# Efficient algorithms for identifying all maximal isothetic empty rectangles in VLSI layout design

## Abstract

In this paper, we consider the following problem of computational geometry which has direct applications to VLSI layout design : given a set of n isothetic solid rectangles on a rectangular floor, identify all maximal-empty-rectangles (MER's). A tighter upper bound on the number of MER's is derived. A new algorithm based on interval trees for identifying all MER's is then presented which runs in *O*(nlogn+* R*) time in the worst case and in

*O*(nlogn) time in the average case, where

*denotes the number of MER's. The space complexity of the algorithm is*

**R***O*(n). Finally, we explore the problem of recognizing the

*maximum*(area)- empty- rectangle without explicitly generating all MER's. Our analysis shows that, on an average, around 70% of MER's need not be examined in order to locate the maximum. The proposed algorithm can readily be tailored to solve the MER problem in an ensemble of points as well as within an isothetic polygon.

## Keywords

Computational geometry VLSI layout placement geometric algorithms complexity interval trees## Preview

Unable to display preview. Download preview PDF.

## References

- 1.D. T. Lee and F. P. Preparata, “Computational geometry — a survey”,
*IEEE Trans. on Computers*, Vol C-33, pp. 1072–1101, Dec. 1984.Google Scholar - 2.A. Asano, M. Sato and T. Ohtsuki, “Computational geometry algorithms”, in Layout Design & Verification,
*Advances in CAD for VLSI*, Vol-4, (Ed. T. Ohtsuki), pp. 295–347, North Holland, 1986.Google Scholar - 3.T. G. Szymanski and C. J. Van Wyk, “Layout analysis and verification”, in
*Physical Design Automation of VLSI Systems*, Ed. B. T. Preas and M. J. Lorenzette, Benjamin Cummings Publishing Company, Inc., 1988.Google Scholar - 4.F. P. Preparata, and M. L. Shamos, Computational Geometry: An Introduction,
*Springer Verlag*, NY, 1985.Google Scholar - 5.Kurt Mehlhorn, Multidimensional Searching and Computational Geometry,
*Springer Verlag*, NY, 1984.Google Scholar - 6.D. Wood, “An isothetic view of computational geometry”, in
*Computational Geometry*(G. T. Toussaint, Edited), North Holland, 1985.Google Scholar - 7.J.L. Bentley and D. Wood, “An optimal worst case algorithm for reporting intersection of rectangles”,
*IEEE Trans. on Computers*, Vol C-29, pp. 571–577, July 1980.Google Scholar - 8.J.L. Bentley and T.A. Ottman, “Algorithms for reporting and counting geometric intersections”,
*IEEE Trans. on Computers*, Vol C-28, pp. 643–647, Sept. 1979.Google Scholar - 9.M. I. Shamos and D. Hoey, “Geometric intersection problems”, Proc. 17th Annual
*IEEE Symp. on Foundations of Computer Science*, pp. 208–215, Oct. 1976.Google Scholar - 10.K. Yoshida, “Layout verification”, in
*Layout Design and Verification*, Ed. T. Ohtsuki, Elsevier Science Publication B.V.(North Holland), 1986.Google Scholar - 11.A. Naamad, D. T. Lee and W. L. Hsu, “On the maximum empty rectangle problem”,
*Discrete Appl. Math.*,8,1984, pp. 267–277.Google Scholar - 12.S. Wimer and I. Koren, “Analysis of strategies for constructive general block placement”,
*IEEE Trans on CAD*, Vol CAD-7, No. 3, pp. 371–377, 1988.Google Scholar - 13.B. Chazelle, R.L. Drysdale and D.T. Lee, “Computing the largest empty rectangle “,
*SIAM J. Comput.*, Vol. 15, No. 1, pp. 300–315, February 1986.Google Scholar - 14.M. J. Atallah and G. N. Frederickson, “A note on finding a maximum empty rectangle”,
*Discrete Applied Math.*, Vol. 13, pp. 87–91, 1986.Google Scholar - 15.A. Aggarwal and S. Suri, “Fast algorithms for computing the largest empty rectangle”, Proc. 3rd Annual
*ACM Symp. on Computational Geometry*, pp. 278–290, 1987.Google Scholar - 16.M. Orlowski, “A new algorithm for the largest empty rectangle problem”,
*Algorithmica*, Vol. 5, pp. 65–73, 1990.Google Scholar - 17.T. Dey, “Two problems in computational geometry”,
*M.E. Thesis*, Indian Institute of Science, Bangalore, India, 1987.Google Scholar - 18.M.J. Atallah and S.R. Kosaraju, “An efficient algorithm for maxdominance, with applications”,
*Algorithmica*, Vol. 4, pp. 221–236, 1989.Google Scholar - 19.J.K. Ousterhout, “Corner stitching: data structuring technique for VLSI layout tools”,
*IEEE Trans. on CAD*, Vol CAD-3No. 1, 1984, pp. 87–100.Google Scholar - 20.D. E. Knuth, The Art of Computer Programming, Vol. 3, Sorting and Searching,
*Addison Wesley*, 1973.Google Scholar - 21.F. Harary, Graph Theory,
*Addison Wesley*, 1972.Google Scholar