Diagnosability of t-Connected Networks
and Product Networks under the
Comparison Diagnosis Model
Chien-Ping Chang, Pao-Lien Lai, Jimmy Jiann-Mean Tan, and Lih-Hsing Hsu
Abstract—Diagnosability is an important factor in measuring the reliability of an interconnection network, while the (node) connectivity is used to measure the fault tolerance of an interconnection network. We observe that there is a close relationship between the connectivity and the diagnosability. According to our results, a t-regular and t-connected network with at least 2t + 3 nodes is t-diagnosable. Furthermore, the diagnosability of the product networks is also investigated in this work. The product networks, including hypercube, mesh, and tori, comprise very important classes of interconnection networks. Herein, different combinations of t-diagnosable and t-connected are employed to study the diagnosability of the product networks.
Index Terms—Diagnosability, comparison diagnosis model, t-diagnosable, connectivity, order graph, product networks.
æ
1
I
NTRODUCTIONM
ANYstudies have proposed and examined thefeasi-bility topologies of multiprocessor interconnection networks. Such a topology is usually modeled as an undirected graph where the set of nodes represents the processors and the set of edges represents the communication links between the processors. Desirable features of an interconnection network include topological properties such as symmetry, regularity, large connectivity, and others. Related studies have investigated a class of graphs called Cayley graphs, with their desirable features. In [1], Cayley graphs are based on permutation groups and are a very useful framework for the design and analysis of interconnection networks. All Cayley graphs are regular, explaining why this study considers regular graphs throughout.
The reliability of an interconnection network system is essential to system design and system maintenance. The reliability of a system is maintained by ensuring that it can discriminate the faulty nodes from the fault-free ones. Then, fault-free nodes must replace the faulty nodes. Identifying the faulty nodes is called the diagnosis of the system. The diagnosability of the system refers to the maximum number of faulty nodes that can be identified by the system. The fault tolerance is another important issue related to interconnection networks. The fault tolerance of an inter-connection network can be measured from the connectivity
of the underlying graph. In an interconnection network with connectivity t, the fault-free node is guaranteed to communicate with any other fault-free node even if ðt 1Þ nodes are faulty. Hence, diagnosability and connectivity are important properties of interconnection networks. A t-regular and t-connected interconnection network may not be t-diagnosable, accounting for why the condition under which a given t-regular and t-connected interconnec-tion network is t-diagnosable is of interest. This study will prove that, given a t-regular and t-connected interconnec-tion network with at least 2t þ 3 nodes, the interconnecinterconnec-tion network is t-diagnosable according to the comparison diagnosis model. Therefore, many well-known interconnec-tion networks are found to be t-diagnosable under the comparison diagnosis model.
A product network is generated by applying the graph Cartesian product operation to factor networks. Combining two known topologies with established properties into a new one with the properties of both would be valuable. The Cartesian product can be used to perform this combining. Product networks are very important classes of intercon-nection networks. Some well-known interconintercon-nection net-works, e.g., hypercubes, meshes, tori, k-ary n-cubes, and generalized hypercubes, are product networks [2], [3], [4]. Motivated by this observation, this work addresses the diagnosability of product networks by applying the comparison diagnosis model. Although related studies have investigated various characteristics of product net-works (e.g., connectivity, diameter, shortest path routing, and embedding) [5], [6], [8], [10], [11], [16], [17], [19], [23], this paper studies some topological properties different from those investigated elsewhere. The diagnosability of hypercubes and enhanced hypercubes was studied in [12], [21], [22] and that of crossed cubes was considered in [9]. The diagnosability of the product networks under the PMC model was investigated in [2]. Lai et al. [13] addressed the diagnosability of matching composition networks. In . C-.P. Chang is with the Department of Electrical Engineering, Chung
Cheng Institute of Technology, Tashi, Taoyuan, Taiwan 335, ROC. E-mail: [email protected].
. P.-L. Lai and J.J.-M. Tan are with the Department of Computer and Information Science, National Chiao Tung University, Hsinchu, Taiwan 300, ROC. E-mail: {baolein.cis90g, jmtan}@cis.nctu.edu.tw.
. L.-H. Hsu is with the Department of Information Engineering, Ta Hwa Institute of Technology, Qionglin, Hsinchu, Taiwan 307, ROC. E-mail: [email protected].
Manuscript received 29 Oct. 2003; revised 21 May 2004; accepted 11 June 2004.
For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number TC-0193-1003.
[13], the matching composition network can be viewed as a
particular product network of G and K2, where G is a
t-connected network. This study examines the
diagnosa-bility of the product network of G1 and G2, where Gi is
ti-diagnosable or ti-connected for i ¼ 1; 2. Moreover, we
apply different combinations of ti-diagnosability and
ti-connectivity to investigate the diagnosability of the
product networks.
