## 4 Algorithm Analysis
This achievement is primarily attributable to the core principle of the MCD-BP algorithm--identifying the backup path with the minimal intersection degree relative to the optimal path. This strategic approach maximizes the bypassing of potential failure zones, consequently diminishing the susceptibility of the network to single points of failure. In terms of PSD, the MCD-BP algorithm consistently demonstrates a minimal PSD. This suggests that the algorithm effectively balances the relationship between path length and fault protection capability when selecting backup paths. Notably, it does not introduce unnecessarily long backup paths in its pursuit of fault protection, thus mitigating any substantial impact on data transmission latency. For example, in real topology Ans, the PSD of the MCD-BP algorithm is 1.05976, which is only surpassed by the NPC algorithm's 1.04754 and significantly lower than the MARA algorithm's 1.54651. This underscores the MCD-BP algorithm's commendable performance in PSD. Furthermore, the MCD-BP algorithm demonstrates a significant advantage in terms of cross-degree means. In the majority of topologies, the cross-degree mean of the MCD-BP algorithm is typically 0, suggesting that there are virtually no common edges between its backup path and the optimal path. This feature substantially reduces the risk of multipath failure due to a single fault point, thereby enhancing network robustness. These findings further confirm the effectiveness and reliability of the MCD-BP algorithm in identifying backup paths. Despite the remarkable performance of the MCD-BP algorithm across various experiments, results indicate that network topology complexity can influence its efficiency. For instance, in real topology Agis, the MCD-BP algorithm's cross-degree mean is 0. However, in topologies A19719, A19723, and A19728, there is noticeable variation in this metric. This likely stems from the inherent complexity of these structures, characterized by a larger number of nodes and links. Such complexity requires the algorithm to account for more factors when identifying backup paths, subsequently impacting the cross-degree mean. In simulated topologies, as the scale increases (e.g., from 20 to 4 to 100-4), the MARA algorithm's cross-degree mean exhibits a consistent increase. Meanwhile, although the MCD-BP algorithm consistently maintains the lowest cross-degree mean, its computational complexity may also escalate with increased topology scale. Thus, for practical deployments, it is imperative to evaluate both the network topology's complexity and the algorithm's computational complexity to ascertain that the optimal backup path can be identified within a reasonable timeframe. Every link failure triggers a fresh Dijkstra search, delaying convergence from milliseconds to seconds. For backbone networks thatmust react within micro-seconds, this is prohibitive. Consequently, scalability beyond a few hundred nodes remains questionable without approximations, caching, or distributed pre-computation. ## 8 Conclusions

This paper provides a comprehensive overview of the intra-domain routing protection scheme, which is predicated on the minimum cross-degree backup path. We focus on the crucial index of the cross-degree between two paths and establish link deletion rules to generate a backup path that shares the fewest common edges with the original path. Through rigorous testing across diverse network topologies, we ascertain that the proposed MCD-BP algorithm exhibits a high fault protection rate, minimal path stretch, and a low average crossing degree. These findings suggest that the routing protection mechanism performs well, thereby mitigating the impact of network failures and bolstering the overall stability of the network. While the MCD-BP algorithm enhances network failure protection and stability, it encounters challenges in real-world implementations. Its time complexity is \(O(|V|-1)\times O(\log|E|+\log|E|\log|K|)\), potentially resulting in significant computational overhead in expansive networks.