Optimal Binary Representation of Mosaic Floorplans and Baxter Permutations

  • Bryan He
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7285)


A floorplan is a rectangle subdivided into smaller rectangular blocks by horizontal and vertical line segments. Two floorplans are considered equivalent if and only if there is a bijection between the blocks in the two floorplans such that the corresponding blocks have the same horizontal and vertical boundaries. Mosaic floorplans use the same objects as floorplans but use an alternative definition of equivalence. Two mosaic floorplans are considered equivalent if and only if they can be converted into equivalent floorplans by sliding the line segments that divide the blocks. The Quarter-State Sequence method of representing mosaic floorplans uses 4n bits, where n is the number of blocks. This paper introduces a method of representing an n-block mosaic floorplan with a (3n − 3)-bit binary string. It has been proven that the shortest possible binary string representation of a mosaic floorplan has a length of (3n − o(n)) bits. Therefore, the representation presented in this paper is asymptotically optimal. Baxter permutations are a set of permutations defined by prohibited subsequences. There exists a bijection between mosaic floorplans and Baxter permutations. As a result, the methods introduced in this paper also create an optimal binary string representation of Baxter permutations.


Binary Representation Mosaic Floorplan Baxter Permutation 


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  1. 1.
    Ackerman, E., Barequet, G., Pinter, R.Y.: A bijection between permutations and floorplans, and its applications. Discrete Applied Mathematics 154, 1674–1684 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Amano, K., Nakano, S., Yamanaka, K.: On the number of rectangular drawings: Exact counting and lower and upper bounds. IPSJ SIG Notes 2007-AL-115-5C, 33–40 (2007)Google Scholar
  3. 3.
    Baxter, G.: On fixed points of the composite of commuting functions. Proceedings American Mathematics Society 15, 851–855 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bonichon, N., Bousquet-Mélou, M., Fusy, É.: Baxter permutations and plane bipolar orientations. Séminaire Lotharingien de Combinatoire 61A (2010)Google Scholar
  5. 5.
    Canary, H.: Aztec diamonds and baxter permutations. The Electronic Journal of Combinatorics 17 (2010)Google Scholar
  6. 6.
    Dulucq, S., Guibert, O.: Baxter permutations. Discrete Mathematics 180, 143–156 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Fujimaki, R., Inoue, Y., Takahashi, T.: An asymptotic estimate of the numbers of rectangular drawings or floorplans. In: Proceedings 2009 IEEE International Symposium on Circuits and Systems, pp. 856–859 (2009)Google Scholar
  8. 8.
    Giraudo, S.: Algebraic and combinatorial structures on baxter permutations. Discrete Mathematics and Theoretical Computer Science, DMTCS (2011)Google Scholar
  9. 9.
    Hong, X., Huang, G., Cai, Y., Gu, J., Dong, S., Cheng, C.-K., Gu, J.: Corner-block list: An effective and efficient topological representation of non-slicing floorplan. In: Proceedings of the International Conference on Computer Aided Design (ICCAD 2000), pp. 8–12 (2000)Google Scholar
  10. 10.
    Lengauer, T.: Combinatorial Algorithms for Integrated Circuit Layout. John Wiley & Sons (1990)Google Scholar
  11. 11.
    Murata, H., Fujiyoshi, K.: Rectangle-packing-based module placement. In: Proceedings of the International Conference on Computer Aided Design (ICCAD 1995), pp. 472–479 (1995)Google Scholar
  12. 12.
    Nakano, S.: Enumerating Floorplans with n Rooms. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 107–115. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Sakanushi, K., Kajitani, Y., Mehta, D.P.: The quarter-state-sequence floorplan representation. IEEE Transactions on Circuits and Systems - I: Fundamental Theory and Applications 50(3), 376–386 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Shen, Z.C., Chu, C.C.N.: Bounds on the number of slicing, mosaic, and general floorplans. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 22(10), 1354–1361 (2003)CrossRefGoogle Scholar
  15. 15.
    Speckmann, B., van Kreveld, M., Florisson, S.: A linear programming approach to rectangular cartograms. In: Proceedings 12th International Symposium on Spatial Data Handling (SDH), pp. 527–546 (2006)Google Scholar
  16. 16.
    Takahashi, T., Fujimaki, R., Inoue, Y.: A (4n − 4)-Bit Representation of a Rectangular Drawing or Floorplan. In: Ngo, H.Q. (ed.) COCOON 2009. LNCS, vol. 5609, pp. 47–55. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    van Kreveld, M., Speckmann, B.: On rectangular cartograms. Computational Geometry: Theory and Applications 37(3), 175–187 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Yamanaka, K., Nakano, S.: Coding floorplans with fewer bits. IEICE Transactions Fundamentals E89(5), 1181–1185 (2006)CrossRefGoogle Scholar
  19. 19.
    Yamanaka, K., Nakano, S.: A Compact Encoding of Rectangular Drawings with Efficient Query Supports. In: Kao, M.-Y., Li, X.-Y. (eds.) AAIM 2007. LNCS, vol. 4508, pp. 68–81. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  20. 20.
    Yao, B., Chen, H., Cheng, C.-K., Graham, R.: Floorplan representation: Complexity and connections. ACM Transactions on Design Automation of Electronic Systems 8(1), 55–80 (2003)CrossRefGoogle Scholar
  21. 21.
    Young, E.F.Y., Chu, C.C.N., Shen, Z.C.: Twin binary sequences: A nonredundant representation for general nonslicing floorplan. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 22(4), 457–469 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Bryan He
    • 1
  1. 1.Department of Computer ScienceCalifornia Institute of TechnologyPasadenaUnited States of America

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