Pao-Lien Lai, Jimmy J.M. Tan, Chang-Hsiung Tsai,
and Lih-Hsing Hsu
Abstract—The classical problem of diagnosability is discussed widely and the diagnosability of many well-known networks have been explored. In this paper, we consider the diagnosability of a family of networks, called the Matching Composition Network (MCN); two components are connected by a perfect matching. The diagnosability of MCN under the comparison model is shown to be one larger than that of the component, provided some connectivity constraints are satisfied. Applying our result, the diagnosability of the Hypercube Qn, the Crossed
cube CQn, the Twisted cube T Qn, and the Mo¨bius cube MQncan all be proven to
be n, for n 4. In particular, we show that the diagnosability of the four-dimensional Hypercube Q4is 4, which is not previously known.
Index Terms—Diagnosability, t-diagnosable, comparison model, Matching Composition Network, MM* model.
æ
1
I
NTRODUCTIONWITHthe rapid development of technology, the need for
high-speed parallel processing systems has been continuously increas-ing. The reliability of the processors in parallel computing systems is therefore becoming an important issue. In order to maintain the reliability of a system, whenever a processor (node) is found faulty, it should be replaced by a fault-free processor (node). The process of identifying all the faulty nodes is called the diagnosis of the system. The maximum number of faulty nodes that the system can guarantee to identify is called the diagnosability of the system.
In this paper, we consider the diagnosability of the system under the comparison model, proposed by Malek and Maeng [16], [17]. The diagnosability of some well-known interconnection networks under the comparison model has been investigated. For example, Wang [21], [22] showed that the diagnosability of an
n-dimensional hypercube Qnis n for n 5 and the diagnosability
of an n-dimensional enhanced hypercube is n þ 1 for n 6. Fan [12] proved that the diagnosability of an n-dimensional crossed cube is n for n 4. Araki and Shibata [1] proposed that the k-ary r-dimensional butterfly network BF ðk; rÞ is 2k-diagnosable for k
2 and r 5. Besides, the diagnosability of the Hypercubes, the
Crossed cubes, and the Mo¨bius cubes under the PMC diagnostic model were also studied in [2], [10], [11], [14].
We study the diagnosability of a family of interconnection networks, called the Matching Composition Networks (MCN), which can be recursively constructed. MCN includes many well-known interconnection networks as special cases, such as the
Hypercube Qn, the Crossed cube CQn, the Twisted cube T Qn, and
the Mo¨bius cube MQn. Basically, MCN and these mentioned cubes
are all constructed from two graphs G1 and G2 with the same
number of nodes by adding a perfect matching between the nodes
of G1 and G2. We shall call these two graphs G1 and G2 the
components of MCN.
for t 2. In other words, the diagnosability of MCN is increased by one as compared with those of the components. Using our result, it is straightforward to see that the diagnosability of the
Hypercube Qn, the Crossed cube CQn, the Twisted cube T Qn, and
the Mo¨bius cube MQn are n for n 4. Some of these particular
applications are previously known results [12], [22], using rather lengthy proofs. Our approach unifies these special cases and our proof is much simpler. We would like to point out that the
diagnosability of the four-dimensional Hypercube Q4is 4, which is
not previously known [12], [22]. The diagnosability of the Twisted
cube T Qnand the Mo¨bius cube MQn, as far as we know, are not
yet resolved until now.
The rest of this paper is organized as follows: Section 2 introduces the comparison model for diagnosis. Section 3 provides preliminaries. In Section 4, we present the Matching Composition Network and discuss its diagnosability. We then discuss the
diagnosability of Qn, CQn, T Qn, and MQnin Section 5. Finally, our
conclusions are given in Section 6.
2
T
HEC
OMPARISONM
ODEL FORD
IAGNOSISFor the purpose of self-diagnosis of a given system, several different models have been proposed in the literature [16], [17], [18]. Preparata et al. [18] first introduced a model, the so-called PMC-model, for system level diagnosis in multiprocessor systems. In this model, it is assumed that a processor can test the faulty or fault-free status of another processor.
The comparison model, called the MM model, proposed by Maeng and Malek [16], [17], is considered to be another practical approach for fault diagnosis in multiprocessor systems. In this approach, the diagnosis is carried out by sending the same testing task to a pair fu; vg of processors and comparing their responses. The compar-ison is performed by a third processor w that has direct communication links to both processors u and v. The third processor w is called a comparator of u and v.
