## 4 Algorithm Design
### Fast Rerouting Mechanism
Meanwhile, the shortest path algorithm searches for a route from _Source_ to _Target_, resulting in the globally minimal latency path. Firstly, we discuss the complexity of the MOLM algorithm. For the filtering process of the continuous neighbor set, we note that its time complexity is approximately O(n), where n represents the cardinality of the continuous neighbor set. Similarly, for the selection of the best node, we employ the Optimum method based on a greedy algorithm, with a time complexity that is also approximately O(n). This is because, under the greedy strategy, selecting nodes from the continuous neighbor set is a relatively efficient operation. Furthermore, the complexity of the shortest path algorithm is also discussed. We use the Dijkstra algorithm to find the shortest path. Due to the +Grid networking method employed, with each satellite maintaining four connections, the topology graph is relatively sparse. Therefore, its time complexity is approximately O(nm), where m represents the number of edges in the network. In summary, the MOLM algorithm exhibits low time complexity in addressing network congestion issues, making it capable of efficient operation in practical applications and effectively enhancing network performance. ### Simulation Result

We measure path stability using the average duration of a path, which is calculated based on the consecutive occurrences of a particular path. Assuming a path persists continuously for \(N\) times, the duration of that path would be \(N*step\). The comparison of different algorithms is shown in Figure 5.

In terms of path stability, when using a shortest-path-first rerouting mechanism to address network congestion, the average path duration is approximately 125 s. By employing the MOLM algorithm in conjunction with multi-objective simulated annealing to address congestion, the average path duration can exceed 225 s. This represents an improvement of around 80% compared to the _Shortest Path_ strategy. In terms of path latency, the _Shortest Path_ strategy achieves minimal latency, approximately around 70 ms. The MOLM algorithm filters out the continuous neighbor set from all satellites, iteratively selects an optimal candidate neighbor using multi-objective simulated annealing, and replaces congested nodes. This strategy, while enhancing path stability, maintains a relatively small latency cost. As shown in Figure 5b, the latency introduced by MOLM is around 80 ms, an increase of less than 15% compared to _Shortest Path_. This is because the _Shortest Path_ strategy prioritizes the selection of the shortest path, sacrificing path stability in pursuit of minimizing latency. In contrast, the nodes selected by the MOLM algorithm in the continuous neighbor set maintain sustained connections with the predecessor and successor nodes of congested nodes, contributing to robust path stability. After undergoing iterations of multi-objective simulated annealing, the nodes selected by MOLM are very close to the global optimum, resulting in a minimal latency cost. Whether considering path latency or path stability, the results obtained with the MOLM algorithm approach those achieved by the _Optimum_ strategy. This highlights thecharacteristic of multi-objective simulated annealing of gradually converging towards the global optimum through iterations. The convergence process of latency and path stability in multi-objective simulated annealing is shown in Figure 6. During the annealing process, when encountering a poorer new solution (i.e., a new path with higher latency or poorer path stability), the algorithm accepts this poorer solution with a probability P. However, when a better solution is encountered (i.e., a new path with both lower latency and better path stability than the original path), the algorithm directly accepts the new solution.