A Novel Approach to Comparison-Based Diagnosis
for Hypercube-Like Systems
*CHIEH-FENG CHIANGAND JIMMY J. M. TAN
Department of Computer Science National Chiao Tung University
Hsinchu, 300 Taiwan
Interconnection network has been an active research area for parallel and distributed computer systems. The diagnosability is one of the important issues in the reliability of interconnection networks. In this paper, a novel idea on system diagnosis called local diagnosability is presented. The concept of local diagnosability is strongly related to the traditional global one. For this local sense, the status of every particular processor can be correctly identified. A sufficient condition is also proposed to determine the local nosability of a given processor. Following this local sense, we prove that the diag-nosability of an n-dimensional hypercube-like network HLn is n for n ≥ 5, and show that in HLn with up to n − 2 faulty links, the local diagnosability of each processor equals to the connection links incident with it.
Keywords: locally t-diagnosable, local diagnosability, comparison model, extended star, hypercube-like system
1. INTRODUCTION
With the rapid development of technology, multiprocessor systems are more and more popular. One of the important issues about multiprocessor systems is the reliability of the processors in it. In order to maintain the reliability of a system, if a processor is found faulty, it should be replaced by a fault-free one. The procedure of identifying faulty processors is called the diagnosis of the system. The maximum number of faulty proces-sors that a system can guarantee to identify is called the diagnosability of the system.
There are several approaches for interconnected processors to perform self-diag- nosis. One major approach is called the comparison model, first proposed by Maeng and Malek [6, 7]. This approach performs diagnosis by sending the same inputs to a pair of adjacent processors and comparing their responses. Under this model, an n-dimensional hypercube is proved to have diagnosability n for n ≥ 5 [11], and an n-dimensional en-hanced hypercube has diagnosability n + 1 for n ≥ 6 [12]; Fan [3, 4] showed that the di-agnosability of an n-dimensional Crossed cube is n, and that of an n-dimensional Möbius cube is n for n ≥ 4; a k-ary n-dimensional butterfly graph has diagnosability 2k − 2 when k ≥ 3 and n ≥ 2 [1]; Lai et al. [5] proposed that the diagnosability of the matching com-position network is n for n ≥ 4; Chang et al. [2] investigated the diagnosabilities of regu-lar networks.
In the previous studies on diagnosis, most investigators focused on the global diag-Received February 2, 2007; accepted July 13, 2007.
nosis ability of a system but ignored some local systematic details. For example, if the diagnosability of a system is t, this system is t-diagnosable. That is, given any syndrome σ, all the faulty processors in a system G can be precisely identified, given that there are at most t faulty processors in system G. But in some situations, it is possible to correctly point out all faulty processors in some part of the system G, even if there are more than t faulty processors in G. Thus, only considering the global status let us lose some local details of a system.
In this paper, we propose a new concept on system diagnosis, which is called local diagnosability. More local information about a system can be retrieved through this con-cept. By our new definition of diagnosability, every processor in a system has its own local diagnosability which states some kind of connection status around it. Moreover, we propose a sufficient condition to easily compute the local diagnosability of each proces-sor based on the comparison model. Finally, we can get back to the original global nosis in the point of view of local diagnosis. As a case study, we prove that the diag-nosability of an n-dimensional hypercube-like network HLn is n for n ≥ 5 via this local
sense, and show that in an n-dimensional hypercube-like network with up to n − 2 faulty links, the local diagnosability of each processor equals to the connection links incident with it.
2. PRELIMINARIES
In this section, the basic graph definitions and notations are given [13]. G = (V, E) is a graph if V is a finite set and E is a subset of {(u, v)⎪(u, v) is an unordered pair of V}. The degree of node v in a graph G is the number of edges incident with v. A node cover of G is a subset Q ⊂ V(G) such that every edge of E(G) has at least one end node in Q. A node cover set with the minimum cardinality is called a minimum node cover.
