search trees,
*Comm. ACM***29**(1986), 669–679.MathSciNetCrossRefGoogle Scholar - [28]J. Schwartz and M. Sharir, On the 2-dimensional Davenport-Schinzel problem,
*J. Symbolic Comput.***10**(1990), 371–393.zbMATHMathSciNetCrossRefGoogle Scholar - [29]R. Wenger, Upper bounds on geometric permutations for convex sets,
*Discrete Comput. Geom.***5**(1990), 27–33.zbMATHMathSciNetCrossRefGoogle Scholar - [30]A. Wiernik and M. Sharir, Planar realization of nonlinear Davenport-Schinzel sequences by segments,
*Discrete Comput. Geom.***3**(1988), 15–47.zbMATHMathSciNetCrossRefGoogle Scholar