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A Tight Lower Bound for Computing the Diameter of a 3D Convex Polytope

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Abstract

The diameter of a set P of n points in ℝd is the maximum Euclidean distance between any two points in P. If P is the vertex set of a 3-dimensional convex polytope, and if the combinatorial structure of this polytope is given, we prove that, in the worst case, deciding whether the diameter of P is smaller than 1 requires Ω(nlog n) time in the algebraic computation tree model. It shows that the O(nlog n) time algorithm of Ramos for computing the diameter of a point set in ℝ3 is optimal for computing the diameter of a 3-polytope. We also give a linear time reduction from Hopcroft’s problem of finding an incidence between points and lines in ℝ2 to the diameter problem for a point set in ℝ7.

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References

  1. Ben-Or, M.: Lower bounds for algebraic computation trees. In: Proceedings of the 15th Annual ACM Symposium on Theory of Computing, pp. 80–86 (1983)

  2. Bespamyatnikh, S.: An efficient algorithm for the three-dimensional diameter problem. Discrete Comput. Geom. 25(2), 235–255 (2000)

    Article  MathSciNet  Google Scholar 

  3. Bürgisser, P., Clausen, M., Shokrollahi, M.: Algebraic Complexity Theory. Springer, Berlin (1997)

    MATH  Google Scholar 

  4. Bürgisser, P., Karpinski, M., Lickteig, T.: On randomized semi-algebraic test complexity. J. Complex. 9(2), 231–251 (1993)

    Article  MATH  Google Scholar 

  5. Chan, T.: Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus. Int. J. Comput. Geom. Appl. 12(1–2), 67–85 (2002)

    Article  MATH  Google Scholar 

  6. Chazelle, B., Devillers, O., Hurtado, F., Mora, M., Sacristán, V., Teillaud, M.: Splitting a Delaunay triangulation in linear time. Algorithmica 34(1), 39–46 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Clarkson, K., Shor, P.: Applications of random sampling in computational geometry, II. Discrete Comput. Geom. 4, 387–421 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 2nd edn. Springer, Berlin (2000)

    MATH  Google Scholar 

  9. Erickson, J.: On the relative complexities of some geometric problems. In: Proceedings of the 7th Canadian Conference on Computational Geometry, pp. 85–90 (1995)

  10. Erickson, J.: New lower bounds for Hopcroft’s problem. Discrete Comput. Geom. 16, 389–418 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Grigoriev, D.: Randomized complexity lower bounds. In: Proceedings of the 30th ACM Symposium on Theory of Computing, pp. 219–223 (1998)

  12. Har-Peled, S.: A practical approach for computing the diameter of a point set. In: Proceedings of the Seventeenth Annual Symposium on Computational Geometry, New York, 3–5 June 2001, pp. 177–186. ACM, New York (2001)

    Chapter  Google Scholar 

  13. Malandain, G., Boissonnat, J.-D.: Computing the diameter of a point set. Int. J. Comput. Geom. Appl. 12(6), 489–510 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Matoušek, J.: Range searching with efficient hierarchical cuttings. Discrete Comput. Geom. 10(2), 157–182 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Matoušek, J., Schwarzkopf, O.: On ray shooting in convex polytopes. Discrete Comput. Geom. 10(2), 215–232 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Preparata, F., Shamos, I.: Computational Geometry: An Introduction, 2nd edn. Texts and Monographs in Computer Science. Springer, New York (1985)

    Google Scholar 

  17. Ramos, E.: An optimal deterministic algorithm for computing the diameter of a three-dimensional point set. Discrete Comput. Geom. 26, 233–244 (2001)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire PRiSM, Université de Versailles Saint-Quentin-en-Yvelines, 45 avenue des États-Unis, 78035, Versailles cedex, France

    Hervé Fournier

  2. INRA, UR341 Mathématiques et Informatique Appliquées, Domaine de Vilvert, 78352, Jouy-en-Josas cedex, France

    Antoine Vigneron

Authors
  1. Hervé Fournier
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  2. Antoine Vigneron
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Correspondence to Hervé Fournier.

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Fournier, H., Vigneron, A. A Tight Lower Bound for Computing the Diameter of a 3D Convex Polytope. Algorithmica 49, 245–257 (2007). https://doi.org/10.1007/s00453-007-9010-0

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  • DOI: https://doi.org/10.1007/s00453-007-9010-0

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