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 HFOk). A floorplan is a hierarchical floorplan of order 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 HFOk 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 HFO5 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 HFOk floorplans with n rooms. Considering its application in VLSI design we also give moves on HFOk 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.