Previous studies have proposed various models for diagnosis [14], [15], [18]. An important approach, first proposed by Malek and Maeng [14], [15], is called the comparison diagnosis model (MM model). In the MM model, the number of faulty nodes is limited and all faults are permanent. The MM model deals with the faulty diagnosis by sending the same input (or task) from a node w to each pair of distinct neighbors, u and v, and then comparing their responses. The node w is called the comparator of the two nodes u and v. Different comparators may examine the same pair of nodes. The result of the comparison is that either the two responses are consistent or two responses disagree. The goal is to use the comparison results to identify the faulty/fault-free status of the nodes in the system. Using the comparison diagnosis model, Sengupta and Dahbura characterized the diagnosable system and presented a polynomial algorithm to determine the set of all faulty nodes [20].
The rest of this paper is organized as follows: Section 2 summarizes some known results on product networks and provides necessary background and notation used herein. Section 3 shows that, under certain conditions, a t-connected network is also t-diagnosable. Section 4 pre-sents the diagnosability of the product networks under the comparison diagnosis model. Conclusions are finally made in Section 5.
2
P
RELIMINARIES ANDN
OTATIONLet G ¼ ðV ; EÞ be a graph. V and E represent the set of nodes and the set of edges of G, respectively. The topology of an interconnection network is usually denoted by a graph
G¼ ðV ; EÞ, where nodes represent processors and edges
represent links between processors. Let V0be a subset of V ;
G V0 represents the subgraph of G induced by V V0.
The (node) connectivity of G is defined as
ðGÞ ¼ minfjV0jjV0 V and G V0is not connectedg:
A graph G is t-connected if ðGÞ t. Given a t-connected graph, Menger’s theorem states there exist t internally node-disjoint (abbreviated as node-disjoint) paths between any two distinct nodes.
The comparison scheme of the system can be modeled as a multigraph M ¼ ðV ; CÞ, where V represents the node set
and C the labeled-edge set. Let ðu; vÞw denote an edge
labeled by w. In M, an edge ðu; vÞw2 C represents the nodes
uand v, which are to be compared by w. The same pair of
nodes may be compared by various comparators, so M is a
multigraph. For ðu; vÞw2 C, rððu; vÞwÞ denotes the results of
comparing nodes u and v by w such that rððu; vÞwÞ ¼ 0 if the
outputs of u and v agree and rððu; vÞwÞ ¼ 1 if the outputs of
u and v disagree. If rððu; vÞwÞ ¼ 0 and w is fault-free, then
both u and v are fault-free. If rððu; vÞwÞ ¼ 1, then at least one
of u, v, and w must be faulty. If w is faulty, then the result of comparison is unreliable and the exact status of u and v are unknown. The complete result of all comparisons, defined as a function s: C ! f0; 1g is called the syndrome of the diagnosis.
A subset F V is said to be consistent with a syndrome s if s can arise from the circumstance that all the nodes in F are faulty and all the nodes in V F are fault-free. A system is said to be diagnosable if a unique F V is consistent with s for every syndrome s. In [20], a system is called a t-diagnosable system if the system is diagnosable as long as the number of faulty nodes therein does not exceed t. Let ðF Þ represent the set of syndromes which could be generated if F is the set of faulty nodes. Two distinct sets
S1; S2 V are said to be indistinguishable if and only if
ðS1Þ \ ðS2Þ 6¼ ;; otherwise, S1 and S2 are said to be
distinguishable. Clearly, a system is t-diagnosable if and only if
each pair of sets S1; S2 V are distinguishable and jS1j t
and jS2j t.
Consider two interconnection networks modeled by two
undirected graphs G1¼ ðV1; E1Þ and G2¼ ðV2; E2Þ. The
Cartesian product, G1 G2, of two factor networks is an
interconnection network, defined as follows:
Definition 1. The Cartesian product G ¼ G1 G2 of two
graphs G1¼ ðV1; E1Þ and G2¼ ðV2; E2Þ is the graph
G¼ ðV ; EÞ, where the set of nodes V and the set of edges E
are given by:
1. V ¼ fhx; yijx 2 V1and y2 V2g, and
2. for u ¼ hxu; yui and v ¼ hxv; yvi in V , ðu; vÞ 2 E if
and only if ðxu; xvÞ 2 E1 and yu¼ yv, or ðyu; yvÞ 2
E2and xu ¼ xv.
Let y be a fixed node of G2. The subgraph Gy1-component
of G1 G2 has node set V
y
1 ¼ fðx; yÞjx 2 V1g and edge set
E1y¼ fðu; vÞj < u ¼ xu; y >; v¼< xv; y >;ðxu; xvÞ 2 E1g.