Vaidya et al. [10] introduced a class of hypercube-like interconnection networks, which can be defined by applying a ⊕ operation repeatedly as follows: HL0 = {K1},
where K1 is a graph with just one node and no edges; for m ≥ 1, HLm = {HL0m−1⊕HL1m−1| 0 1, m HL − 1 1 1}, m m
HL − ∈HL − which has node set 0 1
1 1
( m ) ( m )
V HL − ∪V HL − and edge set 0 1 ( m ) E HL − ∪ 1 1 ( m ) ,
E HL − ∪M where M is an arbitrarily perfect matching between the node sets of 0 1 m HL − and 1 1. m
HL − That is, M is a set of edges connecting the nodes in 0 1 m HL − and in 1 1 m HL − with a bijection.
For the purpose of self-diagnosis, several different models had been proposed. The comparison model, also called the MM model, proposed by Maeng and Malek [7, 8], is considered as a major approach for fault diagnosis in multiprocessor systems. In this ap-proach, every processor performs diagnosis by sending two identical signals to two other linked processors, and comparing their returning responses. The agreement is denoted by 0, whereas the disagreement is denoted by 1. After the completion of diagnosis on every processor, a syndrome is generated by collecting all these testing results. For a given syn-drome σ, a subset of processors F ⊂ V(G) is said to be consistent with σ if the syndrome σ can be produced when all processors in F are faulty and all processors in V − F are fault-free. We say that a system is diagnosable if, for every syndrome σ, an unique set of processors F ⊂ V is consistent with it. A system is defined to be t-diagnosable if the
sys-tem is diagnosable as long as the number of faulty processors does not exceed t. The maximum number t for a system being t-diagnosable is called the diagnosability of the system. Two distinct subsets of processors F1, F2 ⊂ V are distinguishable if and only if
every syndrome consistent with F1 is different to those consistent with F2.
A labeled multigraph M = (V, C) is usually used to model this diagnosis strategy, where V represents the set of all processors in G and C represents the set of labeled edges. Each labeled edge (u, v)w∈ C implies that processors u and V are being compared by
processor w.
The following is a characterization presented by Sengupta and Dahbura to deter-mine the distinguishability of two sets of processors in a system.
Lemma 1 [9] For every two distinct subsets of nodes F1 and F2, (F1, F2) is a
distin-guishable pair if and only if at least one of the following conditions is satisfied: (1) ∃u, w ∈ V − F1 − F2 and ∃v ∈ (F1 − F2) ∪ (F2 − F1) such that (u, v)w ∈ C,
(2) ∃u, v ∈ F1 − F2 and ∃w ∈ V − F1 − F2 such that (u, v)w ∈ C, or
(3) ∃u, v ∈ F2 − F1 and ∃w ∈ V − F1 − F2 such that (u, v)w ∈ C.
3. LOCAL DIAGNOSABILITY
In this section, we define the concept of local diagnosability, and provide some practical theorems about it. Applying these theorems, the traditional global diagnosability of a system can be easily computed.
Definition 1 A system G(V, E) is locally t-diagnosable at node x ∈ V(G) if, given a test syndrome 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 σF and |F′| ≤ t,
must also contain node x.
Definition 2 The local 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 locally t-diagnosable at x.
The concept of a system being locally t-diagnosable at a node x is consistent with the traditional concept of a system being t-diagnosable in the global sense. We can see the relationship between these two concepts by the following theorem.
Theorem 1 A system G(V, E) is t-diagnosable if and only if G is locally t-diagnosable at x, for every x ∈ V(G).
In the following, we propose a sufficient condition for verifying whether a system G is locally t-diagnosable at a given node x. Before discussing it, we need some definitions of terminologies. For any set of nodes S ⊂ V(G), let G − S denote the subgraph of G in-duced by the node subset V(G) − S. Let S be a set of nodes and x be a node not in S, we use Cx,S to denote the connected component which x belongs to in G − S.
Theorem 2 A system G(V, E) is locally 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 cover including x of the component Cx,S is at least 2(t − p) + 1.
Proof: We prove this theorem by ways of contradiction. Suppose on the contrary that G is not locally t-diagnosable at node x. By Definition 1, we know that there exists two distinct sets of nodes F1 ≠ F2 ⊂ V with |Fi| ≤ t, i = 1, 2, and x ∈ (F1 ∪ F2) − (F1 ∩ F2),
such that (F1, F2) is an indistinguishable pair. Let S = F1 ∩ F2, then |S| = p ≤ t − 1.