If the comparator is fault-free, a disagreement between the two responses is an indication of the existence of a faulty processor. To gain as much knowledge as possible about the faulty status of the system, it was assumed that a comparison is performed by each processor for each pair of distinct neighbors with which it can communicate directly. This special case of the MM-model is referred to as the MM*-model. Sengupta and Dahbura [20] studied the MM-model and the MM*-MM-model, gave a characterization of diagnosable systems under the comparison approach, and proposed a poly-nomial time algorithm to determine faulty processors under MM*-model. In this paper, we study the diagnosability of MCN (which will be defined subsequently) under the MM*-model.
In the study of multiprocessor systems, the topology of networks is usually represented by a graph G ¼ ðV ; EÞ, where each node v 2 V represents a processor and each edge ðu; vÞ 2 E represents a communication link. The diagnosis by comparison approach can be modeled by a labeled multigraph, called the comparison graph, M ¼ ðV ; CÞ, where V is the set of all processors
and C is the set of labeled edges. A labeled edge ðu; vÞw2 C, with w
being a label on the edge, connects u and v, which implies that processors u and v are being compared by w. Under the MM-model, processor w is a comparator for processors u and v only if ðw; uÞ 2 E and ðw; vÞ 2 E. The MM*-model is a special case of the MM model; it is assumed that each processor w such that ðw; uÞ 2 E and ðw; vÞ 2 E is a comparator for the pair of processors
uand v. The comparison graph M ¼ ðV ; CÞ of a given system can
. P.-L. Lai, J.J.M. Tan, and L.-H. Hsu are with the Department of Computer and Information Science, National Chiao Tung University, Hsinchu, Taiwan 300, ROC. E-mail: jmtan@cis.nctu.edu.tw.
. C.-H. Tsai is with the Department of Computer Science and Information Engineering, Daohan Institute of Technology, Hualien, Taiwan 971, ROC. Manuscript received 8 Oct. 2002; revised 7 Nov. 2003; accepted 15 Jan. 2004. For information on obtaining reprints of this article, please send e-mail to: tc@computer.org, and reference IEEECS Log Number 117536.
be a multigraph for the same pair of nodes may be compared by several different comparators.
For ðu; vÞw2 C, the output of comparator w of u and v is
denoted by rððu; vÞwÞ, a disagreement of the outputs is denoted by
the comparison results rððu; vÞwÞ ¼ 1, whereas an agreement is
denoted by rððu; vÞwÞ ¼ 0.
In this paper, in order to be consistent with the MM model, we have the following assumptions [20]:
1. All faults are permanent;
2. A faulty processor produces incorrect outputs for each of
its given testing tasks;
3. The output of a comparison performed by a faulty
processor is unreliable; and
4. Two faulty processors with the same input do not produce
the same output.
Therefore, if the comparator w is fault-free and rððu; vÞwÞ ¼ 0,
then u and v are both fault-free. If rððu; vÞwÞ ¼ 1, then at least one of
u, v, and w must be faulty. The set of all comparison results of a
multicomputer system that are analyzed together to determine the faulty processors is called a syndrome of the system.
For a given syndrome , a subset of nodes F V is said to be consistent with if syndrome can be produced from the situation that all nodes in F are faulty and all nodes in V F are fault-free. Because a faulty comparator can lead to unreliable results, a given set F of faulty nodes may produce different syndromes. Let
ðF Þ ¼ f j is consistent with F g.
Two distinct sets S1; S2 V are said to be indistinguishable if and
only if ðS
1ÞTðS2Þ 6¼ ; otherwise, S1; S2 are said to be
distinguishable. A system is said to be t-diagnosable if, for every syndrome, there is a unique set of faulty nodes that could produce the syndrome, provided the number of faulty nodes does not exceed t.
3
P
RELIMINARIESWe need some definitions and previous results for further discussion. Let G ¼ ðV ; EÞ be a graph, if there are ambiguities, we shall write the node set V as V ðGÞ and edge set E as EðGÞ. Assume U V ðGÞ. G½U denotes the subgraph of G induced by the
node subset U of G and UU¼ V ðGÞ U.
The vertex connectivity (simply abbreviated as connectivity) of a network G ¼ ðV ; EÞ, denoted by ðGÞ or , is the minimum number of vertices whose removal leaves the remaining graph
disconnected or trivial. Assume that V1 and V2 are two disjoint
nonempty subsets of V ðGÞ. The neighborhood set of V1in V2, denoted
by NðV2; V1Þ, is defined as fx 2 V2j there exists a node y 2 V1such
that ðx; yÞ 2 EðGÞg. A vertex cover of G is a subset K V ðGÞ such that every edge of EðGÞ has at least one end vertex in K. A vertex cover set with the minimum cardinality is called a minimum vertex cover.
