**Force-directed placement** is used to place center of standard cell clusters onto
the center of the grid cells.
## **Information provided by Google.**
## **Information provided by Google**
The Methods section of the [Nature paper](https://www.nature.com/articles/s41586-021-03544-w.epdf?sharing_token=tYaxh2mR5EozfsSL0WHZLdRgN0jAjWel9jnR3ZoTv0PW0K0NmVrRsFPaMa9Y5We9O4Hqf_liatg-lvhiVcYpHL_YQpqkurA31sxqtmA-E1yNUWVMMVSBxWSp7ZFFIWawYQYnEXoBE4esRDSWqubhDFWUPyI5wK_5B_YIO-D_kS8%3D) provides the following information.
* “(1) We group millions of standard cells into a few thousand clusters using hMETIS, a partitioning technique based
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@@ -17,11 +17,26 @@ according to the weight×distance formula, causing tightly connected nodes to be
We also introduce a repulsive force between overlapping nodes to reduce placement density.
After applying all forces, we move nodes in the direction of their force vector. To reduce oscillations, we set a maximum distance for each move.”
For each pair of nodes $(u, v)$, there are two types of forces between them : attractive force $f_a(u,v)$ and repulsive force $f_r(u,v)$. The magnitude of attractive force can be expressed as $f_a(u, v) = k_a * distance(u, v)$, where $k_a$ is the attractive factor and $distance$ is the euclidean distance between node $u$ and node $v$; the magnitude of repulsive force can be expressed as $f_r(u, v) = k_r$. Both attracive force and repulsive force are along the straight line joining the node $u$ and node $v$.
## **Our implementation**
Our force-directed placer takes a clustered netlist as input and generates the locations for standard-cell clusters. During the force-directed placement, all the hard macros and IO ports are fixed. Only the standard-cell clusters can be moved but the standard-cell clusters are not necessarily placed onto the centers of gridcells. At the beginning, all the standard-cell clusters will be placed at the center of the canvas. \[[code](https://github.com/TILOS-AI-Institute/MacroPlacement/blob/5addfc904527d764ee67429811c868c5eeb605d4/CodeElements/FDPlacement/FD.py#L1130)\]
During force-directed placement, all the hard macros and IO ports are fixed. Only the stanard-cell clusters can be moved. Note that we need to consider the contribution from hard macros and IO ports when we calculate the total force exerted on a standard-cell cluster.
In the force-directed placement, there are two types of forces between nodes: attractive force and repulsive force.
* The attractive force is ONLY applied to the nodes connected by nets. For the two-pin net connecting pin $P1$ of macro $M1$ and $P2$ of macro $M2$, the attractive force applied to $M1$ and $M2$ is calculated as $f_x = k_a * abs(P1.x - P2.x)$ and $f_y = k_a * abs(P1.y - P2.y)$, where $k_a$ is the attractive factor. If one of pins is an IO port, $k_a$ is the attractive factor times io factor. The attractive force is along the straight line joining $P1$ and $P2$. All the multi-pin nets are decomposied into two-pin nets using the start model. \[[code](https://github.com/TILOS-AI-Institute/MacroPlacement/blob/5addfc904527d764ee67429811c868c5eeb605d4/CodeElements/FDPlacement/FD.py#L1105)\]
* The repulsive force is ONLY applied to the nodes overlapped with each other. If two macros are not overlapped, there is no repulsive force between them. For the two macros $M1$ and $M2$ overlapped with each other, the repulsive force applied to $M1$ and $M2$ is calculated as $f_x = F * abs(M1.x - M2.x) / distance(M1, M2)$ and $f_y = F * abs(M1.y - M2.y) / distance(M1, M2)$, where $F$ is the maximum move distance specified by the users. The repulsive force is along the straight line joining $M1$ and $M2$. For the cases where multiple macros are overlapped together, we calculate the repulsive forces between each pair of macros in sequence. \[[code](https://github.com/TILOS-AI-Institute/MacroPlacement/blob/5addfc904527d764ee67429811c868c5eeb605d4/CodeElements/FDPlacement/FD.py#L1082)\]
After calculating all the attractive forces and repulsive forces, all the forces are normalized as following:
* f_x = f_x / f_x_max * max_move_distance
* f_y = f_y / f_y_max * max_move_distance
Here f_x_max (f_y_max) is the absolute value of f_x (f_y) which has the maximum absolute value. max_move_distance is the maximum move distance specified by users. \[[code](https://github.com/TILOS-AI-Institute/MacroPlacement/blob/5addfc904527d764ee67429811c868c5eeb605d4/CodeElements/FDPlacement/FD.py#L1137)\]
After normalization, the standard-cell clusters are moved based on the forces exerted on them. The move which will push the standard-cell clusters outside of canvas will be cancelled.
## **Experimental results**
We have tested our codes on the Ariane133 (NanGate45). The experimental results are presented below.
The standard-cell clusters are not placed onto the centers of gridcells.
