Skip to main content

Optimal Binary Representation of Mosaic Floorplans and Baxter Permutations

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7285)

Abstract

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.

Keywords

  • Binary Representation
  • Mosaic Floorplan
  • Baxter Permutation

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ackerman, E., Barequet, G., Pinter, R.Y.: A bijection between permutations and floorplans, and its applications. Discrete Applied Mathematics 154, 1674–1684 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  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. Baxter, G.: On fixed points of the composite of commuting functions. Proceedings American Mathematics Society 15, 851–855 (1964)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Bonichon, N., Bousquet-Mélou, M., Fusy, É.: Baxter permutations and plane bipolar orientations. Séminaire Lotharingien de Combinatoire 61A (2010)

    Google Scholar 

  5. Canary, H.: Aztec diamonds and baxter permutations. The Electronic Journal of Combinatorics 17 (2010)

    Google Scholar 

  6. Dulucq, S., Guibert, O.: Baxter permutations. Discrete Mathematics 180, 143–156 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  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. Giraudo, S.: Algebraic and combinatorial structures on baxter permutations. Discrete Mathematics and Theoretical Computer Science, DMTCS (2011)

    Google Scholar 

  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. Lengauer, T.: Combinatorial Algorithms for Integrated Circuit Layout. John Wiley & Sons (1990)

    Google Scholar 

  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. Nakano, S.: Enumerating Floorplans with n Rooms. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 107–115. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  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)

    CrossRef  MathSciNet  Google Scholar 

  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)

    CrossRef  Google Scholar 

  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. 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)

    CrossRef  Google Scholar 

  17. van Kreveld, M., Speckmann, B.: On rectangular cartograms. Computational Geometry: Theory and Applications 37(3), 175–187 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. Yamanaka, K., Nakano, S.: Coding floorplans with fewer bits. IEICE Transactions Fundamentals E89(5), 1181–1185 (2006)

    CrossRef  Google Scholar 

  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)

    CrossRef  Google Scholar 

  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)

    CrossRef  Google Scholar 

  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)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

He, B. (2012). Optimal Binary Representation of Mosaic Floorplans and Baxter Permutations. In: Snoeyink, J., Lu, P., Su, K., Wang, L. (eds) Frontiers in Algorithmics and Algorithmic Aspects in Information and Management. Lecture Notes in Computer Science, vol 7285. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29700-7_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-29700-7_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-29699-4

  • Online ISBN: 978-3-642-29700-7

  • eBook Packages: Computer ScienceComputer Science (R0)