# Computing the largest empty rectangle

## 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(N^{2}) worst-case and O(Nlog^{2}N) expected time. This paper presents an O(N log^{3}N) 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.

## References

- 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.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.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.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.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.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.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.Kirkpatrick, D. G., "Optimal search in planar subdivisions," SIAM J. Comput., 12 (1983), 28–35.Google Scholar
- 9.Knuth, D. E., The Art of Computer Programming, Vol. I: Fundamental Algorithms, Addison-Wesley, Reading, Mass., 1968.Google Scholar
- 10.Lee, D. T., "Two dimensional Voronoi diagrams in the L
_{p}-metric," J. ACM, 27,4 (Oct. 1980), 604–618.Google Scholar - 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.Lee, D. T. and C. K. Wong, "Voronoi diagrams in L
_{1}-(L_{∞}-)metrics with 2-dimensional storage applications," SIAM J. Comput. 9 (1980), 200–211.Google Scholar - 13.Naamad, A., W. L. Hsu, and D. T. Lee, "On maximum empty rectangle problem," Disc. Applied Math., to appear.Google Scholar
- 14.Shamos, M. I., "Computational geometry," Ph. D. dissertation, Dept. Computer Sci., Yale Univ., 1978.Google Scholar
- 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