Ac-cording to the condition, the cardinality of a node cover including x of the component Cx,S is at least 2(t − p) + 1. Since |(F1 ∪ F2) − (F1 ∩ F2)| ≤ 2(t − p) and x ∈ (F1 ∪ F2) −
(F1 ∩ F2), there is at least one node in the node cover of Cx,S lying in Cx,S − [(F1 ∪ F2) −
(F1 ∩ F2)], and there is at least one edge of Cx,S lying in Cx,S − [(F1 ∪ F2) − (F1 ∩ F2)],
consequently. Then, (F1, F2) is a distinguishable pair since it satisfies condition 1 of
Lemma 1. Therefore G is locally t-diagnosable at node x, which is a contradiction. We now propose a substructure at a node to guarantee its local diagnosability.
Fig. 1. Extended star sturcture ES(x; n).
Definition 3 Let x be a node in a graph G(V, E). For n ≤ deg(x), an extended star ES(x; n) of order n at node x is defined as ES(x; n) = (V(x; n), E(x; n)), where the set of nodes V(x; n) = {x} ∪ {vij |1 ≤ i ≤ n, 1 ≤ j ≤ 4} and the set of edges E(x; n) = {(x, vk1), (vk1, vk2),
(vk2, vk3), (vk3, vk4)|1 ≤ k ≤ n}.
We say that there is an extended star structure ES(x; n) ⊆ G at node x if G contains an extended star ES(x; n) of order n at node x as a subgraph.
As stated in the following theorem, the extended star is an useful structure to check the local diagnosability of a node in a system. Then, the global diagnosability can be also derived consequently.
Theorem 3 Let x be a node in a system G(V, E) with deg(x) = n. The local diagnosabil-ity of x is n if there exists an extended star ES(x; n) ⊆ G at x.
Proof: For the first part of this proof, we claim that the local diagnosability of x is at most n. Suppose on the contrary that the local diagnosability of x is larger than n, i.e. tl(x)
> n. Let the nodes adjacent to x be vk, for all k, 1 ≤ k ≤ n. Let F1 be the set {x} ∪ {vk | k =
1 to n} and F2 be the set {vk | k = 1 to n}. Then, (F1, F2) is not a distinguishable pair
a contradiction. Therefore, the local diagnosability of x is at most n, that is, tl(x) ≤ n.
Now, we claim that the local diagnosability of x is at least n if there exists an ex-tended star ES(x; n) ⊆ G at x. We use Theorem 2 to prove this result. First, we define lk =
(vk1, vk2, vk3, vk4) to be a quadruple of four consecutive nodes for any k, 1 ≤ k ≤ n, with
respect to ES(x; n). We note that lk is a path of length 3. Accordingly, the cardinality of a
node cover of each lk is at least 2. Let S ⊂ V(G) be a set of nodes in G with |S| = p ≤ n − 1
and x ∉ S. After deleting S from V(G), there are at least (n − p) complete lk’s still
remain-ing in ES(x; n), where the word “complete” means that all vk1, vk2, vk3, and vk4 of an lk
have not been deleted in G − S. Thus, the cardinality of a node cover including x of the connected component Cx,S is at least 1 + 2(n − p). Therefore, the system G with an
ex-tended star ES(x; n) is locally n-diagnosable at x by Theorem 2. By Definition 2, the local diagnosability of x is at least n, that is, tl(x) ≥ n.
Therefore, tl(x) = n, and the proof is complete.
4. DIAGNOSABILITY OF HYPERCUBE-LIKE NETWORKS
For many practical interconnection networks, an extended star structure does exist and can be found without too much effort. As an example, we apply the concept of local diagnosability to the class of hypercube-like networks.
Theorem 4 The diagnosability of an n-dimensional hypercube-like network HLn is n
for n ≥ 5.
Proof: We will find an extended star structure at every node, and then the result follows from Theorems 1 and 3. We now prove it by induction on n, the dimension of a hyper-cube-like network HLn.
