## 3 Methods  
### Calculation of Polygon Shape Complexity  
Previous scholars have used the number of vertices to represent the complexity of polygons, but this method cannot accurately measure the complexity of polygons and cannot represent the relationship between shape complexity and algorithm operation efficiency. To calculate the shape complexity of the subject polygon and the clipping polygon, on the basis of Reference [35], this paper selects 6 polygon shape indicators to calculate the shape complexity of the subject polygon and the clipping polygon. The selected indicators are as follows: number of vertices (NOV), amplitude of the vibration (Ampl), number of holes (NOH), average nearest neighbor (ANN), concavity (Conv), and equivalent rectangular index (ER). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**CPU** & **Memory/GB** & \begin{tabular}{c} **External** \\ **Memory/TB** \\ \end{tabular} & **GPU** & **Development Tools** \\ \hline AMD Ryzen 9 5950X & 128 & 4.5 & \
\begin{tabular}{c} NVIDIA GeForce \\ RTX 3090 \\ \end{tabular} &
\begin{tabular}{c} Visual Studio \\ 2019 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental environment. Figure 1: Experimental data. (**a**) Chinese land use patch data. (**b**) Clipping polygon. (**c**) Subject polygon. NOV refers to the number of all vertices on the boundary of the polygon. The more vertices a polygon has, the more complex it becomes. Ampl refers to the difference in perimeter between the vector polygon and its convex hull, which can measure the irregularity of the vector polygon. NOH refers to the number of inner rings of the polygon, which is also an important indicator used to measure the complexity of the polygon. ANN refers to the shortest distance between the current vertex and the remaining vertices, which can measure the degree of spatial clustering of vertices. Conv is the area difference between a vector polygon and its convex hull, which can reflect the degree of concavity inside the polygon. ER refers to the perimeter difference between a vector polygon and a rectangle with the same area. More complex polygons often have larger perimeter differences. The formulas for calculating the six polygon shape indices are shown in Table 2.  
In this paper, 100,000 polygons are randomly selected as the subject polygons in the Chinese land use patch data, and the six index values of NOV, Ampl, NOH, ANN, Conv and ER of the target polygons are calculated as independent variables. A complex polygon is selected as the clipping polygon and overlaid with the 100,000 target polygons. The time of the overlay analysis is used as the dependent variable, and the algorithm selects the intersection operator of the Greiner-Hormann algorithm. Before the overlay analysis, the centroid of the subject polygon and the clipping polygon need to coincide to ensure the success of the overlay analysis. The selected indicators were diagnosed for collinearity using IBM SPSS Statistics 27 software to ensure that there was no correlation between the variables, and then a multivariate linear stepwise regression analysis was performed. After screening and analysis, the final six independent variable indicators were retained, and they were all closely related to the running time of the overlay analysis algorithm. The constructed shape complexity model is shown in Formula (1). \[C=0.966x_{1}+0.041x_{2}+0.027x_{3}-0.019x_{4}+0.016x_{5}+0.011x_{6}, \tag{1}\]  
where \(C\) is the shape complexity of the intersection operator of the Greiner-Hormann algorithm, \(x_{1}\) is NOV, \(x_{2}\) is Ampl, \(x_{3}\) is NOH, \(x_{4}\) is ANN, \(x_{5}\) is Conv, and \(x_{6}\) is ER. After the shape complexity model is constructed, the six indicators of the subject polygon and the clipping polygon are substituted into the constructed shape complexity model to obtain the shape complexity of the subject polygon and the clipping polygon. The vertices and complexity information of the eight subject polygons and the clipping polygon are shown in Table 3.