Abstract
In a randomized incremental construction of the minimization diagram of a collection of n hyperplanes in ℝd, for d≥2, the hyperplanes are inserted one by one, in a random order, and the minimization diagram is updated after each insertion. We show that if we retain all the versions of the diagram, without removing any old feature that is now replaced by new features, the expected combinatorial complexity of the resulting overlay does not grow significantly. Specifically, this complexity is O(n ⌊d/2⌋log n), for d odd, and O(n ⌊d/2⌋), for d even. The bound is asymptotically tight in the worst case for d even, and we show that this is also the case for d=3. Several implications of this bound, mainly its relation to approximate halfspace range counting, are also discussed.
Article PDF
Similar content being viewed by others
References
Afshani, P., Chan, T.M.: On approximate range counting and depth. Discrete Comput. Geom. 42, 3–21 (2009)
Agarwal, P.K., Erickson, J., Guibas, L.: Kinetic binary space partitions for intersecting segments and disjoint triangles. In: Proc. 9th Annu. ACM-SIAM Sympos. Discrete Algo, pp. 107–116 (1998)
Agarwal, P.K., Guibas, L.J., Murali, T.M., Vitter, J.S.: Cylindrical static and kinetic binary space partitions. Comput. Geom. Theory Appl. 16, 103–127 (2000)
Aronov, B., Har-Peled, S.: On approximating the depth and related problems. SIAM J. Comput. 38, 899–921 (2008)
Aronov, B., Har-Peled, S., Sharir, M.: On approximate halfspace range counting and relative epsilon approximations. In: Proc. 23rd Annu. ACM Sympos. on Comput. Geom., pp. 327–336 (2007)
Aronov, B., Sharir, M.: Approximate halfspace range counting. SIAM J. Comput. 39, 2704–2725 (2010)
de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 2nd edn. Springer, Heidelberg (2000)
Boissonnat, J.-D., Yvinec, M.: Algorithmic Geometry (English edn.). Cambridge University Press, Cambridge (1998)
Clarkson, K., Shor, P.: Applications of random sampling in computational geometry, II. Discrete Comput. Geom. 4, 387–421 (1989)
Cohen, E.: Size-estimation framework with applications to transitive closure and reachability. J. Comput. Syst. Sci. 55, 441–453 (1997)
Edelsbrunner, H., Seidel, R.: Voronoi diagrams and arrangements. Discrete Comput. Geom. 1, 25–44 (1986)
Guibas, L., Knuth, D.E., Sharir, M.: Randomized incremental construction of Voronoi and Delaunay diagrams. Algorithmica 7, 381–413 (1992)
Har-Peled, S.: Constructing planar cuttings in theory and practice. SIAM J. Comput. 29, 2016–2039 (2000)
Kaplan, H., Ramos, E., Sharir, M.: Range minima queries with respect to a random permutation, and approximate range counting. Discrete Comput. Geom. (in press); published online November 11, 2010. doi:10.1007/s00454-010-9308-6
Kaplan, H., Sharir, M.: Randomized incremental constructions of three-dimensional convex hulls and planar Voronoi diagrams, and approximate range counting. In: Proc. 17th Annu. ACM-SIAM Sympos. Discrete Algo, pp. 484–493 (2006)
Sharir, M., Agarwal, P.K.: Davenport–Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York (1995)
Ziegler, G.: Lectures on Polytopes, Graduate Texts in Math., vol. 152. Springer, New York (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Work by Haim Kaplan was partially supported by Grant 975/06 from the Israel Science Foundation (ISF) and by Grant 2006/204 from the U.S.–Israel Binational Science Foundation. Work by Micha Sharir was partially supported by NSF Grants CCR-05-14079 and CCF-08-30272, by Grant 2006/194 from the U.S.–Israel Binational Science Foundation, by Grants 155/05 and 338/09 from the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.
Rights and permissions
About this article
Cite this article
Kaplan, H., Ramos, E. & Sharir, M. The Overlay of Minimization Diagrams in a Randomized Incremental Construction. Discrete Comput Geom 45, 371–382 (2011). https://doi.org/10.1007/s00454-010-9324-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-010-9324-6