A Simple Optimal Binary Representation of Mosaic Floorplans and Baxter Permutations

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.

💡 Research Summary

The paper addresses the problem of compactly encoding n‑block mosaic floorplans, a combinatorial structure that models the relative placement of rectangular modules in VLSI physical design. A mosaic floorplan is defined as a rectangle partitioned by horizontal and vertical line segments, where two floorplans are considered equivalent if one can be transformed into the other by sliding the dividing line segments; equivalently, they share identical horizontal and vertical constraint graphs. The number of n‑block mosaic floorplans equals the n‑th Baxter number B(n), which grows as Θ(8ⁿ / n⁴), implying an information‑theoretic lower bound of log₂ B(n) = 3n − o(n) bits for any encoding.

Existing binary representations—such as the Quarter‑State Sequence (4n bits), Sequence‑Pair (≈2n log n bits), Corner‑Block list, Twin‑Binary‑Sequences, and various tree‑based encodings—either require more than 4n bits or need an additional n log n term because they explicitly name each block. Consequently, a gap remained between the best known 4n‑4‑bit representation and the lower bound.

The authors introduce a dramatically simpler and optimal encoding that uses exactly 3n − 3 bits. The key steps are:

1. Standard Form Conversion – Every mosaic floorplan can be transformed, in linear time, into a canonical “standard form” where, for each horizontal segment, all vertical segments above it appear to the right of those below it, and for each vertical segment, all horizontal segments to its right appear above those to its left. This canonicalization preserves the equivalence class because it only slides line segments.

2. Staircase Representation – In standard form, the floorplan can be viewed as a “staircase”: a polygon whose boundary contains the x‑axis and y‑axis, followed by a non‑increasing chain of alternating horizontal and vertical edges. A mosaic floorplan is a special staircase with a single “step”.

3. Deletable Rectangles – Within a staircase, a block whose top and right edges lie on the outer boundary is called a deletable rectangle. Lemma 1 proves that removing such a rectangle yields another staircase (unless only one block remains). This property enables a recursive decomposition: repeatedly delete a deletable rectangle, record its position, and continue on the reduced staircase.

4. Encoding Scheme – For each of the n − 1 deletions, two bits encode whether the rectangle is attached to the top or bottom and to the left or right of the current boundary; a third bit (or the implicit order of deletions) records the identity of the rectangle. The first block is fixed as a reference, so the total length is 3n − 3 bits.

The authors prove that this length matches the lower bound log₂ B(n) up to lower‑order terms, establishing optimality. Moreover, because there exists a well‑known bijection between mosaic floorplans and Baxter permutations (as shown in earlier work), the same 3n − 3‑bit representation can be directly translated into an optimal encoding for Baxter permutations and, by extension, for every object in the Baxter combinatorial family (alternating sign matrices, plane bipolar orientations, etc.).

Compared with prior methods, the new representation is strikingly simple: it requires no explicit naming of blocks, no auxiliary permutations, and no large auxiliary trees. The encoding and decoding algorithms both run in O(n) time, making the approach practical for VLSI floorplanning tools that need fast, memory‑efficient representations.

In summary, the paper makes three major contributions: (1) a linear‑time conversion to a canonical standard form; (2) a recursive deletable‑rectangle decomposition that yields a 3n − 3‑bit encoding; (3) a proof of optimality and an immediate extension to optimal encodings of Baxter permutations and all related combinatorial structures. This resolves a long‑standing gap between known representations and the information‑theoretic lower bound for mosaic floorplans.