A system is defined to be diagnosable if, for every syndrome σ, an unique set of nodes F ⊆ V is consistent with it. In addition, we call a system t-diagnosable if the system is diagnosable as long as the number of faulty processors is at most t. The maximum number t for a system to be t-diagnosable is called the diagnosability of the system. Two distinct subsets of nodes F1, F2 ⊂ V are distinguishable if and only if σF1 ∩ σF1 = φ;
otherwise, F1 and F2 are indistinguishable.
The following is an useful characterization, proposed by Sengupta and Dahbura [30], for the distinguishability of two sets of nodes under the comparison model.
Lemma 1 [30] For every two distinct subsets of nodes F1 and F2, that is, F1 = F2 and F1, F2 ⊂ V , (F1, F2) is a distinguishable pair if and only if at least one of the following conditions is satisfied (as illustrated in Figure 2.1):
1) ∃ u, w ∈ V − F1− F2 and ∃ v ∈ F1ΔF2 such that (u, v)w ∈ C,
Figure 2.1: illustration of Lemma 1 — the distinguishability of two distinct subsets of nodes.
The detailed proof of this lemma was demonstrated by Sengupta and Dahbura [30]. For the completeness of this thesis, we sketch the proof briefly. If one of the three conditions holds, the distinguishability is absolutely determined:
i) Suppose condition 1) is satisfied. If v∈ F1− F2 then r((u, v)w) = 0 for each syndrome in σF2, and r((u, v)w) = 1 for each syndrome in σF1. Similarly, if v ∈ F2− F1 then r((u, v)w) = 0 for each syndrome in σF1, and r((u, v)w) = 1 for each syndrome in σF2. Either case implies σF1 ∩ σF2 = φ.
ii) Suppose condition 2) is satisfied. Then r((u, v)w) = 0 for each syndrome in σF2 and r((u, v)w) = 1 for each syndrome in σF1, which lead to σF1∩ σF2 = φ.
iii) Suppose condition 3) is satisfied, a similar argument is used as condition 2).
On the contrary, if none of the three conditions holds. We consider a syndrome such that for each (u, v)w ∈ C, the comparison result can be classified to the following nine situations [30]:
i) If u, v, w∈ V − F1− F2 then r((u, v)w) = 0.
ii) If w ∈ V − F1− F2 and u, v∈ F1 then r((u, v)w) = 1.
iii) If w∈ V − F1− F2 and u, v ∈ F2 then r((u, v)w) = 1.
iv) If w∈ V − F1− F2 and u∈ F1 and v ∈ F2 then r((u, v)w) = 1.
v) If w∈ F1− F2 and v∈ V − F2 and u∈ V − F1− F2 then r((u, v)w) = 0.
vi) If w∈ F2− F1 and v ∈ V − F1 and u∈ V − F1− F2 then r((u, v)w) = 0.
vii) If w∈ F1− F2 and u∈ F2 then for all v, r((u, v)w) = 1.
viii) If w∈ F2− F1 and u∈ F1 then for all v, r((u, v)w) = 1.
ix) Other arbitrary comparison results.
Then the syndrome above belongs to σF1 ∩ σF2, and therefore F1 and F2 are indis-tinguishable. For example, if w ∈ V − F1 − F2, u ∈ F1 ∩ F2, and v ∈ F1 − F2, then r((u, v)w) = 1 whenever the faulty set of nodes is either F1 or F2. In such circumstance, pair (F1, F2) can not be distinguished only with such few information.
Let G = (V, E) be a graph and let M = (V, C) be the comparison graph of G. Define the order graph [30] of a node u∈ V to be a digraph Gu = (Xu, Yu), where Xu ={v | either (u, v)∈ E or (u, v)w∈ C for some w} and Yu ={(v, w) | v, w ∈ Xu and (u, v)w ∈ C}.
For a given node u ∈ V , the order of u, order(u), is defined as the cardinality of a minimum node cover of Gu. For a subset of nodes U ⊂ V , define T (G, U) to be the set {v | (u, v)w∈ C and u, w ∈ U and v ∈ V − U}.
Next is a characterization proposed by Sengupta and Dahbura which gives a sufficient condition for a system being t-diagnosable.
Lemma 2 [30] A system G(V, E) with N nodes is t-diagnosable if 1) N ≥ 2t + 1,
2) each node has order at least t, and
3) for each U ⊂ V such that | U |= N − 2t + p and 0 ≤ p ≤ t − 1, | T (G, U) |> p.
Chapter 3
The Local Approach to Determining System Diagnosability
There were some studies on system diagnosability of some well-known networks under the comparison model. For example, Wang [31][32] presented that the diagnosability of an n-dimensional hypercube Qn is n for n ≥ 5 and the diagnosability of an n-dimensional enhanced hypercube is n + 1 for n ≥ 6. Fan [16][17] showed that the diagnosability of an n-dimensional Crossed cube is n, and the diagnosability of an n-dimensional M¨obius cube is n, for n ≥ 4. Lai et al. [24] proposed that the diagnosability of the matching composition network is n for n≥ 4.