BASIS For the basis of this theorem, we claim that HL5 is 5-diagnosable. For every
node x in HL5, we can easily find an extended star subgraph ES(x, 5) at x since n is
greater than or equal to five. By Theorem 3, the system HL5 is locally 5-diagnosable at
every node. So HL5 is 5-diagnosable by Theorem 1.
HYPOTHESIS Suppose HLn−1 contains an extended star of order n − 1 at every node
as a subgraph, which implies that this theorem holds for HLn−1.
INDUCTION Consider an n-dimensional hypercube-like network HLn. We want to
show that there is an extended star structure at each node in HLn. Consider an arbitrary
node x in HLn, we can separate HLn into two HLn−1, denoted by HL0n−1 and HL1n−1. With-out loss of generality, we may assume that x is in 0
1. n
HL− By hypothesis, there is an ex-tended star subgraph ES(x, n − 1) at x in 0
1. n
HL− Consider the corresponding node x′ in 1
1, n
HL− that is, the node in 1 1 n
HL− which is adjacent to x. There is also an extended star subgraph ES(x′, n − 1) at x′ in 1
1. n
HL− Hence there is an extended star subgraph ES(x, n) at x in HLn. So, the system HLn is locally n-diagnosable at node x by Theorem 3, and HLn
Furthermore, we consider the case of an interconnection network with some com-munication link faults, and study the effect on its local diagnosability.
Theorem 5 In an n-dimensional hypercube-like network HLn (n ≥ 5) with at most n − 2
edge faults, the local diagnosability of each node equals to its degree.
Fig. 2. An indistinguishable pair.
Proof: First we explain why the local diagnosability of each node in HLn may not equal
to its degree if there are n − 1 faulty edges. We give an example in Fig. 2. Let b be an arbitrary node and a be another node adjacent to it. Suppose all edges exclusive {(a, b)} incident with node a are faulty. We can see that (F1, F2) is an indistinghushable pair by
Lemma 1. In this situation, |F1| = |F2| = n, the degree of b equals to n, but the local
diag-nosability of node b does not equal n.
Now we are going to prove this theorem by induction on n. For the base case n = 5, if the number of edge faults in an HLn is less than or equal to three (n − 2 = 3), for each
node with degree k where k ≤ n, it is straightforward though tedious to find an extended star structure of order k at this node. As a consequence, the local diagnosability of each node can be easily checked by this extended star structure, and it will be the same as its degree. For induction hypothesis, we suppose for each node with degree k on an HLn-1
with n − 3 edge faults ((n − 1) − 2 = n − 3), there exists an extended star of order k at this node, which implies that the local diagnosability of every node equals to its degree. Con-sider an HLn which has n − 2 faulty edges and can be constructed with two copies, that is,
0 1 n HL− and 1 1. n
HL− We say an edge is crossed edge if its one end is in 0 1 n
HL− and the other end is in HL1n−1. Let x be an arbitrary node. Without loss of generality, we let x be in 0
1 n
HL− and deg(x) = m, and x′ be its corresponding node in 1 1 n
HL− and deg(x′) = m′. Our proof is divided into the following two major cases.
(a) (b) Fig. 3. Case1 of the proof in Theorem 5.
Case 1: There are k faulty edges that are crossed edges, where 1 ≤ k ≤ n − 2. Case 1.1: Edge (x, x′) is faulty. See Fig. 3 (a).
Since there are k faulty edges in the crossed edge, where 1 ≤ k ≤ n − 2, the number of faulty edges in 0
1 n
HL− is at most n − 3. By induction hypothesis, there exists an ex-tended star of order m at node x, and then the local diagnosability of x in this HLn with n
− 2 faulty edges equals to its degree m.
Case 1.2: Edge (x, x′) is fault-free. See Fig. 3 (b).
As in Case 1.1, the number of faulty edges in both HL0n−1 and HL1n−1 are at most n − 3. Hence there is a subgraph ES(x, m − 1) at x in 0
1 n
HL− and a subgraph ES(x′, m′ − 1) in 1
1. n
HL− So there is a subgraph ES(x, m) at x in HLn. By Theorem 3, the local
diag-nosability of x in HLn with n − 2 faulty edges equals to its degree m. Case 2: None of the faulty edges are crossed edges.
