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Abstract

We show that the expected number of maximal empty axis-parallel boxes amidst n random points in the unit hypercube [0,1] d in ℝ d is \((1 \pm o(1))\allowbreak \frac{(2d-2)!}{(d-1)!} \, n \ln^{d-1} n\), if d is fixed. This estimate is relevant for analyzing the performance of any exact algorithm for computing the largest empty axis-parallel box amidst n points in a given axis-parallel box R, that proceeds by examining all maximal empty boxes. While the Θ(n log d − 1 n) bound has been claimed for d = 3 for more than ten years by now, and has been recently used for all d ≥ 3 in the analysis of algorithms for computing the largest empty box, it did not rely on a valid proof. Here we present the first valid proof for the Θ(n log d − 1 n) bound; only an O(n log d − 1 n) bound was previously proved.

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References

  1. 1.
    Aggarwal, A., Suri, S.: Fast algorithms for computing the largest empty rectangle. In: Proceedings of the 3rd Annual Symposium on Computational Geometry, pp. 278–290 (1987)Google Scholar
  2. 2.
    Atallah, M., Frederickson, G.: A note on finding the maximum empty rectangle. Discrete Applied Mathematics 13, 87–91 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Atallah, M., Kosaraju, S.R.: An efficient algorithm for maxdominance, with applications. Algorithmica 4, 221–236 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Backer, J., Keil, M.: The bichromatic rectangle problem in high dimensions. In: Proceedings of the 21st Canadian Conference on Computational Geometry, pp. 157–160 (2009)Google Scholar
  5. 5.
    Backer, J., Keil, J.M.: The Mono- and Bichromatic Empty Rectangle and Square Problems in All Dimensions. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 14–25. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Bentley, J.L., Kung, H.T., Schkolnick, M., Thompson, C.D.: On the average number of maxima in a set of vectors and applications. Journal of the ACM 25, 536–543 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chazelle, B., Drysdale, R., Lee, D.T.: Computing the largest empty rectangle. SIAM Journal on Computing 15, 300–315 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chuan-Chong, C., Khee-Meng, K.: Principles and Techniques in Combinatorics. World Scientific, Singapore (1996)Google Scholar
  9. 9.
    Datta, A.: Efficient algorithms for the largest empty rectangle problem. Information Sciences 64, 121–141 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Datta, A., Soundaralakshmi, S.: An efficient algorithm for computing the maximum empty rectangle in three dimensions. Information Sciences 128, 43–65 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dumitrescu, A., Jiang, M.: On the largest empty axis-parallel box amidst n points. Algorithmica (2012), doi:10.1007/s00453-012-9635-5Google Scholar
  12. 12.
    Edmonds, J., Gryz, J., Liang, D., Miller, R.: Mining for empty spaces in large data sets. Theoretical Computer Science 296, 435–452 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Giannopoulos, P., Knauer, C., Wahlström, M., Werner, D.: Hardness of discrepancy computation and ε-net verification in high dimension. Journal of Complexity (2011), doi:10.1016/j.jco.2011.09.001Google Scholar
  14. 14.
    Kaplan, H., Rubin, N., Sharir, M., Verbin, E.: Efficient colored orthogonal range counting. SIAM Journal on Computing 38, 982–1011 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Klein, R.: Direct dominance of points. International Journal of Computer Mathematics 19, 225–244 (1986)zbMATHCrossRefGoogle Scholar
  16. 16.
    Kudryavtsev, L.D.: The method of undetermined coefficients. In: Hazewinkel, M. (ed.) Encyclopaedia of Mathematics. Springer (2001)Google Scholar
  17. 17.
    Kurosh, A.: Higher Algebra. Mir Publishers, Moscow (1975)Google Scholar
  18. 18.
    Marx, D.: Parameterized complexity and approximation algorithms. Computer Journal 51, 60–78 (2008)CrossRefGoogle Scholar
  19. 19.
    McKenna, M., O’Rourke, J., Suri, S.: Finding the largest rectangle in an orthogonal polygon. In: Proceedings of the 23rd Annual Allerton Conference on Communication, Control and Computing, Urbana-Champaign, Illinois (October 1985)Google Scholar
  20. 20.
    Naamad, A., Lee, D.T., Hsu, W.-L.: On the maximum empty rectangle problem. Discrete Applied Mathematics 8, 267–277 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Orlowski, M.: A new algorithm for the largest empty rectangle problem. Algorithmica 5, 65–73 (1990)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Minghui Jiang
    • 2
  1. 1.Department of Computer ScienceUniversity of Wisconsin–MilwaukeeUSA
  2. 2.Department of Computer ScienceUtah State UniversityLoganUSA

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