Computing the largest empty rectangle

  • B. Chazelle
  • R. L. DrysdaleIII
  • D. T. Lee
Contibuted Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 166)

Abstract

We consider the following problem: Given a rectangle containing N points, find the largest area subrectangle with sides parallel to those of the original rectangle which contains none of the given points. If the rectangle is a piece of fabric or sheet metal and the points are flaws, this problem is finding the largest-area rectangular piece which can be salvaged. A previously known result[13] takes O(N2) worst-case and O(Nlog2N) expected time. This paper presents an O(N log3N) time, O(N log N) space algorithm to solve this problem. It uses a divide-and-conquer approach similar to the ones used by Strong and Bentley[1] and introduces a new notion of Voronoi diagram along with a method for efficient computation of certain functions over paths of a tree.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bentley, J. L., "Divide-and-conquer algorithms for closest point problems in multidimensional space," Ph. D. thesis, Dept. Comput. Sci., Univ. of North Carolina, Chapel Hill, NC., 1976.Google Scholar
  2. 2.
    Bentley J. L. and D. Wood, "An optimal worst case algorithm for reporting intersections of rectangles," IEEE Trans. Comput., (July 1980), 571–577.Google Scholar
  3. 3.
    Boyce, J. E., D. P. Dobkin, R. L. Drysdale III, and L. J. Guibas, "Finding extremal polygons," Proc. ACM Symp. Theory of Comput., (May 1982), 282–289.Google Scholar
  4. 4.
    Chazelle, B. M., R. L. Drysdale III, and D. T. Lee, "Computing the largest empty rectangle," Dept. Comput. Sci. Brown Univ. Tech. Rep. CS 83-20, Sept. 1983.Google Scholar
  5. 5.
    Dobkin, D. P., R. L. Drysdale III, and L. J. Guibas, "Finding smallest polygons," in Advances of Computing Research, Vol. 1, (F. P. Preparata, ed.), JAI Press Inc. (1983), 181–224.Google Scholar
  6. 6.
    Gowda, I. G., D. G. Kirkpatrick, D. T. Lee and A. Naamad, "Dynamic Voronoi diagrams," IEEE Trans. Information Theory., IT-29,5 (Sept. 1983), 724–731.Google Scholar
  7. 7.
    Hwang, F. K., "An O(n log n) algorithm for rectilinear minimal spanning trees," J. ACM, 26,2 (April 1979), 177–182.Google Scholar
  8. 8.
    Kirkpatrick, D. G., "Optimal search in planar subdivisions," SIAM J. Comput., 12 (1983), 28–35.Google Scholar
  9. 9.
    Knuth, D. E., The Art of Computer Programming, Vol. I: Fundamental Algorithms, Addison-Wesley, Reading, Mass., 1968.Google Scholar
  10. 10.
    Lee, D. T., "Two dimensional Voronoi diagrams in the Lp-metric," J. ACM, 27,4 (Oct. 1980), 604–618.Google Scholar
  11. 11.
    Lee, D. T. and R. L. Drysdale III, "Generalization of Voronoi diagrams in the plane," SIAM J. Comput. 10,1 (Feb. 1981), 73–87.Google Scholar
  12. 12.
    Lee, D. T. and C. K. Wong, "Voronoi diagrams in L1-(L-)metrics with 2-dimensional storage applications," SIAM J. Comput. 9 (1980), 200–211.Google Scholar
  13. 13.
    Naamad, A., W. L. Hsu, and D. T. Lee, "On maximum empty rectangle problem," Disc. Applied Math., to appear.Google Scholar
  14. 14.
    Shamos, M. I., "Computational geometry," Ph. D. dissertation, Dept. Computer Sci., Yale Univ., 1978.Google Scholar
  15. 15.
    Shamos, M. I. and D. Hoey, "Closest-point problem," Proc. 16th IEEE Symp. on Foundations of Computer Science, (Oct. 1975), 151–162.Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • B. Chazelle
    • 1
  • R. L. DrysdaleIII
    • 2
  • D. T. Lee
    • 3
  1. 1.Dept. Computer ScienceBrown UniversityUSA
  2. 2.Dept. Mathematics and Computer ScienceDartmouth CollegeUSA
  3. 3.Dept. Electrical Engineering/Computer ScienceNorthwestern UniversityUSA

Personalised recommendations