## 3 Proposed RBAVO-DE for GS
### The Complete RBAVO-DE Algorithm
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#### 4.7.2 Computational Complexity of the Memory Usage of the RBAVO-DE Algorithm

This involves measuring the amount of memory required by an algorithm to tackle a problem as the quantity of the input increases. It is frequently expressed as the additional memory needed by the algorithm beyond the input. It entails merging the following two primary elements:

1. **Memory usage of input variables**: This is the amount of memory required for the algorithm to store the input data. There are 13 input variables related to the proposed RBAVO-DE algorithm, as follows: \(N\), \(G_{max}\), \(D\), \(LB\), \(UB\), \(C_{R}\), \(W_{M}\), \(L_{1}\), \(L_{2}\), \(w\), \(P_{1}\), \(P_{2}\), and \(P_{3}\). Since each variable stores numerical values, 4 bytes of memory are utilized by each one. Consequently, the total memory usage complexity of these 13 input variables is 52 bytes (\(13\times 4\) bytes \(=52\) bytes). The memory usage complexity of the input values is constant. 2. **Additional memory usage**: This shows how much more memory the algorithm needs in addition to the input. It includes the memory required for data structures, internal variables, and other components of the algorithm. Regardless of the size of the input, the RBAVO-DE algorithm requires a specific amount of extra memory. The following variables are involved: * The memory usage complexity consumed by \(X_{initial}\) is \((4\times N\times D)\) bytes. This is because each position in the positions vector \(X_{initial}\) requires 4 bytes of memory, and its size is \((N\times D)\), proportionate to the initial population of \(N\) positions with dimension size \(D\). Because the amount of memory required rises linearly with the value \((N\times D)\), its memory use complexity is linear. * The variables \(pr\), \(F_{i}\), \(t\), \(R\), \(D_{i}\), \(d_{t}\), \(S_{1}\), \(S_{2}\), \(A_{1}\), \(A_{2}\), \(sigma\), \(fit(X_{i})\), \(fit(X_{Best_{1}})\), \(fit(X_{SecondBest_{2}})\), and \(fit(X_{Best})\) require 4 bytes of memory space each since they only represent numerical values. As a result, the total memory usage complexity of these 16 variables is 64 bytes (\(16\times 4\) bytes \(=64\) bytes). This memory usage complexity is constant. * The position vectors \(BestVulture_{1}\), \(SecondBestVulture_{2}\), \(X_{i}\), \(Levy_{d}\), \(\mu\), \(\sigma\), \(X_{i}^{lin}\), \(v_{i}\), \(X_{r_{1}}\), \(X_{r_{2}}\), \(X_{r_{3}}\), \(u_{i}\), and \(X_{Best}\) each require 4 bytes of memory space, and the size of each vector is \(D\), proportionate to the dimension size of the acquired positions. As a result, the vectors at these 13 points have a total memory usage complexity of \((52\times D)\) bytes (\(13\times 4\times D\) bytes). Since the memory needed grows linearly with the value \(D\), its memory consumption complexity is linear. As a result, the overall memory usage complexity for all of the additional variables listed above is \((4\times N\times D)+64+(52\times D)\) bytes. Lastly, the following formula can be used to determine the computational complexity of the overall memory usage of the proposed RBAVO-DE algorithm:

Memory usage complexity(RBAVO-DE) = Input values memory usage\(+\)

Additional memory usage = \(52+\left((4\times N\times D)+64+(52\times D)\right)\) bytes. Keep in mind that there are constant bytes that are not taken into account. In big-O notation, the computational complexity of the total memory consumption of RBAVO-DE can be represented as \(O_{memory\_usage}(RBAVO-DE)\).