Maximal and Convex Layers of Random Point Sets

  • Meng HeEmail author
  • Cuong P. Nguyen
  • Norbert Zeh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


We study two problems concerning the maximal and convex layers of a point set in d dimensions. The first is the average-case complexity of computing the first k layers of a point set drawn from a uniform or component-independent (CI) distribution. We show that, for \(d \in \{2,3\}\), the first \(n^{1/d-\epsilon }\) maximal layers can be computed using \(dn + o(n)\) scalar comparisons with high probability. For \(d \ge 4\), the first \(n^{1/2d-\epsilon }\) maximal layers can be computed within this bound with high probability. The first \(n^{1/d-\epsilon }\) convex layers in 2D, the first \(n^{1/2d-\epsilon }\) convex layers in 3D, and the first \(n^{1/(d^2+2)}\) convex layers in \(d \ge 4\) dimensions can be computed using \(2dn + o(n)\) scalar comparisons with high probability. Since the expected number of maximal layers in 2D is \(2\sqrt{n}\), our result for 2D maximal layers shows that it takes \(dn + o(n)\) scalar comparisons to compute a \(1/n^\epsilon \)-fraction of all layers in the average case. The second problem is bounding the expected size of the kth maximal and convex layer. We show that the kth maximal and convex layer of a point set drawn from a continuous CI distribution in d dimensions has expected size \(O(k^d \log ^{d-1} (n/k^d))\).


Maximal layers Skyline Convex layers Average-case analysis 


  1. 1.
    Bentley, J.L., Clarkson, K.L., Levine, D.B.: Fast linear expected-time algorithms for computing maxima and convex hulls. Algorithmica 9(2), 168–183 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bentley, J.L., Kung, H.T., Schkolnick, M., Thompson, C.D.: On the average number of maxima in a set of vectors and applications. J. ACM 25(4), 536–543 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008). CrossRefzbMATHGoogle Scholar
  4. 4.
    Blunck, H., Vahrenhold, J.: In-place algorithms for computing (layers of) maxima. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 363–374. Springer, Heidelberg (2006). CrossRefGoogle Scholar
  5. 5.
    Börzsönyi, S., Kossmann, D., Stocker, K.: The skyline operator. In: Proceedings of the 17th International Conference on Data Engineering, pp. 421–430 (2001)Google Scholar
  6. 6.
    Buchsbaum, A.L., Goodrich, M.T.: Three-dimensional layers of maxima. Algorithmica 39(4), 275–286 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chazelle, B.: An optimal convex hull algorithm in any fixed dimension. Discret. Comput. Geom. 10(4), 377–409 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chazelle, B.: On the convex layers of a planar set. IEEE Trans. Inf. Theor. 31(4), 509–517 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dalal, K.: Counting the onion. Random Struct. Algorithms 24(2), 155–165 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Frieze, A.: On the length of the longest monotone subsequence in a random permutation. Ann. Appl. Probab. 1(2), 301–305 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Golin, M.J.: A provably fast linear-expected-time maxima-finding algorithm. Algorithmica 11(6), 501–524 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kung, H.T., Luccio, F., Preparata, F.P.: On finding the maxima of a set of vectors. J. ACM 22(4), 469–476 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Matoušek, J., Plecháč, P.: On functional separately convex hulls. Discret. Comput. Geom. 19(1), 105–130 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, New York (1995)CrossRefzbMATHGoogle Scholar
  15. 15.
    Nielsen, F.: Output-sensitive peeling of convex and maximal layers. Inf. Process. Lett. 59(5), 255–259 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Okabe, A., Boots, B., Sugihara, K., Chiu, S., Kendall, D.G.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, Hoboken (2008)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Computer ScienceDalhousie UniversityHalifaxCanada

Personalised recommendations