## 4 Methods
### Relevance of Different Size and Shape Occupancy Grids
\[M_{1}=1\ and\ M_{r}=M_{r-1}+8*(r-1),\ if\ \ r>1 \tag{11}\]  
We then describe the area covered by the occupancy grid also as a function of \(r\), based on the area of the elementary regions (polygon) \((A_{e})\) of the investigated grid structure. \[A_{r}=A_{e}*M_{r}\,\ \forall r \tag{12}\]  
In light of the above, we describe the complexity of the square occupancy grid as follows in Equation (13). \[\lim_{r\to\infty}UC_{r}=\lim_{r\to\infty}\frac{N_{r}}{A_{r}} \tag{13}\]  
**Hexagon grid**  
For the hexagon-based occupancy grid, the total number of connections between the elementary polygons as a function of the covering steps (\(r\)) has been described in Equation (14). \[N_{1}=0\text{ and }\ N_{r}=N_{r-1}+[2*2*(6*(r-1)-6)+2*6]+2*(6*(r-1)),\ if\ r>1 \tag{14}\]  
where the expression in the square bracket expresses the number of connections between the kernel and the layer, distinguishing the elementary polygons of the layer according to whether they are adjacent to one (first member of the square bracket) or two (second member of the square bracket) of the elementary polygons of the kernel. The third term of the formula expresses the number of connections between the elementary polygons within the layer. The number of elementary polygons forming the occupancy grid is described in Equation (15). \[M_{1}=1\text{ and }M_{r}=M_{r-1}+6*(r-1),\ if\ r>1 \tag{15}\]  
The area covered by the occupancy grid (\(A_{r}\)), and the related computational complexity (\(\lim\limits_{r\to\infty}UC_{r}\)) then can be calculated as presented in Equations (12) and (13), in this case using the data of the hexagon-based structure. If the size of the elementary regions in the examined square and hexagonal grid structures is determined such that the diameter of their inscribed circles is uniformly 5 units [48], then based on the presented equations, the value of \(\lim\limits_{r\to\infty}UC_{r}\) is 1.6 for the square grid, while for the hexagonal grid it is 0.277. This clearly demonstrates the higher computational complexity associated with the hexagonal grid compared to the square grid. Based on the presented considerations, the modelling process and the calculations related to computational complexity can be carried out according to the same principle using any regular polygon. **Computational complexity analysis**  
Below we examine the effect of increasing resolution on computational complexity. Accordingly, we investigate what happens when the model is built up from smaller and smaller elementary regions (polygons), gradually reaching higher and higher resolutions. The number of connections between elementary regions characterizes the evolution of computational complexity as a function of the number of elementary regions included in the model. In line with the above, the x-axis of Figure 1 shows the number of elementary regions in the case of the hexagon- and square-based grids. Figure 1: Illustrating the increase in computational complexity with the increase in resolution. The number of connections increases slower for the square grid reaching 160 k connections for 40 k elementary regions. In comparison, the complexity of the hexagonal grid (blue circle) grows faster, reaching 180 k connections for 30 k elementary regions.