## 3 Algorithmic Mathematical Analysis
### Simulation and Analysis  
Firstly, the computational complexity of the above algorithm is simulated.  
Figure 1 on the left is an actual simulation, and the data on the right are the logarithmic processing. As can be seen from the left figure, with the increase in the number of satellites, the computational complexity of the flooding and backtracking algorithm increases exponentially, while other algorithms increase slowly. From the figure on the right, in general, the flooding and backtracking algorithm has the highest computational complexity, followed by the two partition routing algorithms, and the greedy algorithm has the lowest computational complexity. When the number of inter-star nodes is not very large, the computational complexity of the balanced partition routing algorithm is slightly higher than that of an ordinary partition algorithm.  
The computation force of a single star node under different algorithms is simulated:  
As can be seen from the analysis of Figure 2, with the increase in the number of star nodes in the system, the requirement of the flooding and backtracking algorithm on the computing capacity of a single star node increases rapidly. However, the computing capacity of the two partition routing algorithms increases slowly for a single star node, even approaching the level of a greedy algorithm. This shows that a balanced partition routing algorithm can well distribute computing pressure to multiple star nodes, with stronger destruction resistance and better scalability.  
Figure 1: Simulation of computational complexity of three algorithms.  
Figure 2: Simulation of calculated pressure for a single node.  
Next, the efficiency of the above algorithms is simulated, and the inter-satellite node number is set to 15.  
There were a total of 20 simulations. The average path delay of the greedy algorithm is the highest, and the variance of the average delay in each experiment is large. However, the efficiency of the flooding and backtracking algorithm is relatively stable, and the average delay calculated by this algorithm is significantly less than that of the two partition routing algorithms. Figure 3 shows that the average route length of the balanced partition routing algorithm is shorter than that of the classical partition routing algorithm. In other words, the path calculated by the balanced partition algorithm is better than that of the classical partition routing algorithm.  
Twenty simulation experiments were conducted here. It can be seen from Figure 4 that the variance of each shortest path length calculated by the flooding and backtracking algorithm is nearly 0, followed by the two partition routing algorithms. The variance calculated by the greedy algorithm is the largest and the stability is the worst. However, considering that the computation complexity of the balanced partition routing algorithm is much lower than that of the flooding and backtracking algorithm, the balanced partition routing algorithm still has high application significance.  
Figure 4: Simulation of the variance of average path length for four algorithms.  
Figure 3: Simulation of average path length for four algorithms.