Given a graph G, let M be the comparison graph of G. For a
node v 2 V ðGÞ, we define Xv to be the set of nodes fu j ðv; uÞ 2
EðGÞgSfu j ðv; uÞw2 EðMÞ for some wg and Yv to be the set of
edges fðu; wÞ j u; w 2 Xv and ðv; uÞw2 EðMÞg. In [20], the order
graph of node v is defined as Gv¼ ðXv; YvÞ and the order of the
node v, denoted by orderGðvÞ, is defined to be the cardinality of a
minimum vertex cover of Gv. Let U V ðGÞ, we use T ðG; UÞ to
denote the set fv j ðu; vÞw2 EðMÞ and w; u 2 U; v 2 UUg. We observe
that T ðG; UÞ ¼ Nð UU; UÞ if G½U is connected and jUj > 1. This
observation can be extended to the following lemma.
Lemma 1. Let U be a subset of V ðGÞ and G½Ui, 1 i k, be the
connected components of the subgraph G½U such that U ¼Ski¼1Ui.
Then, T ðG; UÞ ¼Ski¼1fNð UU; UiÞ j jUij > 1g.
In Fig. 1, taking Q3as an example, we have T ðG; UÞ ¼ f4; 5; 6; 7g,
where U ¼ f0; 1; 2; 3g.
The next lemma follows directly from the definition of connectivity of G.
Lemma 2 [10].Let G be a connected graph and U be a subset of V ðGÞ.
Then, jNð UU; UÞj ðGÞ if j UUj ðGÞ and Nð UU; UÞ ¼ UU if
j UUj < ðGÞ.
There are several different ways to verify a system to be t-diagnosable under the comparison approach. In this paper, we need three theorems given by Sengupta and Dahbura [20]. The first two are necessary and sufficient conditions for ensuring distin-guishability, the third one is a sufficient condition for verifying a system to be t-diagnosable.
Theorem 1 [20].For any S1; S2where S1; S2 V and S16¼ S2, ðS1; S2Þ
is a distinguishable pair if and only if at least one of the following conditions is satisfied (see Fig. 2):
1. 9i; k 2 V S1 S2 and 9j 2 ðS1 S2ÞSðS2 S1Þ such
that ði; jÞk2 C,
2. 9i; j 2 S1 S2and 9k 2 V S1 S2such that ði; jÞk2 C,
or
3. 9i; j 2 S2 S1and 9k 2 V S1 S2such that ði; jÞk2 C.
Theorem 2 [20].A system is t-diagnosable if and only if each node has
order at least t and, for each distinct pair of sets S1; S2 V such that
jS1j ¼ jS2j ¼ t, at least one of the conditions of Theorem 1 is satisfied.
Theorem 3 [20].A system G with N nodes is t-diagnosable if
1. N 2t þ 1;
2. orderGðvÞ t for every node v in G;
3. jT ðG; UÞj > p for each U V ðGÞ such that jUj ¼ N 2t þ
pand 0 p t 1.
According to the above three theorems, we observe that Condition 3 of Theorem 3 restricts G, satisfying the first condition of Theorem 1, and ignores Conditions 2 and 3. Hence, we present a hybrid theorem to test whether a system is t-diagnosable.
Theorem 4.A system G with N nodes is t-diagnosable if
1. N 2t þ 1;
2. orderGðvÞ t for every node v in G;
3. for any two distinct subsets S1, S2 V ðGÞ such that jS1j ¼
jS2j ¼ t either
a. jT ðG; UÞj > p, where U ¼ V ðGÞ ðS1SS2Þ, and
jS1TS2j ¼ p or
b. the pair ðS1; S2Þ satisfies Condition 2 or 3 of Theorem 1.
Fig. 1. An example for TðG; UÞ of Q3.