Si-milarly, let x be a fixed node of G1; the subgraph
Gx
2-componentof G1 G2has node set V2x¼ fðx; yÞjy 2 V2g
and edge set
E2x¼ fðu; vÞj u ¼< x; yu>; v¼< x; yv>;ðyu; yvÞ 2 E2g:
Clearly, the Gy1-component (abbreviated as G
y
1) and the
Gx
2-component(abbreviated as Gx2) are isomorphic with G1
and G2, respectively (as illustrated in Fig. 1). The following
lemma lists a set of known results [5], [6], [8], [10], [23] related to the topological properties of the Cartesian
product of G1 G2of two graphs G1and G2.
Lemma 1. Let u ¼ hxu; yui and v ¼ hxv; yvi be two nodes in
G1 G2. The following properties hold:
1. G1 G2is isomorphic to G2 G1,
2. jG1 G2j ¼ jG1j jG2j, where jGj is the number of
nodes in G,
3. degG1G2ðuÞ ¼ degG1ðxuÞ þ degG2ðyuÞ,
4. distG1G2ðu; vÞ ¼ distG1ðxu; xvÞ þ distG2ðyu; yvÞ,
where distGðu; vÞ is the distance between u and v in G,
5. DðG1 G2Þ ¼ DðG1Þ þ DðG2Þ, where DðGÞ is the
diameter of G,
6. ðG1 G2Þ ðG1Þ þ ðG2Þ, where ðGÞ is the
Let G ¼ ðV ; EÞ be a graph. A node cover of G is a subset
Q V such that every edge of E has at least one end node
in Q. A node cover with the minimum cardinality is called a minimum node cover. Let NðuÞ be the set of neighbors of u,
NðuÞ ¼ fvjðu; vÞ 2 Eg. Let V1; V2 be two subsets of nodes,
V16¼ ;, V26¼ ;; the neighbor set of V1 in V2 is defined as
NðV2; V1Þfxjx 2 V2; x62 V1; a node y2 V1andðx; yÞ 2 Eg.
Given an interconnection network G, let M ¼ ðV ; CÞ represent the comparison scheme of G. For a node u 2 V , let
Xu¼ fvjðu; vÞ 2 E or ðu; vÞw2 C; for some wg;
Yu¼ fðv; wÞjv; w 2 Xuandðu; vÞw2 Cg:
In [20], the order graph of u is defined as GðuÞ ¼ ðXu; YuÞ and
the order of the node u is defined as the cardinality of a minimum node cover of GðuÞ. Given a network G and a
comparison scheme M, for a subset of nodes V0 V ,
TðG; V0Þ denotes the set of all nodes in V V0, which are
compared to some node of V0 by the other nodes of V0.
Therefore,
TðG; V0Þ ¼ fvjðu; vÞw2 C and u; w 2 V0and v2 V V0g:
For V0 V and jV0j > 1, if the subgraph induced by V0is
connected, then T ðG; V0Þ ¼ NðV V0; V0Þ, where NðV
V0; V0Þ is the neighbor set of V0in V V0. Fig. 2 shows an
example of T ðG; V0Þ. For V0¼ f0; 1; 5g, T ðG; V0Þ ¼ f2; 3; 4; 6g;
for V0¼ f0; 2; 6g, T ðG; V0Þ ¼ ;.
Five theorems presented by Sengupta and Dahbura [20] must be applied to characterize whether a system is t-diagnosable. The results of these theorems are as follows:
Lemma 2 [20].For any S1; S2 where 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 (as shown in Fig. 3):
1. 9 u; w 2 V S1 S2 and 9 v 2 ðS1 S2Þ [ ðS2
S1Þ such that ðu; vÞw2 C,
2. 9 u; v 2 S1 S2 and 9 w 2 V S1 S2 such that
ðu; vÞw2 C,
3. 9 u; v 2 S2 S1 and 9 w 2 V S1 S2 such that
ðu; vÞw2 C.
Lemma 3 [20].If a system with N nodes is t-diagnosable, then
N 2t þ 1.
Lemma 4 [20].If, in a system, each node has order at least t,
then, for each S1; S2 V such that jS1[ S2j t, ðS1; S2Þ is a
distinguishable pair.
Lemma 5 [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 and at least one of the conditions of
Lemma 2 is satisfied.
From condition 1 of Lemma 2 and Lemma 5, the following is a sufficient condition for a system to be a t-diagnosable.
Lemma 6 [20]. A system with N nodes is t-diagnosable if
1) N 2t þ 1, 2) each node has order at least t, 3) for each
V0 V such that jV0j ¼ N 2t þ p and 0 p t 1,
jT ðG; V0Þj > p.
3
D
IAGNOSABILITY OFt-C
ONNECTEDN
ETWORKSThis section considers the problem that, under suitable conditions, a t-regular and t-connected interconnection network is also t-diagnosable. A t-regular and t-connected interconnection network with at least 2t þ 3 nodes is first proven also to be t-diagnosable. Moreover, the product
network of G1 and G2 is shown to be ðt1þ t2Þ-diagnosable,
where Gi is ti-connectedwith regularity ti for i ¼ 1; 2.
