## 2 Improved Symmetric Algorithm
### Improved Symmetrical Arrangement Algorithm
#### 2.3.2 Calculation of Remaining Effective Area  
The remaining effective area calculation is the last step after each round of arrangement is completed. This function is to determine whether the remaining effective area is enough to be arranged again. If the remaining effective area can continue to be arranged, a new effective area will be generated and represented by \(\mathrm{P_{x1}}\) and \(\mathrm{P_{y1}}\). If the new effective area can be re-arranged, the step will return to the object arrangement part to find the best arrangement to continue the arrangement. If the equation result shows that the new effective area can no longer be arranged, the step will move to the next step \"Object Coordinate Marking\". At this step, according to the four factors at the beginning, whether the horizontal area is larger or the vertical area is larger, and whether they are overlapped, the length and width of the remaining effective area are calculated by using the following four different equations:  
1. Block 1 and Block 3 are larger in area and non-overlapped When this horizontal area is larger and does not overlap, define the new effective area as \(\mathrm{m_{1}}\) and \(\mathrm{m_{2}}\). It can be found that \(\mathrm{m_{1}}\) determines the length of \(\mathrm{P_{x1}}\), and \(\mathrm{m_{2}}\) determines the length of \(\mathrm{P_{y1}}\). The equations are as Equations (7) and (8). After calculation, the area when Block 1 and Block 3 are non-overlapped is as shown in Figure 11. \[\mathrm{P_{x1}}=\mathrm{P_{x}}-2\times\left(\mathrm{m_{1}}\times\mathrm{B_{y}}\right)\] (7) \[\mathrm{P_{y1}}=\mathrm{P_{y}}-2\times\left(\mathrm{m_{2}}\times\mathrm{B_{y}}\right)\] (8)
2. Block 2 and Block 4 are larger in area and non-overlapped In the case of non-overlapped, the calculated new effective area is roughly the same. The difference is that the key factors that determine \(\mathrm{P_{x1}}\) and \(\mathrm{P_{y1}}\) are different. When this vertical area is larger and does not overlap, define the new effective area as \(\mathrm{n_{1}}\) and \(\mathrm{n_{2}}\). It can be found that \(\mathrm{n_{1}}\) determines the length of \(\mathrm{P_{x1}}\), and \(\mathrm{n_{2}}\) determines the length of \(\mathrm{P_{x1}}\). Figure 10: Single object overlap. (**a**) Before removal. (**b**) After removal. Figure 9: Vertical multi-object overlap. (**a**) Before removal. (**b**) After removal. of \(P_{y1}\). The equations are as Equations (9) and (10). After calculation, the area when Block 2 and Block 4 are non-overlapped is as shown in Figure 12. After confirming the length and width of the new area, it can be known whether the area can continue to be arranged. \[P_{x1}=P_{x}-2\times(n_{1}\times B_{x}) \tag{9}\]  
\[P_{y1}=P_{y}-2\times(n_{2}\times B_{x}) \tag{10}\]
3. Block 1 and Block 3 overlapped In the case of overlap, the remaining effective area is the area remaining after the overlapped object is removed. It can be known that when overlap occurs, there will be areas that cannot be used. The remaining area calculation will determine the new area size according to \(n^{\prime}\), \(m^{\prime}\), \(n_{1}\), \(m_{2}\). Take \(P_{x1}\) as an example. Its effective length is the length remaining after removing the overlapped objects, and \(P_{y1}\) is the length remaining after \(m_{2}\) minus \(m^{\prime}\). The equations are as shown in Equations (11) and (12). After calculation, the area when Block 1 and Block 3 are overlapped is as shown in Figure 13.  
\[P_{x1}=P_{x}-2\times((n_{1}-\frac{n^{\prime}}{2})\times B_{x}) \tag{11}\]  
\[\mathrm{P_{x1}}=\mathrm{P_{x}}-2\times\left(\mathrm{m_{1}}\times\mathrm{B_{y}}\right)\]0
4. Block 2 and Block 4 overlapped The area calculation of horizontal overlap is roughly the same as that of vertical overlap. The remaining area calculation will determine the new area size according to \(n^{\prime}\), \(m^{\prime}\), \(n_{1}\), \(m_{2}\). Take \(P_{x1}\) as an example.