Proof.Conditions 1 and 2 are the same as Conditions 1 and 2 of
Theorem 3. Consider Condition 3a. S1 and S2 are two distinct
subsets of V ðGÞ with jS1j ¼ jS2j ¼ t, U ¼ V ðGÞ ðS1SS2Þ, and
jS1TS2j ¼ p. Then, 0 p t 1 and jUj ¼ N 2t þ p. If
jT ðG; UÞj > p, it implies that the pair ðS1; S2Þ satisfies
Condition 1 of Theorem 1. Combining Conditions 3a and 3b,
by Theorems 1 and 2, this theorem follows. tu
4
D
IAGNOSABILITY OFM
ATCHINGC
OMPOSITIONN
ETWORKSNow, we define the Matching Composition Network (MCN) as
follows: Let G1 and G2 be two graphs with the same number of
nodes. Let L be an arbitrary perfect matching between the nodes of
G1and G2, i.e., L is a set of edges connecting the nodes of G1and
G2 in a one to one fashion; the resulting composition graph is
called a Matching Composition Network (MCN). For convenience, G1
and G2are called the components of the MCN. Formally, we use the
notation GðG1; G2; LÞ to denote an MCN, which has node set
VðGðG1; G2; LÞÞ ¼ V ðG1Þ
[
VðG2Þ
and edge set EðGðG1; G2; LÞÞ ¼ EðG1Þ [ EðG2Þ [ L. See Fig. 3.
What we have in mind is the following: Let G1and G2be two
t-connected networks with the same number of nodes and
orderGiðvÞ t for every node v in Gi, where i ¼ 1; 2, and let L be
an arbitrary perfect matching between the nodes of G1 and G2.
Then, the degree of any node v in GðG1; G2; LÞ as compared with
that of node v in Gi, i ¼ 1; 2, is increased by one. We expect that the
diagnosability of GðG1; G2; LÞ is also increased to t þ 1. For
example, the Hypercube Qnþ1 is constructed from two copies of
Qn by adding a perfect matching between the two and the
diagnosability is increased from n to n þ 1 for n 5. Other
examples, such as the Twisted cube T Qnþ1, the Crossed cube
CQnþ1, and the Mo¨bius cube MQnþ1, are all constructed
recursively using the same method as above.
Theorem 5.Let G1 and G2 be two networks with the same number of
nodes and t be a positive integer. Suppose that orderGiðvÞ t for
every node v in Gi, where i ¼ 1; 2. Then, orderGðG1;G2;LÞðvÞ t þ 1
for node v in GðG1; G2; LÞ.
Proof.See Fig. 3. Let v be a node of GðG1; G2; LÞ. Without loss of
generality, we assume that v 2 V ðG1Þ, v02 V ðG2Þ, and
ðv; v0Þ 2 L. Of course, node v0 is connected to at least one other
node v00 in V ðG
2Þ. Let Gðv; G1Þ and Gðv; GðG1; G2; LÞÞ be the
order graph of v in graph G1and GðG1; G2; LÞ, respectively. We
observe that Gðv; G1Þ is a proper subgraph of Gðv; GðG1; G2; LÞÞ,
both v0and v00are in the latter, none of them in the former, and
ðv0; v00Þ is an edge in Gðv; GðG
1; G2; LÞÞ. Therefore, every vertex
cover of the order graph Gðv; GðG1; G2; LÞÞ contains a vertex
cover of the order graph Gðv; G1Þ. Besides, any vertex cover of
Gðv; GðG1; G2; LÞÞ has to include at least one of v0and v00. Thus,
orderGðG1;G2;LÞðvÞ orderGiðvÞ þ 1 for any node v in Gi, i ¼ 1; 2.
This completes the proof. tu
We need the following lemma later in Theorem 6.
Lemma 3. Let G be a t-connected network, jV ðGÞj t þ 2 and
orderGðvÞ t for every node v in G, where t 2. Suppose that U
is a subset of nodes of V ðGÞ with j UUj t. Then, T ðG; UÞ ¼ UU.
Proof.By assumption j UUj t and ðGÞ t, we prove the lemma by
two cases; the first for j UUj < ðGÞ and the second for j UUj ¼ ðGÞ.
If j UUj < ðGÞ, the induced graph G½U is connected. By
Lemma 1, T ðG; UÞ ¼ Nð UU; UÞ. By Lemma 2, Nð UU; UÞ ¼ UU. This
case holds.
Suppose that j UUj ¼ ðGÞ. We observe that, adding any node v
of UUto U, the induced subgraph G½USfvg forms a connected
graph. It implies that every node v of UU is adjacent to every
connected components of G½U. We claim that the subgraph induced by U contains a connected component A with cardinality at least two (see Fig. 4a). Then, the connected
component A is adjacent to all nodes in UUand, so, T ðG; UÞ ¼ UU.
