# An Accelerated Maximum Flow Algorithm with Prediction Enhancement in Dynamic LEO Networks
## 4 Predictive Acceleration Algorithm for e-TEG Satellite Networks
### Predictive Accelerated e-TEG Network Maximum Flow Algorithm  
Here, we present an optimized maximum flow algorithm for dynamic e-TEG networks, leveraging predictive flow and residual network updates to enhance computational efficiency. The algorithm builds on the predictive flow obtained from the predictive flow acquisition algorithm (Algorithm 1) and the residual network structure maintained using the residual network update algorithm (Algorithm 2). By combining these components, the algorithm efficiently computes the maximum flow for a given time slot, adapting to changes in the network topology. The predictive flow provides a \"warm start\", reducing the number of iterations required, while the residual network ensures that updates reflect the current network state. The predictive accelerated maximum flow algorithm (Algorithm 3) integrates predictive flow and residual network updates to efficiently compute the maximum flow. The algorithm begins with initialization (Step 1), where the residual graph \(G^{*}\) is set to the current network \(G\), and the total flow \(f^{*}\) is initialized by combining the predictive flow \(\hat{f}\) and the maximum flow from the previous time slot, \(f^{*}_{\text{prev}}\). This initialization incorporates historical and predictive information to provide a strong starting point. Next, in Steps 2-4, the algorithm adjusts the capacities in \(G\) by subtracting the capacity already occupied by \(\hat{f}\). This step ensures that the residual graph reflects the remaining capacity available for flow augmentation, effectively using the residual network update process from the previous algorithm. The algorithm then performs the Ford-Fulkerson computation on the adjusted residual graph \(G\) (Step 5). This step computes the additional maximum flow \(f_{\text{res}}\) that can be achieved given the residual capacities. By starting with a graph already adjusted for \(\hat{f}\), the number of iterations required to converge is significantly reduced. In Steps 6-8, the algorithm updates the residual capacities in \(G^{*}\) by subtracting the flows from both \(f_{\text{res}}\) and \(\hat{f}\). This ensures that the final residual network \(G^{*}\) is consistent with the total flow. Finally, the algorithm combines \(f_{\text{res}}\) and \(\hat{f}\) to compute the total maximum flow \(f^{*}\) (Step 9). The updated \(f^{*}\) satisfies flow conservation and capacity constraints and is returned along with the residual graph \(G^{*}\). ```
0: Updated network \(G(\mathcal{V},\mathcal{A})\), maximum flow \(f^{*}_{\text{prev}}(\mathcal{V}^{*},\mathcal{A}^{*})\) from the previous time slot, predictive flow \(\hat{f}(\mathcal{V}_{\hat{f}},\mathcal{A}_{\hat{f}})\), source \(s\), and sink \(g\)
0: Maximum flow \(f^{*}\) and updated residual network \(G^{*}\)
1: Set residual graph \(G^{*}\gets G\), residual flow \(f_{\text{res}}\leftarrow\emptyset\), and total flow \(f^{*}\gets f^{*}_{\text{prev}}+\hat{f}\) [Initialize the residual graph and total flow by combining historical and predictive flows]
2:for all\((\mu,\
u)\in\mathcal{A}_{\hat{f}}\)do
3: Update residual capacity: \(c_{G}(\mu,\
u)\gets c_{G}(\mu,\
u)-c_{\hat{f}}(\mu,\
u)\) [Adjust residual capacities to reflect the flow already occupied by the predictive flow]
4:endfor
5: Run Ford-Fulkerson algorithm on \(G\) to compute residual flow \(f_{\text{res}}\) from \(s\) to \(g\) [Compute additional flow using the Ford-Fulkerson algorithm on the adjusted residual graph]
6:for all\((\mu,\
u)\in\mathcal{A}\)do
7: Update residual capacities: \(c_{G^{*}}(\mu,\
u)\gets c_{G}(\mu,\
u)-c_{f_{\text{res}}}(\mu,\
u)-c_{f}( \mu,\
u)\) [Update residual capacities to reflect both residual and predictive flows]
8:endfor
9: Set total flow: \(f^{*}\gets f_{\text{res}}+\hat{f}\) [Combine residual and predictive flows to obtain the total maximum flow]
10:return\(f^{*}\), \(G^{*}\)
```  
**Algorithm 3** Predictive accelerated maximum flow algorithm for e-TEG networks.