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A Study on Hierarchical Floorplans of Order k

A floorplan is a rectangular dissection which describes the relative placement of electronic modules on the chip. It is called a mosaic floorplan if there are no empty rooms or cross junctions in the rectangular dissection. We study a subclass of mosaic floorplans called hierarchical floorplans of order $k$ (abbreviated HFO-${k}$). A floorplan is HFO-$k$ if it can be obtained by starting with a single rectangle and recursively embedding mosaic floorplans of at most $k$ rooms inside the rooms of intermediate floorplans. When $k=2$ this is exactly the class of slicing floorplans as the only distinct floorplans with two rooms are a room with a vertical slice and a room with a horizontal slice respectdeively. And embedding such a room is equivalent to slicing the parent room vertically/horizontally. In this paper we characterize permutations corresponding to the Abe-labeling of HFO-$k$ floorplans and also give an algorithm for identification of such permutations in linear time for any particular $k$. We give a recurrence relation for exact number of HFO-5 floorplans with $n$ rooms which can be easily extended to any $k$ also. Based on this recurrence we provide a polynomial time algorithm to generate the number of HFO-$k$ floorplans with $n$ rooms. Considering its application in VLSI design we also give moves on HFO-$k$ family of permutations for combinatorial optimization using simulated annealing etc. We also explore some interesting properties of Baxter permutations which have a bijective correspondence with mosaic floorplans.