Now, we prove the claim. Suppose, on the contrary, that every connected component of the subgraph induced by U is an
isolated node. Let v be an arbitrary node in UU and let Gv¼
ðXv; YvÞ be the order graph of v in G. Then, UU fvg is a vertex
cover of Gvbecause every connected component of G½U is an
isolated node. Since j UUj t, we have j UU fvgj t 1.
There-fore, even if the induced graph G½ UU fvg is a complete graph
(see Fig. 4b), the cardinality of a minimum vertex cover of the
order graph Gv is at most t 1. However, this contradicts the
hypothesis of orderGðvÞ t for every node v in G. So, G½U has a
connected component A with cardinality at least two. This
proves the claim, and the lemma follows. tu
We are now ready to state and prove the following theorem about the diagnosability of Matching Composition Network under the comparison model. As an illustration, the conditions of the following theorem are applicable to some well-known
interconnec-tion networks, such as Qn, CQn, T Qn, and MQn for n ¼ t 3.
Theorem 6. For t 2, let G1 and G2 be two graphs with the same
number of nodes N, where N t þ 2. Suppose that orderGiðvÞ t
for every node v in Giand the connectivity ðGiÞ t, where i ¼ 1; 2.
Then, MCN GðG1; G2; LÞ is ðt þ 1Þ-diagnosable.
Proof. Since jV ðG1Þj ¼ jV ðG2Þj ¼ N, 2N 2ðt þ 2Þ > 2ðt þ 1Þ þ 1.
By Theorem 5, orderGðG1;G2;LÞðvÞ t þ 1 for any node v in
GðG1; G2; LÞ. It remains to prove that GðG1; G2; LÞ satisfies
Condition 3 of Theorem 4.
Let S1and S2be two distinct subsets of V ðGÞ with the same
number t þ 1 of nodes and let jS1TS2j ¼ p, then 0 p t. In
order to prove this theorem, we will prove that S1and S2 are
distinguishable, i.e., this pair ðS1; S2Þ satisfies either Condition
3a or 3b of Theorem 4.
Let G¼ GðG1; G2; LÞ and U¼ V ðGÞ ðS1SS2Þ, then
jUj ¼ 2N 2ðt þ 1Þ þ p. L e t U¼ U1SU2 w i t h Ui¼
UTVðGiÞ and UUi¼ V ðGiÞ Ui, i ¼ 1; 2. Without loss of
generality, we assume that jU1j jU2j. Let j UU1j ¼ n1,
Fig. 3. An example of GðG1; G2; LÞ.
j UU2j ¼ n2, n1þ n2¼ 2ðt þ 1Þ p, a n d n1 n2. S i n c e
0 n12ðtþ1Þp2 , the maximum value of n1 is equal to t þ 1
when p ¼ 0 and n2¼ t þ 1. According to different values of n1
and n2, we divide the proof into two cases. The first case is
n2 t, which implies n1 t. The second case is n2> tand this
case is further divided into three subcases n1< t, n1¼ t, and
n1> t.
Case 1: n1 t and n2 t.
By Lemma 3, we have
jT ðG; UÞj jT ðG1; U1Þj þ jT ðG2; U2Þj ¼ j UU1j þ j UU2j
¼ n1þ n2¼ 2ðt þ 1Þ p:
We know that 0 < p t, jT ðG; UÞj 2ðt þ 1Þ p > p, and Condition 3a of Theorem 4 is satisfied.
Case 2: n2> t.
We discuss the case according to the following three
subcases, 2a) n1< t, 2b) n1¼ t, and 2c) n1> t.
Subcase 2a: n1< t.
Since ðG1Þ t and j UU1j ¼ n1< t, G½U1 is connected. By
Lemmas 1 and 2, T ðG1; U1Þ ¼ Nð UU1; U1Þ ¼ n1. There are n1and
n2 nodes in UU1 and UU2, respectively, and n2¼ 2t þ 2 p n1
(see Fig. 5). If all the nodes in UU1are adjacent to some n1nodes
in UU2, there are still at least n2 n1¼ 2t þ 2 p 2n1nodes in
U
U2such that each of them is adjacent to some node in U1under
the matching L. So,
jT ðG; UÞj jT ðG1; U1Þj þ ðn2 n1Þ ¼ n1þ ðn2 n1Þ ¼ n2:
Because n2> t p, the proof of this subcase is complete.
Subcase 2b: n1¼ t.
