## 4 Algorithm Analysis  
In the intra-domain routing protection scheme proposed in this paper, which is based on minimal cross-degree backup path, the backup path discovery process involves link removal and path recomputation. This procedure aims to minimize the cross-degree between the backup path and the optimal path, thereby mitigating the impact of single-point failures on routing protection. The link removal rules are formally defined below. **Theorem 4** (link deletion rule). : _In the topology graph \(G=(V,E,W)\), \(sp(s,d)\) represents the optimal path from the source node \(s\) to the destination node \(d\), and the backup path is denoted as \(sp^{\prime}(s,d)\). Let \(K(sp(s,d),sp^{\prime}(s,d))=\{(s,s_{1}),(s_{1},s_{2}),(s_{2},s_{3})\}\) be the cross-degree between the optimal path and the backup path. When deleting links, sequentially remove the links closer to the source node \(s\) and re-plan the backup path._  
Proof. : (Reversal method). Assume the optimal path is \(sp(s,d)=\{s,s_{1},s_{2},s_{3},d\}\), and according to [PERSON]'s algorithm, the backup path obtained is \(sp^{\prime}(s,d)=\{s,s_{1},s_{2},s_{3},s_{5},d\}\), and the cross-degree between them is \(K(sp(s,d),sp^{\prime}(s,d))=\{(s,s_{1}),(s_{1},s_{2}),(s_{2},s_{3})\}\). Assuming that the links in the cross-degree are sorted in order, starting with the removal of the link closest to the destination node, such as removing the \((s_{2},s_{3})\) link. The backup path is then recalculated, resulting in a sequence \(sp^{\prime}(s,d)=\{s,s_{1},s_{2},s_{4},s_{5},d\}\). At this point, the cross-degree between the two paths remains \(K(sp(s,d),sp^{\prime}(s,d))=\{(s,s_{1}),(s_{1},s_{2})\}\). Next, again starting with the removal of the link closest to the destination node, the \((s_{1},s_{2})\) link is deleted; assume that the recalculated backup path at this point is \(sp^{\prime}(s,d)=\{s,s_{1},s_{6},s_{7},s_{2},s_{3},s_{5},d\}\), \(K(sp(s,d),sp^{\prime}(s,d))=\{(s,s_{1}),(s_{2},s_{3})\}\) represents the current cross-degree, and link \((s_{2},s_{3})\) reappears in the backup path, which fails to ensure that the backup path and the optimal path have the minimum cross-degree. Therefore, we choose to remove links from the source node within the cross-degree and recalculate the backup path. The subsequent pseudocode (Algorithm 1) elucidates the method of proof by contradiction:```
input :\(G\), optimal path \(sp(s,d)\), possible initial backup \(sp^{\prime}(s,d)\)
1\(K\gets sp(s,d)\cap sp^{\prime}(s,d)\)
2 removed \(\leftarrow\emptyset\)
3 for\(i\leftarrow|K|\)dovnto 1 do
4\(e\gets K[i]\)
5 removed \(\leftarrow\) removed \(\cup\{e\}\)
6\(G^{\prime}\leftarrow\)remove_link\((G,e)\)
7\(sp^{\prime}\leftarrow\) [PERSON]\((G^{\prime},s,d)\)
8\(K^{\prime}\gets sp(s,d)\cap sp^{\prime}(s,d)\)
9 if\(\exists\)link \(\in K^{\prime}\cap\)removedthen
10 break//reversedelete fails
11 end if
12
13 end for
```

**Algorithm 1**Reverse link delete explanation

Under the established link deletion rules, the following key steps are used to search for a backup path. Step 1: Initialize the best path and possible backup paths. First, [PERSON]'s algorithm is utilized to compute the best path \(sp(s,d)\) from the source node \(s\) to the destination node \(d\). The advantage of this algorithm is that it gives preference to the path with the minimum cumulative weight, which results in the path with the lowest cost. Then, the computed best path \(sp(s,d)\) is initially assigned to the initial backup path \(sp^{\prime}(s,d)\). Step 2: Determine the cross-degrees between the backup and optimal paths. Remove the links within this intersection set from the topology graph. Compute \(K(sp(s,d),sp^{\prime}(s,d))=\{sp(s,d)\cap sp^{\prime}(s,d)\}\), which identifies shared links between the two paths, referred to as common edges. Subsequently, eliminate these shared links from the topology graph. Step 3: Determine the backup path for the node.