## 3 Improved Snake Optimization Algorithm for Melting Point Optimization Solution
### Improved Snake Optimization Algorithm
#### 3.2.1 Halton Sequence Population Initialization  
The original SO algorithm employs a common random initialization formula to initialize the population, as shown in Equation (13):  
\[X_{i}=X_{\text{min}}+r\times(X_{\text{max}}-X_{\text{min}}) \tag{13}\]  
This method, although providing certain randomness in the initial population, does not guarantee a uniform distribution of individuals across the search space. Often, there is a problem of either excessive concentration or dispersion of individuals in the population, leading to a significant portion of the search space being overlooked and reducing the efficiency of population search. Therefore, this paper introduces the Halton sequence to uniformly initialize the population, ensuring a more even distribution of individuals throughout the entire solution space. This enhances the algorithm's search efficiency, improves the speed of discovering high-quality solutions, and accelerates the convergence of the algorithm.  
The Halton sequence is a low-discrepancy sequence constructed using a deterministic method based on prime numbers. It ensures a more uniform distribution of the population across the entire solution space, thereby increasing the diversity of the population. This accelerates the convergence speed and improves the convergence accuracy of the algorithm. The implementation process of the two-dimensional Halton sequence involves selecting two prime numbers as bases corresponding to two dimensions. Each dimension iteratively divides and zigzags within the range (0,1) based on the respective base, forming a set of non-repeating and evenly spaced points. The mathematical model for the division process is represented by Equations (14)-(16):  
\[n=\sum_{i=0}^{m_{0}}a_{i}p_{1}^{i}=a_{0}+a_{1}p_{1}^{1}+\cdots+a_{m_{0}}p_{1}^ {m_{0}},\ a_{i}\in\{0,1,2,\cdots,p_{1}-1\} \tag{14}\]  
\[\phi_{p_{1}}(n)=a_{0}p_{1}^{-1}+a_{1}p_{1}^{-2}+\cdots+a_{m_{0}}p_{1}^{-m_{0}-1} \tag{15}\]  
\[H(n)=\big{[}\phi_{p_{1,1}}(n),\phi_{1,2}(n)\big{]} \tag{16}\]  
In the equations, \(n\) represents the ordinal number of the Halton sequence; \(p_{1}\) denotes the base of the Halton sequence, which is a prime number greater than or equal to 2; \(\phi_{p1(n)}\) represents the defined sequence function; and \(H(n)\) denotes the two-dimensional uniform sequence.  
The distributions of the initial population generated by the random initialization function and the Halton sequence are shown in Figure 3.  
Comparing the initial population generated by the random initialization function with the distribution map of the Halton sequence, it was found that under the condition of an initial population size of 1000, there was a significant dispersion phenomenon in the left part of the initial population generated by the random initialization function, resulting in  
Figure 3: The Random Initialization and Halton Sequence Initialization.  
large gaps, and a clear clustering phenomenon in the right part. Many individuals had obvious overlapping occlusion. The population generated by the Halton sequence is more evenly distributed in the entire space, and compared to the random initialization function, there is no obvious clustering or dispersion phenomenon, presenting an overall uniform trend. Therefore, by comparing the two graphs, it can be concluded that the initialization population generated by the Halton sequence produces better results.