We know that n1þ n2¼ 2ðt þ 1Þ p, 0 p t, n2> t, and
n1¼ t, the only two valid values for n2 are t þ 1 and t þ 2.
n2¼ t þ 1 implies p ¼ 1, and n2¼ t þ 2 implies p ¼ 0. By
Lemma 3, jT ðG1; U1Þj ¼ j UU1j ¼ t 2 > p for p ¼ 0 or 1. Then,
the subcase holds.
Subcase 2c: n1> t.
Observing that 0 n12ðtþ1Þp2 , where 0 p t and
n2 n1> t, so n1¼ n2¼ t þ 1. It also implies p ¼ 0. Here, we
will prove that the subcase satisfies either Condition 3a or Condition 3b of Theorem 4.
First, if the subgraph induced by U contains a connected
component A1 with cardinality at least two (see Fig. 6), then it
must be adjacent to some node in UU. Thus, we know that
jT ðG; UÞj > 0 ¼ p and Condition 3a of Theorem 4 is satisfied. Otherwise, every connected component of U contains a
single node only. By Theorem 1, we know that S1 and S2 are
distinguishable if there exists a path hu1! u ! u2i such that
u2 U, and u1; u22 S1 S2 or u1; u22 S2 S1. If p ¼ 0, it
implies S1TS2¼ , any node u in G½U with degree more than
two must be connected to at least two nodes in S1 or S2 (see
Fig. 6). By Theorem 5, orderGðG1;G2;LÞðvÞ t þ 1 for every node v
in GðG1; G2; LÞ, therefore degðvÞ t þ 1 for every node v in
GðG1; G2; LÞ. Since t 2, Condition 3b of Theorem 4 is satisfied.
Hence, the subcase holds and the theorem follows. tu
By Theorem 3 and Theorem 6, we have the following corollary.
Corollary 1.Let G1 and G2 be two graphs with the same number of
nodes N. Suppose that both G1 and G2 are t-diagnosable and have
connectivity ðG1Þ ¼ ðG2Þ t, where t 2. Then, MCN
GðG1; G2; LÞ is ðt þ 1Þ-diagnosable.
5
A
PPLICATIONSIn this section, we demonstrate the usefulness of our proposed construction scheme for some well-known networks. For example,
the diagnosability of the Hypercube Qn[19], the Crossed cube CQn
[6], [7], [8], the Twisted cube T Qn [9], [13], and the Mo¨bius cube
MQn[5] can all be proven to be n, for n 4.
The Hypercube is a popular topology for interconnection networks. The Crossed cube, the Twisted cube, and the Mo¨bius cube are variations of the Hypercube. For each of these cubes, an n-dimensional cube can be constructed from two copies of ðn 1Þ-dimensional subcubes by adding a perfect matching between the two subcubes. The main difference is that each of these cubes has various perfect matching between its subcubes. An
n-dimensional cube has 2n nodes, connectivity n, and each node
has the same degree n. In the following, we briefly state the recursive definitions of these cubes and prove that they are all n-diagnosable.
The nodes of these n-dimensional cubes are usually represented by the n-bit binary strings. A binary string u of length n will be
written as u ¼ un1un2un3. . . u0, where ui2 f0; 1g, 0 i n 1.
The classical n-dimensional Hypercubes Qnis recursively defined
as follows.
Definition 1.Let n 1 be an integer. The Hypercube Qnof dimension n
has 2nnodes. Q
1is a complete graph with two nodes labeled by 0 and
1, respectively. For n 2, an n-dimensional Hypercube Qn is
obtained by taking two copies of ðn 1Þ-dimensional subcubes
Qn1, denoted by Q0n1and Q1n1. For each v 2 V ðQnÞ, insert a 0 to
the front of ðn 1Þ-bit binary string for v in Q0
n1and a 1 to the front
of ðn 1Þ-bit binary string for v in Q1
n1. There are 2n1 edges
between Q0 n1and Q1n1as follows: Let V ðQ0 n1Þ ¼ f0un2un3. . . u0: ui¼ 0 or 1g and V ðQ1n1Þ ¼ f1vn2vn3. . . v0: vi¼ 0 or 1g, where 0 i n 2. A node u ¼ 0un2un3. . . u0of V ðQ0n1Þ is joined to a node v ¼ 1vn2vn3. . . v0 of V ðQ1
n1Þ if and only if ui¼ vifor 0 i n 2.
In [22], Wang has proven that the diagnosability of hypercube-structured multiprocessor systems under the comparison model is
nwhen n 5. However, the diagnosability of Q4is not known to
be 4. Using our Theorem 6, we can strengthen the result as follows.
