Unions of Onions: Preprocessing Imprecise Points for Fast Onion Layer Decomposition

  • Maarten Löffler
  • Wolfgang Mulzer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)

Abstract

Let \(\mathcal{D}\) be a set of n pairwise disjoint unit disks in the plane. We describe how to build a data structure for \(\mathcal{D}\) so that for any point set P containing exactly one point from each disk, we can quickly find the onion decomposition (convex layers) of P.

Our data structure can be built in O(n logn) time and has linear size. Given P, we can find its onion decomposition in O(n logk) time, where k is the number of layers. We also provide a matching lower bound.

Our solution is based on a recursive space decomposition, combined with a fast algorithm to compute the union of two disjoint onion decompositions.

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References

  1. 1.
    Ailon, N., Chazelle, B., Clarkson, K.L., Liu, D., Mulzer, W., Seshadhri, C.: Self-improving algorithms. SIAM J. Comput. 40(2), 350–375 (2011)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Alon, N., Katchalski, M., Pulleyblank, W.R.: Cutting disjoint disks by straight lines. Discrete Comput. Geom. 4(3), 239–243 (1989)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bruce, R., Hoffmann, M., Krizanc, D., Raman, R.: Efficient update strategies for geometric computing with uncertainty. Theory of Computing Systems 38(4), 411–423 (2005)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Buchin, K., Löffler, M., Morin, P., Mulzer, W.: Preprocessing imprecise points for Delaunay triangulation: simplified and extended. Algorithmica 61(3), 675–693 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chan, T.M.: A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries. J. ACM 57(3), Art. 16, 15p. (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chazelle, B.: On the convex layers of a planar set. IEEE Trans. Inform. Theory 31(4), 509–517 (1985)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Chazelle, B., Guibas, L.J., Lee, D.T.: The power of geometric duality. BIT 25(1), 76–90 (1985)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Devillers, O.: Delaunay triangulation of imprecise points: preprocess and actually get a fast query time. J. Comput. Geom. 2(1), 30–45 (2011)MathSciNetGoogle Scholar
  9. 9.
    Eddy, W.F.: Convex hull peeling. In: Proc. 5th Symp. Comp. Statistics (COMPSTAT), pp. 42–47 (1982)Google Scholar
  10. 10.
    Ezra, E., Mulzer, W.: Convex hull of points lying on lines in o(nlogn) time after preprocessing. Comput. Geom. 46(4), 417–434 (2013)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Franciosa, P.G., Gaibisso, C., Gambosi, G., Talamo, M.: A convex hull algorithm for points with approximately known positions. Internat. J. Comput. Geom. Appl. 4(2), 153–163 (1994)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Held, M., Mitchell, J.S.B.: Triangulating input-constrained planar point sets. Inform. Process. Lett. 109(1), 54–56 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hoffmann, M., Erlebach, T., Krizanc, D., Mihalák, M., Raman, R.: Computing minimum spanning trees with uncertainty. In: Proc. 25th Sympos. Theoret. Aspects Comput. Sci. (STACS), pp. 277–288 (2008)Google Scholar
  14. 14.
    Huber, P.J.: Robust statistics: A review. Ann. Math. Statist. 43, 1041–1067 (1972)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Kirkpatrick, D., Snoeyink, J.: Computing common tangents without a separating line. In: Sack, J.-R., Akl, S.G., Dehne, F., Santoro, N. (eds.) WADS 1995. LNCS, vol. 955, pp. 183–193. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  16. 16.
    van Kreveld, M., Löffler, M., Mitchell, J.S.B.: Preprocessing imprecise points and splitting triangulations. SIAM J. Comput. 39(7), 2990–3000 (2010)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Löffler, M., Snoeyink, J.: Delaunay triangulation of imprecise points in linear time after preprocessing. Comput. Geom. 43(3), 234–242 (2010)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Matoušek, J.: Efficient partition trees. Discrete Comput. Geom. 8(3), 315–334 (1992)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Motwani, R., Raghavan, P.: Randomized algorithms. Cambridge University Press (1995)Google Scholar
  20. 20.
    Nielsen, F.: Output-sensitive peeling of convex and maximal layers. Inform. Process. Lett. 59, 255–259 (1996)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Overmars, M.H., van Leeuwen, J.: Maintenance of configurations in the plane. J. Comput. System Sci. 23(2), 166–204 (1981)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Suk, T., Flusser, J.: Convex layers: A new tool for recognition of projectively deformed point sets. In: Solina, F., Leonardis, A. (eds.) CAIP 1999. LNCS, vol. 1689, pp. 454–461. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  23. 23.
    Tseng, K.-C.R., Kirkpatrick, D.: Input-thrifty extrema testing. In: Asano, T., Nakano, S.-i., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 554–563. Springer, Heidelberg (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Maarten Löffler
    • 1
  • Wolfgang Mulzer
    • 2
  1. 1.Department of Information and Computing SciencesUniversiteit UtrechtThe Netherlands
  2. 2.Institut für InformatikFreie Universität BerlinGermany

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