An asymptotic estimate of the numbers of rectangular drawings or floorplans

Fujimaki, Ryo; Inoue, Youhei

doi:10.1109/iscas.2009.5117891

Cited by 9 publications

(8 citation statements)

References 3 publications

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“…After this paper was accepted, the author became aware of a substantial literature studying generic rectangulations under the name rectangular drawings. This literature includes some results on asymptotic enumeration as well as computations of the exact cardinality of gRec n for many values of n. See, for example, [3,11,14]. In particular, the main result of this paper answers an open question posed in [3, Section 5].…”

Section: Introductionmentioning

confidence: 89%

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Generic rectangulations

Reading¹

2012

European Journal of Combinatorics

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A rectangulation is a tiling of a rectangle by a finite number of rectangles. The rectangulation is called generic if no four of its rectangles share a single corner. We initiate the enumeration of generic rectangulations up to combinatorial equivalence by establishing an explicit bijection between generic rectangulations and a set of permutations defined by a pattern-avoidance condition analogous to the definition of the twisted Baxter permutations. arXiv:1105.3093v3 [math.CO]

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Section: Introductionmentioning

confidence: 89%

“…Primary 05A05, 05A19, 05B45. 186,042 10 1,395,008 11 10,948,768 Table 1. The number of generic rectangulations with n rectangles…”

Section: Introductionmentioning

confidence: 99%

Generic rectangulations

Reading¹

2012

European Journal of Combinatorics

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“…This lower bound implies that any representation of a floorplan with f inner faces requires at least 3:53 f bits, on average. Inoue et al [15] improved the upper bound to Fð f Þ ¼ Oð2 4:64 f Þ, and then Fujimaki et al [9] proposed a tighter upper bound of Fð f Þ ¼ Oð2 3:75 f Þ. From the above theorem, we obtain the lower and upper bounds of the information-theoretic lower bound of floorplans with f inner faces.…”

Section: The Number Of Floorplans and The Information-theoretic Lowermentioning

confidence: 87%

Recent Developments in Floorplan Representations

Yamanaka

2015

IIS

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A floorplan is a partition (dissection) of a rectangle into smaller rectangles by horizontal and vertical line segments such that no four rectangles meet at the same point. Floorplans are used to design the layout of verylarge-scale integration (VLSI) circuits. Since modern VLSI circuits are extremely large, it is necessary to design compact floorplans (VLSI layouts). In 2004, Feng et al. [8] surveyed ways of representing floorplans. However, over the past decade, various new methods have been developed, and in this paper, we survey these recent developments in floorplan representations.

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“…In [2], it was shown that there exists a constant c = lim n→∞ (F (n)) 1/n and 11.56 < c < 28.3. This means that 11.56 n ≤ F (n) ≤ 28.3 n for large n. The upper bound of F (n) is reduced to F (n) ≤ 13.5 n in [7].…”

Section: Floorplansmentioning

confidence: 97%

A simple optimal binary representation of mosaic floorplans and Baxter permutations

2014

Theoretical Computer Science

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A floorplan is a rectangle subdivided into smaller rectangular sections by horizontal and vertical line segments. Each section in the floorplan is called a block. Two floorplans are considered equivalent if and only if there is a one-to-one correspondence between the blocks in the two floorplans such that the relative position relationship of the blocks in one floorplan is the same as the relative position relationship of the corresponding blocks in another floorplan. The objects of Mosaic floorplans are the same as floorplans, but an alternative definition of equivalence is used. Two mosaic floorplans are considered equivalent if and only if they can be converted to each other by sliding the line segments that divide the blocks.Mosaic floorplans are widely used in VLSI circuit design. An important problem in this area is to find short binary string representations of the set of n-block mosaic floorplans. The best known representation is the Quarter-State Sequence which uses 4n bits. This paper introduces a simple binary representation of n-block mosaic floorplan using 3n − 3 bits. It has been shown that any binary representation of n-block mosaic floorplans must use at least (3n − o(n)) bits. Therefore, the representation presented in this paper is optimal (up to an additive lower order term).Baxter permutations are a set of permutations defined by prohibited subsequences. Baxter permutations have been shown to have one-to-one correspondences to many interesting objects in the so-called Baxter combinatorial family. In particular, there exists a simple one-to-one correspondence between mosaic floorplans and Baxter permutations. As a result, the methods introduced in this paper also lead to an optimal binary representation of Baxter permutations and all objects in the Baxter combinatorial family.

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An asymptotic estimate of the numbers of rectangular drawings or floorplans

Cited by 9 publications

References 3 publications

Generic rectangulations

Generic rectangulations

Recent Developments in Floorplan Representations

A simple optimal binary representation of mosaic floorplans and Baxter permutations

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