Case 2.1: All faulty edges are in HL0n−1 (respectively, or HL1n−1). That is, there are n − 2 faulty edges in 0 1 n HL− (respectively, or 1 1). n HL− (a) (b) Fig. 4. Case 2.1 of the proof in Theorem 5.
If there is a faulty edge s incident with x (see Fig. 4 (a)), we may treat edge s as fault-free temporarily. Hence there are n − 3 faulty edges in 0
1. n
HL− By hypothesis, we can find an extended star ES(x, m − 1) in 0
1. n
HL− Consider the corresponding node x′ in 1
1, n
HL− we can also find an ES(x′, m′ − 1) in 1 1. n
HL− Therefore, we can easily find an ex- tended star ES(x, m) of order m in HLn. Since the edge s is faulty, there is still an ES(x, m
− 1) in this HLn. So the local diagnosability of x in HLn with n − 2 faulty edges equals to
its degree by Theorem 3.
Now we follow the notations in Definition 3. If there is a faulty edge s belonging to {(v11, v12), (v21, v22), …, (vn1, vn2)} (see Fig. 4 (b)), we may treat edge s as fault-free
tem-porarily, and then there are n − 3 faulty edges in 0 1. n
HL− By hypothesis, there is an ES(x, m − 1) at x in 0
1. n
HL− Consider the corresponding node x′ in 1 1, n
HL− we can also find an ES(x′, m′ − 1) at x′ in 1
1. n
HL− Consider node y′ in 0 1, n
HL− we can find an ES(y′, deg(y′) − 1) in 0
1. n
HL− Therefore, we can easily find an ES(x, m) in HLn. Although the edge s is
faulty, the local diagnosability of x in HLn with n − 2 faulty edges equals to its degree by
If there is a faulty edge s belonging to {(v12, v13), (v22, v23), …, (vn2, vn3)} or {(v13,
v14), (v23, v24), …, (vn3, vn4)}, it can be proved by using the same argument.
Case 2.2: There are k0 faulty edges in HL0n−1 and k1 faulty edges in HL1n−1, where 1 ≤ k0,
k1 ≤ n − 3.
Because the number of faulty edges in HL0n−1 is at most n − 3 by induction hypothe-sis, there exists an extended star ES(x, m − 1) in 0
1, n
HL− and there exists an extended star ES(x′, m′ − 1) in 0
1. n
HL− So we can find an ES(x, m) in this HLn. Therefore the local
di-agnosability of x in HLn with n − 2 faulty edges equals to its degree by Theorem 3.
In Cases 1 and 2, we proved all possible distributions of faulty edges. Therefore, the proof is complete.
5. CONCLUSIONS
The reliability of interconnection networks is an important issue. The diagnosability is also an important factor in measuring the reliability of interconnection networks. In this paper, we proposed a new concept called the local diagnosability, and presented an useful local structure called extended star structure to find the local diagnosability of multiprocessor systems under the comparison-based diagnosis model. Then we prove the diagnosability of an n-dimensional hypercube-like network is n for n ≥ 5, and show that the local diagnosability of each node in an n-dimensional hypercube-like network equals to its degree if there are n − 2 faulty edges in it. An interesting problem worth studying will be to identify the faulty or fault-free status of a given node using this useful local structure.
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Chieh-Feng Chiang (江玠峰) received the B.S. and M.S. degrees in Computer and Information Science from National Chiao Tung University in 2002 and 2004, respectively. He is cur-rently a doctoral researcher in the Department of Computer Sci-ence, National Chiao Tung University. His research interests in-clude interconnection network, analysis algorithm and graph the-ory.
Jimmy J. M. Tan (譚建民) received the B.S. and M.S. de-grees in Mathematics from National Taiwan University in 1970 and 1973, respectively, and the Ph.D. degree from Carleton Uni-versity, Ottawa, Canada, in 1981. He has been on the faculty of the Department of Computer Science, National Chiao Tung Uni-versity, since 1983. His research interests include design and analysis of algorithms, combinatorial optimization, and intercon-nection networks.
