## 4 Algorithm Analysis
Consequently, the execution duration might impede fault recovery efficiency. Furthermore, the intricacy of network topologies can influence the MCD-BP algorithm's performance. In specific topological configurations, the algorithm's crossover mean displays fluctuations, suggesting its optimization effects might differ across various network contexts. Though the MCD-BP algorithm surpasses many contemporary routing protection strategies, it might necessitate integration with other methods, such as LFA or ECMP, to leverage their strengths comprehensively. Given the dynamic nature of real-world networks--characterized by evolving topologies, link additions or deletions, and node failures and recoveries--the relevance of pre-computed backup paths may be questioned. Additionally, deploying the algorithm entails addressing the performance constraints of network devices and ensuring protocol compatibility. Future research can be pursued in several directions, outlined as follows: Firstly, the time complexity of the algorithm can be reduced either by enhancing the computational methods inherent in the algorithm or by introducing heuristic strategies. This would particularly improve its applicability in large-scale networks. Secondly, it is worth exploring how to dynamically adjust the algorithm's strategy according to the characteristics of network topology in order to achieve superior performance optimization. Lastly, investigating the integration of the MCD-BP algorithm with other routing protection schemes could lead to a more efficient fault recovery mechanism. In conclusion, while the MCD-BP algorithm has demonstrated commendable enhancements in network fault protection rate and stability, it is not without certain challenges in real-world implementation. Through continued research and refinement, this algorithm can offer a more robust assurance for the high availability of the network. Conceptualization, H.G. and X.L. ; methodology, H.G., X.L., W.H., L.X., and L.W. ; software, X.L., W.H., and L.W. ; validation, X.L. ; formal analysis, X.L. and L.X. ; investigation, X.L., W.H., and L.X. ; resources, X.L. ; data curation, X.L. ; writing--original draft preparation, H.G., X.L., and L.W. ; writing--review and editing, H.G., X.L., W.H., L.X., and L.W. ; visualization, X.L. ; supervision, H.G. and L.W. ; project administration, H.G. ; funding acquisition, H.G. All authors have read and agreed to the published version of the manuscript. This research was funded, in part, by the National Natural Science Foundation of China (NSFC) under Grant No. 62472267 and by the Postdoctoral Fellowship Program and China Postdoctoral Science Foundation under Grant Number 2025M774384. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The raw data supporting the conclusions of this article will be made available by the authors on request. **Conflicts of Interest:** The authors declare no conflicts of interest. ## References

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