Computing the largest empty rectangle
 B. Chazelle,
 R. L. Drysdale III,
 D. T. Lee
 … show all 3 hide
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 largestarea rectangular piece which can be salvaged. A previously known result[13] takes O(N^{2}) worstcase 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 divideandconquer 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.
 Bentley, J. L. (1976) Divideandconquer algorithms for closest point problems in multidimensional space. Dept. Comput. Sci.. Univ. of North Carolina, Chapel Hill, NC.
 Bentley J. L. and D. Wood, "An optimal worst case algorithm for reporting intersections of rectangles," IEEE Trans. Comput., (July 1980), 571–577.
 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.
 Chazelle, B. M., R. L. Drysdale III, and D. T. Lee, "Computing the largest empty rectangle," Dept. Comput. Sci. Brown Univ. Tech. Rep. CS 8320, Sept. 1983.
 Dobkin, D. P., Drysdale, R. L., Guibas, L. J. (1983) Finding smallest polygons. Advances of Computing Research 1: pp. 181224
 Gowda, I. G., Kirkpatrick, D. G., Lee, D. T., Naamad, A. (1983) Dynamic Voronoi diagrams. IEEE Trans. Information Theory. IT29: pp. 724731
 Hwang, F. K. (1979) An O(n log n) algorithm for rectilinear minimal spanning trees. J. ACM 26: pp. 177182
 Kirkpatrick, D. G. (1983) Optimal search in planar subdivisions. SIAM J. Comput. 12: pp. 2835
 Knuth, D. E. (1968) The Art of Computer Programming, Vol. I: Fundamental Algorithms. AddisonWesley, Reading, Mass.
 Lee, D. T. (1980) Two dimensional Voronoi diagrams in the Lpmetric. J. ACM 27: pp. 604618
 Lee, D. T., Drysdale, R. L. (1981) Generalization of Voronoi diagrams in the plane. SIAM J. Comput. 10: pp. 7387
 Lee, D. T., Wong, C. K. (1980) Voronoi diagrams in L1(L∞)metrics with 2dimensional storage applications. SIAM J. Comput. 9: pp. 200211
 Naamad, A., W. L. Hsu, and D. T. Lee, "On maximum empty rectangle problem," Disc. Applied Math., to appear.
 Shamos, M. I., "Computational geometry," Ph. D. dissertation, Dept. Computer Sci., Yale Univ., 1978.
 Shamos, M. I. and D. Hoey, "Closestpoint problem," Proc. 16th IEEE Symp. on Foundations of Computer Science, (Oct. 1975), 151–162.
 Title
 Computing the largest empty rectangle
 Book Title
 STACS 84
 Book Subtitle
 Symposium of Theoretical Aspects of Computer Science Paris, 11–13, 1984
 Pages
 pp 4354
 Copyright
 1984
 DOI
 10.1007/3540129200_4
 Print ISBN
 9783540129202
 Online ISBN
 9783540388050
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 166
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 eBook Packages
 Editors
 Authors

 B. Chazelle ^{(1)}
 R. L. Drysdale III ^{(2)}
 D. T. Lee ^{(3)}
 Author Affiliations

 1. Dept. Computer Science, Brown University, USA
 2. Dept. Mathematics and Computer Science, Dartmouth College, USA
 3. Dept. Electrical Engineering/Computer Science, Northwestern University, USA
Continue reading...
To view the rest of this content please follow the download PDF link above.