## 4 Algorithm Analysis
Within the topology, excluding the link removed in Step 2, ascertain if a backup path exists from the source node \(s\) to the destination node \(d\). If such a path is identified, denote it as the final backup path \(bp(s,d)\). If not, proceed to Step 4. Step 4: Conduct a secondary search for backup paths utilizing the link deletion rule. Within the cross-degree set, identify and remove the edge nearest to the source node \(s\), taking precautions to avoid any disconnections post-deletion. Subsequently, employ the shortest path algorithm to determine a potential backup path \(sp^{\prime}(s,d)\) from the source node to the destination node. Repeat Steps 2 through 4 until no further links can be removed within the cross-degree, signifying the discovery of the backup path \(bp(s,d)\) with the minimal cross degree. By following the aforementioned steps, one can ensure that the backup path has minimal intersection with the optimal path during the search process for a backup path between two nodes. This subsequently augments the network's availability in the event of node or link failures. Consider the following topology, which comprises 11 nodes and 13 edges, as an illustration. The specific calculation process is elucidated below, using the source node \(b\) and the destination node \(k\) as an example. In Figure 2, the optimal backup path \(sp(b,k)=\{b,d,e,h,g,j,k\}\) from node \(b\) to node \(k\) is calculated according to [PERSON]'s algorithm. As described in Step 1 above, this path is then assigned to the previously initialized backup path \(sp^{\prime}(b,k)=\{b,d,e,h,g,j,k\}\). Subsequently, the cross-degree, denoted as \(K(sp(b,k),sp^{\prime}(b,k))=\{(b,d),(d,e),(e,h),(h,g),(g,j),(j,k)\}\), is determined, representing a sequence of links. Then, delete links in the cross-degree. As per the guidelines outlined in Step 3, there is no backup path from node \(b\) to node \(k\). Therefore, transition to Step 4 as previously mentioned. Delete link \((b,d)\) from the original topology diagram, and the corresponding backup path \(sp^{\prime}(s,d)\) can be obtained as \(\{b,a,d,e,h,g,j,k\}\). Subsequently, revisit and execute Steps 2 to 4. As per Step 2, recalculate the crossing degree \(K(sp(b,k),sp^{\prime}(b,k))=\{(d,e),(e,h),(h,g),(g,j),(j,k)\}\). Then, eliminate the set of links in the crossing degree from the original topology diagram; it can be seen that node \(b\) cannot reach the destination node \(k\). In the subsequent phase, pursuant to Step 4, it is imperative to eliminate edge \((d,e)\). Upon this elimination, utilization of the shortest path algorithm will ascertain that node \(b\) cannot establish a connection with the target node \(k\), thereby inducing a network disconnection. Consequently, we persist in our search for links within the intersection set that are proximate to the source point and can be removed without causing a disconnection. Upon the removal of link \((e,h)\), \(sp^{\prime}(s,d)\) is updated to \(\{b,a,d,e,f,j,k\}\). Upon recalculating the cross-degree of the backup path with the optimal path, we obtain \(K(sp(b,k),sp^{\prime}(b,k))=\{(d,e)\}\). Given that there is now only one link \((d,e)\) in the cross-degree set and its removal would render the destination node \(k\) unreachable, the algorithm concludes. The final backup path is denoted as \(bp(b,k)=\{b,a,d,e,f,i,k\}\), having a cross-degree of \(K(sp(b,k),sp^{\prime}(b,k))=\{(d,e)\}\) with a corresponding value of \(1\). ## 5 MCD-BP

The Minimum Crossover Degree Backup Path (MCD-BP) algorithm is an intra-domain routing protection mechanism that leverages the Minimum Cross-Degree Backup Path principle. The primary function of this algorithm is to compute and establish a backup path in advance, thereby enabling a swift response to failures [17]. This approach significantly reduces the duration and consequence of data transmission interruptions. The fundamental premise of the algorithm is to choose a backup path that exhibits minimal cross-degree with the optimal path.