As we observe, the traditional system diagnosability describes the global status of a system. The purpose of this dissertation for considering the node diagnosability is to keep the local connective detail of a system that we might neglect. For example, for any two integers m and n with m >> n ≥ 4, the diagnosability of two hypercube systems Qm and Qn is m and n [31][24], respectively. Combining these two systems with few communication links in some way may cause the diagnosability of the new topology to become n. In this situation, the strong diagnosis ability of some part of the entire system,
the substructure Qm, is ignored. Therefore, the need of keeping local information of each node is concerned.
In the previous studies on diagnosis, most results focused on the diagnosis ability of a system in a global sense: a system is t-diagnosable if all the faulty nodes can be identified given that there are at most t faulty nodes. In contrast to the global sense, we emphasize more on a single node x in a local sense: we require only identifying the status of one particular node correctly. More specifically, if x belongs to a set of faulty nodes, we must correctly identify x to be faulty; or, x is identified to be fault-free if x is indeed fault-free.
In a word, we are only concerned about the status of the node x.
3.1 Node Diagnosability
We now introduce the concept of a system being t-diagnosable at a given node.
Definition 1 A system G(V, E) is t-diagnosable at node x ∈ V (G) if, given a test syn-drome σ∈ σF produced by the system under the presence of a set of faulty nodes F con-taining node x with |F | ≤ t, every set of faulty nodes F consistent with σ and |F| ≤ t, must also contain node x.
An equivalent way of stating the above definition is given below.
Proposition 1 A system G(V, E) is t-diagnosable at node x∈ V (G) if, for each pair of distinct sets F1, F2 ⊂ V (G) such that F1 = F2, |F1|, |F2| ≤ t, and x ∈ F1ΔF2, (F1, F2) is a distinguishable pair.
Then, we define the node diagnosability of a given node as follows.
Definition 2 The node diagnosability tl(x) of a node x ∈ V (G) in a system G(V, E) is defined to be the maximum number of t for G being t-diagnosable at x, that is, tl(x) = max{t | G is t − diagnosable at x}.
The concept of a system being t-diagnosable at a node is consistent with the traditional concept of a system being t-diagnosable in the global sense. The relationship between these two is as follows.
Proposition 2 A system G(V, E) is t-diagnosable if and only if G is t-diagnosable at every node.
Proof. We prove the necessary condition first. Suppose that there exists a node y ∈ V (G) such that G is not t-diagnosable at y. By Proposition 1, there exists an indistinguishable pair (F1, F2) with |Fi| ≤ t, i = 1, 2, and y ∈ F1ΔF2. This contradicts that G is t-diagnosable. Next, we prove the sufficiency. Suppose G is not t-t-diagnosable. Then there exists an indistinguishable pair (F1, F2) with|Fi| ≤ t, i = 1, 2. Pick any node y in F1ΔF2, the system is not t-diagnosable at y by Proposition 1, which is a contradiction. 2
Proposition 3 The diagnosability t(G) of a system G(V, E) is equal to the minimum value among the node diagnosability of every node in G, that is, t(G) = min{tl(x) | x ∈ V (G)}.
Proof. The result follows trivially from Definition 2 and Proposition 2. 2
From Proposition 2 and 3, the relationship between the traditional diagnosability and the node diagnosability was pointed out. Through this concept, the system diagnosability
can be determined by testing the node diagnosability of each node. Especially in some well-known regular networks, the diagnosability can be easily identified because of the system symmetry. For example, in some graphs like hypercubes, cube-connected cycles, or complete graphs, the system diagnosability and the node diagnosability of each node in the system are the same, and such result can be applied in other applications.
In the following, we propose a sufficient condition for verifying whether a system G is t-diagnosable at a given node x.
Theorem 1 A system G(V, E) is t-diagnosable at a given node x ∈ V (G) if, for every set of nodes S ⊂ V (G), |S| = p, 0 ≤ p ≤ t − 1, and x /∈ S, the cardinality of every node
Proof. We prove it using contradiction. Suppose system G is not t-diagnosable at node x. According to Proposition 1, there exists an indistinguishable pair of distinct node set (F1, F2) with |F1| ≤ t, |F2| ≤ t, and x ∈ F1ΔF2. Let S be the intersection of node sets F1 and F2, then the cardinality of S is less than or equal to t−1. (Otherwise, if |S| = t, then F1 = F2.) According to the condition that x /∈ S and 0 ≤ |S| ≤ t − 1, the cardinality of every node cover including x of the component Cx,S is at least 2(t− p) + 1. Comparing this number with|F1ΔF2| ≤ 2(t−p), and x ∈ F1ΔF2, we get the fact that F1ΔF2 can not
be a node cover of Cx,S. In other words, at least one member (a node) of the node cover of Cx,S is outside F1ΔF2 (and also outside S according to the definition of component Cx,S). Consequently, by the property of node cover, there exists an edge e = (u, v) in Cx,S but outside F1ΔF2. Since edge e, nodes u, v, and node x belong to the same connected component Cx,S, there is a path leading from edge e to node x through set F1 (as shown in Figure 3.1(a)) or F2 (as shown in Figure 3.1(b)). Then by condition 1 of Lemma 1, (F1, F2) is a distinguishable pair. This is a contradiction, and the result follows. 2