Theorem 7.The Hypercube Qnis n-diagnosable for n 4.
Proof.We observe that Q3is 3-connected, orderQ3ðvÞ ¼ 3 for every
node v in Q3, and the number of nodes of Q3is 8, 8 t þ 2 ¼ 5
for t ¼ 3. It is well-known that Q4can be constructed from two
copies of Q3by adding a perfect matching between these two
copies. Therefore, by Theorem 6, Q4 is 4-diagnosable.
Fig. 5. Illustration of Subcase 2a of Theorem 6.
Then, the proof is by induction on n. We have shown that Q4
is 4-diagnosable. Assume that it is true for n ¼ m 1.
Considering n ¼ m, Qmis obtained from two copies G1, G2 of
Qm1 by adding a perfect matching joining corresponding
nodes in G1 and G2. It is well-known that Qm1 is ðm
1Þ-connected. By Corollary 1, Qmis m-diagnosable. This completes
the induction proof. tu
However, Q3 is not 3-diagnosable. In Fig. 7, there is a Q3, let
S1¼ f0; 5; 7g and S2¼ f2; 5; 7g. Then, by Theorem 1, S1and S2are
not distinguishable, as shown in Fig. 7.
As we observe, most of the related results on diagnosability of multiprocessors systems [12], [22] are based on a sufficient theorem, namely, Theorem 3. Not satisfying this sufficient
condition, such as in the case of Q4, does not necessarily imply
that the network is not 4-diagnosable. Therefore, we propose a hybrid condition, 3a and 3b of Theorem 4, to check the diagnosability of multiprocessor systems under the comparison model. It is more powerful to use. Applying our Theorem 4 and
Theorem 6, we show that the diagnosability of Q4is indeed 4.
The following is the recursive definition of the n-dimensional
Crossed cube CQn.
Definition 2 [6].The Crossed cube CQ1 is a complete graph with two
nodes labeled by 0 and 1, respectively. For n 2, an n-dimensional
Crossed cube CQnconsists of two ðn 1Þ-dimensional sub-Crossed
cubes, CQ0
n1and CQ1n1, and a perfect matching between the nodes
of CQ0
n1and CQ1n1according to the following rule:
L e t VðCQ0
n1Þ ¼ f0un2un3. . . u0: ui¼ 0 or 1g a n d
VðCQ1
n1Þ ¼ f1vn2vn3. . . v0: vi¼ 0 or 1g. T h e n o d e u¼
0un2un3. . . u02 V ðCQ0n1Þ and the node v ¼ 1vn2vn3. . . v02
VðCQ1
n1Þ are adjacent in CQnif and only if
1. un2¼ vn2if n is even and
2. ðu2iþ1u2i; v2iþ1v2iÞ 2 fð00; 00Þ; ð10; 10Þ; ð01; 11Þ; ð11; 01Þg,
for 0 i < bn1
2 c.
Hilbers et al. [13] defined the Twisted cubes using the parity
f u n c t i o n . L e t u ¼ un1un2. . . u0, w h e r e ui2 f0; 1g a n d
0 i n 1, t h e p a r i t y f u n c t i o n i s d e f i n e d a s
PiðuÞ ¼ uiLui1L Lu0, whereLis the exclusive-or operation.
Definition 3 [13].The Twisted cube T Q1 is a complete graph with two
nodes, 0 and 1. Let n be an odd integer and n 3. The nodes of an
n-dimensional Twisted cube T Qnare decomposed into four sets S0;0,
S0;1, S1;0 and S1;1. The set Si;j consists of those nodes u ¼
un1un2. . . u0 w i t h un1¼ i a n d un2¼ j, w h e r e
ði; jÞ 2 fð0; 0Þ; ð0; 1Þ; ð1; 0Þ; ð1; 1Þg. The induced subgraph of Si;jin
T Qn is isomorphic to T Qn2. Edges which connect these four ðn
2Þ-dimensional subtwisted cubes can be described as follows: Any
n o d e un1un2. . . u0 w i t h Pn3ðuÞ ¼ 0 i s c o n n e c t e d t o
un1un2. . . u0 and un1un2. . . u0; and to un1un2. . . u0 and
un1un2. . . u0, if Pn3ðuÞ ¼ 1.
As stated in [13], for even integer n, the Twisted cube T Qncan
also be defined recursively in a similar way starting from T Q2,
where T Q2 is isomorphic to Q2. In order to see the recursive
to hu; vi if and only if 1) u ¼ u and ðv; vÞ 2 EðHÞ, or 2) v ¼ v and
ðu; u0Þ 2 EðGÞ.
