1. Mixed Non-Rectangular Block Packing for Non-Manhattan Layout Architectures M. Wu, H. Chen and J. Jou Department of EE, NCTU HsinChu, Taiwan ISQED 2011

2. Outline Introduction Review of B*-trees Problem formulation Floorplanning with isosceles right triangular blocks Floorplanning with the trapezoidal blocks Algorithm Experimental results Conclusions

3. Introduction The X architecture is an IC wiring architecture based on the pervasive use of diagonal wires. Compared with the Manhattan architecture, the X architecture shows a wirelength and reduction of more than 20% and a via reduction of more than 30%. In order to take full advantage of the X architecture, it is essential to develop new physical design tools for this architecture.

4. Introduction Besides rectangular blocks, we can add some blocks which have 45 and 135 degree angle. By using these flexible blocks, we can obtain more choices for pin assignment and more shapes can be used in floorplans.

5. Introduction X-half-perimeter wirelength (XHPWL) Manhatten bounding boxX bounding box

6. Review of B*-trees The B*-tree is an ordered binary tree for modeling a non- slicing floorplan. The root of B*-tree represents the block on the bottom-left corner. If node nj is the left child of node ni, block bj is placed on the right-hand side and adjacent to block bi. If node nj is the right child of node ni, block bj is placed above block bi.

7. Problem Formulation Input:  A set of rectangular blocks B  A set of isosceles right triangular blocks T  Some blocks from B and T will form a trapezoidal block Output:  A floorplan F for each block in set B and set T such that no two blocks overlap and the shapes of trapezoidal blocks can be maintained Objective:  Optimize a predefined cost metric, such as the area or XHPWL minimization

8. Floorplanning with Isosceles Right Triangular Blocks Feasibility condition for mixed isosceles right triangular and rectangular blocks Compact floorplan for (a) and (b)The deadspaces of (a) and (b) are quite large

9. The packing with isosceles right triangular blocks The isosceles right triangular blocks are classified into four kinds according to the position of right angles.

10. The packing with isosceles right triangular blocks Case BR: b BR H tBR x b, y b x tBR, y tBR BR b x b +W b -x tBR WbWb W tBR x b, y b x tBR, y tBR

11. The packing with isosceles right triangular blocks Case BL: BL b x b -x tBL WbWb H tBL BL b H tBL H tBL -(x b -x tBL )

12. The packing with isosceles right triangular blocks Case TR: TR b HbHb b W tTR (x tTR +W tTR )- (x b +W b ) HbHb WbWb H b -[(x tTR +W tTR )- (x b +W b )]

13. The packing with isosceles right triangular blocks Case TL: b HbHb TL b x b -x tTL HbHb TL H b -(x b -x tTL )

14. The packing with isosceles right triangular blocks Case TR vs BL: Case TL vs BR: BL TR x tbu, y tbu x tbd, y tbd TL BR H tbd W tbu x tbu +W tbu -x tbd x tbu -x tbd x tbu, y tbu x tbd, y tbd

15. Floorplanning with the Trapezoidal Blocks Feasiblity condition for mixed trapezoidal and rectangular blocks Horizontal trapezoid blocks Vertical trapezoid blocks Packing with B*-tree scheme, tL and tR have falling down problems

16. Floorplanning with the Trapezoidal Blocks B*-trees and corresponding packing scheme with trapezoidal blocks

17. Floorplanning with the Trapezoidal Blocks For falling down problems, we need to calculate the heights of the corresponding dummy blocks:

18. Floorplanning with the Trapezoidal Blocks Vertical trapezoidal block

19. Algorithm

20. The B*-tree is perturbed to another by the following operations: Op1: Rotate a block Op2: Flip a block Op3: Move a block to another place Op4: Swap two blocks Op5: Move a trapezoidal block to another place

21. Experimental Results

24. Conclusions This paper presented an efficient algorithm to handle the floorplanning with isosceles right triangular blocks based on the B*-tree representation. The proposed algorithm can deal with all shapes which are the combination of rectangle and isosceles right triangle.