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.
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References
Ackerman, E., Barequet, G., Pinter, R.Y.: A bijection between permutations and floorplans, and its applications. Discrete Applied Mathematics 154, 1674–1684 (2006)
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)
Baxter, G.: On fixed points of the composite of commuting functions. Proceedings American Mathematics Society 15, 851–855 (1964)
Bonichon, N., Bousquet-Mélou, M., Fusy, É.: Baxter permutations and plane bipolar orientations. Séminaire Lotharingien de Combinatoire 61A (2010)
Canary, H.: Aztec diamonds and baxter permutations. The Electronic Journal of Combinatorics 17 (2010)
Dulucq, S., Guibert, O.: Baxter permutations. Discrete Mathematics 180, 143–156 (1998)
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)
Giraudo, S.: Algebraic and combinatorial structures on baxter permutations. Discrete Mathematics and Theoretical Computer Science, DMTCS (2011)
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)
Lengauer, T.: Combinatorial Algorithms for Integrated Circuit Layout. John Wiley & Sons (1990)
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)
Nakano, S.: Enumerating Floorplans with n Rooms. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 107–115. Springer, Heidelberg (2001)
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)
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)
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)
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)
van Kreveld, M., Speckmann, B.: On rectangular cartograms. Computational Geometry: Theory and Applications 37(3), 175–187 (2007)
Yamanaka, K., Nakano, S.: Coding floorplans with fewer bits. IEICE Transactions Fundamentals E89(5), 1181–1185 (2006)
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)
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)
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)
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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
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DOI: https://doi.org/10.1007/978-3-642-29700-7_1
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