It is known that the n-dimensional hypercube can be defined as
Qn¼ Qn1 K2when n 2, where K2is the complete graph with
two nodes.
In additions, the connectivity of the network G H is listed as follows:
Lemma 4 [4]. ðG HÞ ðGÞ þ ðHÞ.
In Definition 3, let T Q0
n1(T Q1n1, respectively) be the subgraph
of T Qninduced by S0;0SS1;0 (S1;1SS0;1, respectively). It follows
directly from the definition that both T Q0
n1 and T Q1n1 are
isomorphic to the Cartesian product T Qn2 K2. Then, T Qn is
constructed from T Q0
n1 and T Q1n1 by joining them with a
particular perfect matching. The connectivity of T Qn is n [3].
Replacing n by n 2, the connectivity of T Qn2 is n 2. So, by
Lemma 4, both T Q0
n1 and T Q1n1are ðn 1Þ-connected.
By Theorem 6, we observe that both T Q0
4 and T Q14 are
4-diagnosable. Then, by Corollary 1, T Q5is 5-diagnosable. Applying
induction on n, suppose that T Qn2 is ðn 2Þ-diagnosable, by
Corollary 1, both T Q0
n1 and T Q1n1are ðn 1Þ-diagnosable. Then,
we can prove that T Qnis n-diagnosable by induction.
Now, we present the definition of the Mo¨bius cubes MQn[5].
There are two types of MQn, namely, 0 MQnand 1 MQn.
Definition 5 [5]. 0 MQ1 and 1 MQ1 are both the complete
graph on two nodes whose labels are 0 and 1. For n 2, both
0 MQn and 1 MQn contain one 0-type sub-Mo¨bius cube
MQ0
n1 and one 1-type sub-Mo¨bius cube MQ1n1. The first bit of
every node of MQ0
n1 is 0 and the first bit of every node of
MQ1
n1 is 1. For two nodes u ¼ 0un2un3. . . u02 V ðMQ0n1Þ
and v ¼ 1vn2vn3. . . v02 V ðMQ1n1Þ,
1. uconnects to v in 0 MQnif and only if ui¼ vi, for every i,
0 i n 2,
2. uconnects to v in 1 MQnif and only if ui¼ vi, for every i,
0 i n 2.
It is known [11], [15] that the Crossed cube CQnand the Mo¨bius
cube MQnare both n-connected. By Theorem 5, we can prove that
the order of each node in these two cubes is n. We observe that the two cubes are both constructed recursively using a similar way satisfying the requirements of Theorem 6 and Corollary 1.
Therefore, we can prove that CQn and MQn are both
n-diagnosable for n 4. Then, we list the following three theorems.
Theorem 8 [12].The Crossed cube CQnis n-diagnosable for n 4.
Theorem 9.The Twisted cube T Qn is n-diagnosable for n 4.
Theorem 10.The Mo¨bius cube MQnis n-diagnosable for n 4.
6
C
ONCLUSIONSIn this paper, we propose a sufficient theorem to verify the diagnosability of multiprocessor systems under the comparison-based model. The conditions of this theorem include all the cases of the original necessary and sufficient condition stated in Theorem 1. Therefore, it is more suitable for verifying the diagnosability of a system. Then, we propose a family of interconnection networks which are recursively constructed, called the Matching Composition Networks.
Each member GðG1; G2; LÞ of this family is constructed from a
pair G1 and G2 of lower dimensional networks with the same
number of nodes, joining by a perfect matching L between the two. Applying Theorem 6 in this paper, we show that the diagnosability
of GðG1; G2; LÞ is one larger than those of the G1and G2, provided
some regular conditions, as stated in Theorem 6, are satisfied. Many well-known interconnection networks, such as the
Hyper-cubes Qn, the Crossed cubes CQn, the Twisted cubes T Qn, and the
Mo¨bious cubes MQn, belong to our proposed family.
We note here that these special cases all satisfy the condition of Theorem 6 for n 4. Thus, their diagnosabilities are n, for n 4. In
particular, the diagnosability of the 4-dimensional Hypercube Q4is
4. Also, Theorems 9 and 10 are proposed for the first time to
describe the diagnosability of the Twisted cube T Qn and the
Mo¨bious cubes MQn.
A
CKNOWLEDGMENTSThis work was supported in part by the National Science Council of the Republic of China under Contract NSC 90-2213-E-009-149.
R
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