1. Force-Directed Placement Heuristics A Comparative Study

2. Concepts An attempt is made to place cells in their ideal locations by using a mechanical analogy. Two cells separated by a distance D and connected by a net of length L are said to have a force D x L acting upon them. The averaged sum of all such forces on a single cell yields the zero-force target location of the cell.

3. Concepts The net weights are treated like spring constants, and the net lengths are treated like spring lengths in this analogy to Hooke’s Law.

4. Heuristics Once a cell’s zero-force target location is calculated, we must decide where it will be placed in the layout. If the target location is empty, simply place it in the empty location. If the target location is already occupied, we must determine how to proceed.

5. Heuristics 1. The selected cell is moved to the free location nearest to its zero-force target location. 2. The selected cell is swapped with the cell at its zero-force location if the swap will reduce the solution cost. 3. The selected cell replaces the cell at its zero- force location and the replaced cell has its zero-force location computed. The replaced cell is then placed in a similar fashion, inciting a "ripple" of replacements until an end condition is met.

6. General Algorithm fdpl( iterations, cells, nets, width, type ) Initialize the layout according to the layout type and layout width. Sort the cells in decreasing order by total weight of connected nets. i = 0 while (i < iterations) { for each net apply force-directed heuristic i = i + 1 }

7. Initial Layout Types Three types of initial layouts are explored for each heuristic: 1. Solid Layout 2. Donut Layout 3. Random Layout

8. Solid Layout xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx----------------

9. Donut Layout xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx-------xxxxxxxxx-------xxxxxxxxx-------xxxxxxxxx-------xxxxxxxxx-------xxxxxxxxx-------xxxxxxxxx-------xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx---------------

10. Random Layout xxx-xxxxxx-xxxxxxxxxx--xxxx--xxxxxx-xxxx--xxxxx-xxxxxxxxxxxx-xxxx-xxxxx-xxxx--xxxxxxxxxxx-xxxxxx-xxxx-xx-xx-xxxxxxxxxxx-xxxxxxxxxxxxxxxx-x-xxxxxx--xxxxx-x-x-xxx-xxxxxxx-xxxxxxxxxxxxxxx-x-xxxxxxx-x------xx-x-xx-x-x-x-xxxxxx--x-x-xxxxxx-xxxxxxxxxxx-xx--xxxxxxxxxxxx-xxxxxxxxxxx----xxx-xxx-xxxxx-xxxxxxxxxxxxxxxx-xx-xxxxxx-

11. Nearest Location Selection If we can’t place the cell at the zero-force location, how do we find the nearest location?

12. Search Distance is 1 0

13. Manhattan Distance is 4 Search Distance is 1 1

14. Manhattan Distance is 4 Search Distance is 1 0

15. Manhattan Distance is 4 Search Distance is 1 0

16. Manhattan Distance is 2 Search Distance is 2 Search Distance == Manhattan Distance, so Quit 1

17. Simulation Results

21. Problems and Possible Solutions Problem: Force-Directed Heuristics tend to gather all cells about the center in a circular fashion. ------xxx------------xxxxxx---------xxxxxxxx-------xxxxxxxxxx-----xxxxxxxxxxxx---xx-xxxxxxxxxxx-xxxxxxxxxxxxxx-xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx-xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx-xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx-xxxxxxxxxxxxxxxx--xxxxxxxxxx-x----xxxxxxxxxxx------xxxxxxxxxx---

22. Problems and Possible Solutions Possible Solutions: Lock the position of cells that should be on the periphery of the layout. This will help pull cells away from the center. Make outer rows and columns off-limits for placement after a number of iterations. Use negative spring constants for nets with small weights. This should push cells with lightweight connections away from the more densely connected cells.