In Circuit Training, the purpose of gridding is to control the size of the macro placement solution space, thus allowing RL to train within reasonable runtimes. Gridding enables hard macros to find locations consistent with high solution quality, while allowing soft macros (standard-cell clusters) to also find good locations.
Gridding determines a dissection of the layout canvas into some number of rows (n_rows) and some number of columns (n_cols) of gridcells.
The choice of n_rows and n_cols is made once for each design. Once the dimensions (n_rows, n_cols) have been chosen, their values define a gridded canvas, or grid, and remain fixed throughout Circuit Training for the given design. The detailed algorithm is shown as following.
The gridding algorithm starts with the dimensions canvas_width and canvas_height of the layout canvas, as well as a list of macros, where each macro has a width and a height. Macros are not rotatable. The area of a macro is the product of its width and height. Then, the gridding searches over combinations (n_rows, n_cols), with constraints
- min_n_rows <= n_rows < max_n_rows
- min_n_cols <= n_cols < max_n_cols
- min_num_gridcells <= n_rows * n_cols <= max_num_grid_cells
- grid_w / grid_h <= max_aspect_ratio
- grid_h / grid_w <= max_aspect_ratio
- The macros can be packed sequentially on the gridcells. There are n_rows * n_cols gridcells in the canvas. [Algorithm 1 Lines 11-22]
where each gridcell has width of grid_w = canvas_width / n_cols and height of grid_h = canvas_height / n_row. The main idea is to search for a particular (n_rows, n_cols) combination that maximize the metric related to wasted space.
To evaluate metric for a given grid (n_rows, n_cols), all macros are packed into the gridcells, and several terms (empty_ratio, ver_waste and hor_waste) that reflect wasted space are evaluated.
Macro packing is performed as follows [Algorithm 1 Lines 11-22]:
- Macros are placed in order of non-increasing macro area.
- All macros are placed, one by one, into the (n_rows, n_cols) gridcells. If the current macro cannot be placed, then the grid is infeasible and the next candidate grid is considered.
- A macro is placed at the first (according to row-major order) gridcell where it can be legally placed.
- Placement of a macro means placing that macro's center at the center of some gridcell.
- The placement of a macro's center at the center of some gridcell is legal if (1) no part of the macro is outside of the canvas, and (2) no overlap of the macro with any previously-placed macro is induced.
After macro packing, we can calculate the empty_ratio of current grid, i.e.,
the number of empty gridcells over the total number of gridcells (n_rows * n_cols).
A gridcell is claimed as an empty gridcell if the intersection area of placed macros with it is less than 0.00001 times its area.
Next we calculate the hor_waste and ver_waste as described in following algorithm.
To calculate horizontal waste hor_waste, we calculate
- width_tot_macros = the sum of widths of all macros in the design
- width_tot_used_gridcells = the sum of widths of all used gridcells if we pack the macros along the x-axis one by one.
Then, hor_waste = 1.0 - width_tot_macros / width_tot_used_gridcells.
To calculate vertical waste ver_waste, we calculate
- height_tot_macros = the sum of heights of all macros in the design
- height_tot_used_gridcells = the sum of heights of all used gridcells if we pack the macros along the y-axis one by one.
Then, ver_waste = 1.0 - height_tot_macros / height_tot_used_gridcells.
After calculating empty_ratio, hor_waste and ver_waste, the metric is defined as metric = empty_ratio + 2.0 - hor_waste - ver_waste. The grid with best metric is noted as n_rows_opt and n_cols_opt.
Once we have found n_rows_opt and n_cols_opt as described above,
we seek a smaller grid that has similar metric properties. [Algorithm 1 Lines 33-39]
Specifically, we find values of n_rows_actual and n_cols_actual such that
its metric is within some tolerance (5% in Circuit Training) of the optimal metric,
and n_rows_actual * n_cols_actual is minimized.
This is the grid that is used in Circuit Training.
To our understanding, the foregoing procedure results in grids that are of similar sizes, e.g., with ~25 <= n_rows_actual , n_cols_actual <= ~40.
We thank Google engineers for May 19, 2022 discussions that explained the gridding method used in Circuit Training. All errors of understanding and implementation are the authors'. We will rectify such errors as soon as possible after being made aware of them.