# IFDA: Intermittent Fault Diagnosis Algorithm for Augmented Cubes Under the PMC Model
## 5 IFDA for Augmented Cubes  
In this section, we aim to design a quick and accurate intermittent fault diagnosis algorithm applicable to augmented cube networks, which is named the intermittent fault diagnosis algorithm (IFDA). The augmented cube is one instance of a Hamiltonian network, thus all nodes within this network can form a Hamiltonian cycle. The proposed algorithmemploys the cycle partitioning strategy introduced in [23] to decompose the target network into a series of subpaths. Then, it determines the states of nodes within certain subpaths based on their inherent properties. Finally, the DFS strategy is invoked to diagnose the states of unknown nodes, with isolated nodes that without a 0-path being identified as faulty. The cycle partitioning algorithm is detailed in Algorithm 1, followed by an analysis of the subpath properties. ```
0: A \(N\)-node cycle with syndrome based on PMC model. 0: The set of subpaths \(\{P_{1},P_{2},\ldots,P_{s}\}\). Step 1: Choose a 0-path \(a_{0}\) followed by a 1-path in the clockwise direction. Step 2: Let \(a\) be the edge following \(a_{0}\). If \(a\) is 0-path, updating \(a_{0}\) to \(a\) and continue execute Step 2; otherwise, go to Step 3. Step 3: Mark with an \(X\) the edge following \(a\). If it was not previously marked, set \(a_{0}\) as the next edge of the \(X\)-marked and go to Step 2; otherwise, the algorithm terminates. ```  
**Algorithm 1** Cycle-Partitioning Algorithm Based on the PMC Model [23]  
It should be noted that if no starting node satisfies the given conditions in Step 1, all nodes will be deemed to be in a fault-free state. Therefore, the aforementioned scenario will not be considered in the subsequent sections of this paper. For the algorithm, Property 3 below is consistent with the corresponding property in [23]. The test result between adjacent nodes corresponds to a 1-path; it indicates the presence of at least one faulty node between these two nodes. Consequently, the following Property 4 also holds. **Property 3** ([23]). : _The test results for all subpaths always follow the following pattern: \(\{0\}_{i}1\), where \(0\leq i\leq 2^{n}-1\)._  
**Property 4**. : _In every subpath created by the cycle-partition algorithm, there exists at least one intermittent faulty node._  
**Property 5** ([22]). : _There are subpaths with lengths greater than or equal to \(\lceil N/s\rceil\) by the Pigeonhole principle, where \(s\) is the number of subpaths._  
It is observed that the states of all nodes in the subpaths, excluding the tail node, must exhibit consistency. Therefore, the subpaths are categorized into two types based on the state of the head node: Type-I subpaths and Type-II subpaths. The head node in a Type-I subpath is in a fault-free state, whereas the head node in a Type-II subpath is in a faulty state. **Lemma 15** ([23]). : _Let \(t\) be the number of intermittent faulty nodes in an \(N(N\geqslant 3)\) node cycle system, and \(s\) is the number of subpaths. If a subpath satisfies the inequality \(|P_{i}|-1>t-(s-1)\), then it is classified as a Type-I subpath._  
In [22; 23; 24], the evaluation methods for the fault bound are essentially consistent, yet [24] presents a more concise evaluation approach. According to the fault bound evaluation method in [24], Lemma 16 restates the relationship between the fault bound \(T\) and the number of nodes \(N\). **Lemma 16** ([22]). : _Let \(n\) and \(l\) be non-negative integers. In a path with \(N\) (\(N\geq 4\)) nodes, the fault bound under the PMC model is calculated using the following formula:_  
\[T=2n+[l]-3 \tag{1}\]_where \(N=n^{2}+l\), and \(n=\lfloor\sqrt{N}\rfloor\). If \(0<l\leq n\), then, \([l]=1\); if \(n<l\), then \([l]=2\); otherwise, \([l]=0\)._  
Subsequently, we introduce the Algorithm 2 for diagnosing the status of all nodes in the system. In the first step of the intermittent fault diagnosis algorithm, the cycle-partition strategy is invoked to decompose the cycle into a series of subpaths. And then, Lemma 16 is applied to pick out Type-I subpaths that meet the criteria.