Update README.md

CodeElements/Gridding/README.md

@@ -26,6 +26,57 @@ 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 \textbf{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.
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+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** \times **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.
+<img src="./Calculate Waste Ratio.png" width= "1600"/>
+
+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**, **ver_waste** and **ver_waste**, the **metric** is defined as
+**metric** = **empty_ratio** + 2.0 - **ver_waste** - **ver_waste**.
+The _grid_ with best **metric** is noted as **n_rows_opt** and **n_cols_opt**.
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+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. 
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 The gridding process starts with the dimensions **H_canvas** and **W_canvas** of the layout canvas, as well as a list of hard **macro blocks**, where each block has a width and a height. Blocks are not rotatable. The area of a block is the product of its width and height.