Optimal Binary Representation of Mosaic Floorplans and Baxter Permutations

摘要

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 An 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.