Lemma 7. Let G be a t-regular and t-connected network with
N 2t þ 1 nodes and t > 2. Then, each node v of G has order t.
Proof.Let v be a node of G and let GðvÞ be the order graph
of v in G. Assume that node v has order k < t. Since G contains N 2t þ 1 nodes and the order of v is k < t, there exists at least one node y 2 V , y 6¼ v, y 62 NðvÞ, and
y62 Q. The distance between v and y is at least 2. Each
edge of GðvÞ has at least one endpoint in Q, so all paths from v to y in G must be from v via z, which is a node in Fig. 1. The product network G1 G2.
Fig. 2. Example of TðG; V0Þ with V0.
Q. Deleting all the nodes of Q in G ensures that no path exists from v to y. However, exactly k nodes are deleted, contradicting the assumption that G is a t-connected network, so k t. NðvÞ is a node cover of GðvÞ, so the
node v must have order k ¼ t. tu
Given a t-diagnosable system, by Lemma 3, the number of nodes must exceed or be equal to 2t þ 1. However, a t-regular and t-connected network with N ¼ 2t þ 1 nodes is not necessarily t-diagnosable. The graph shown in Fig. 4 is a 4-regular and 4-connected network with N ¼ 9 nodes since any two arbitrarily distinct nodes in Fig. 4 are contained in two disjoint cycles. For example, two distinct nodes 4 and 5 are present in cycles <4, 9, 8, 5> and <4, 1, 6, 5, 2, 7, 3>.This graph can be easily seen to be not 4-diagnosable, since {4, 5, 6, 7} and {6, 7, 8, 9} constitute an indistinguishable pair. With regard to N ¼ 2t þ 2, the three-dimensional crossed
cube CQ3 and the three-dimensional hypercube Q3 are
3-regular, 3-connected networks and each node has order
t¼ 3. However, [9], [22] demonstrated that CQ3and Q3are
not 3-diagnosable under the comparison diagnosis model. The t-regular and t-connected network G with N 2t þ 3 nodes is thus considered in the following theorem.
Theorem 1. Let G ¼ ðV ; EÞ be a t-regular and t-connected
network with N nodes and t > 2. G is t-diagnosable if
N 2t þ 3.
Proof. Let S1 and S2 be two distinct subsets of V with
jS1j ¼ jS2j ¼ t, jS1\ S2j ¼ p, and 0 p t 1. By
Lem-mas 5 and 7, G can be shown to be t-diagnosable by
showing that ðS1; S2Þ is a distinguishable pair. Let V00¼
S1[ S2 and V0¼ V V00. Then, jV00j ¼ 2t p > t.
Nota-bly, V0may not be connected.
The case in which all connected components of the
subgraph induced by V0are isolative nodes is considered
first. For 0 p t 1, the following cases are considered: Case 1. 0 p t 3. Since 0 p t 3 and G is a
t-regular graph, each node of V0 has at least two
neighbors in S1 S2 or S2 S1 for t > 2. Thus, either
condition 2 or condition 3 in Lemma 2 is satisfied.
Case 2. p ¼ t 2. In this case, jV00j ¼ t þ 2, N 2t þ 3,
and jV0j ¼ N ðt þ 2Þ t þ 1. Assume that the pair
S1; S2 are indistinguishable. Therefore, conditions 2 and
3 in Lemma 2 cannot be satisfied, implying that each
node of V0 must be connected to t 2 nodes in S
1\ S2,
one node in S1 S2, and one node in S2 S1. Therefore,
at most t nodes in V0 satisfy this assumption,
contra-dicting the condition jV0j t þ 1. Hence, either condition
2 or condition 3 in Lemma 2 must be satisfied.
Case 3. p ¼ t 1. jV00j ¼ t þ 1 and jV0j ¼ N t 1.
The subgraph induced by V0consists of isolative nodes
and G is a t-regular graph, so ðN t 1Þ t edges are
adjacent to the nodes of V0 and V00. However, G has
exactly Nt=2 edges. For N 2t þ 3, we have ðN t 1Þ t > Nt=2, which is a contradiction, so p ¼ t 1 is impossible.
Now, consider that the subgraph induced by V0
contains a connected component R with cardinality of
at least 2. Let u 2 R and v 2 ðS1 S2Þ [ ðS2 S1Þ. G is
t-connected, so there exist t disjoint paths from u to v. However, at most p disjoint paths exist from u to v via the
nodes of S1\ S2. Therefore, there exists at least one path
from u to v such that no node of the path belongs to
S1\ S2. Since u is a node in R, there exists another node
w adjacent to u. Hence, condition 1 in Lemma 2 is
satisfied, completing the proof of the theorem. tu
Corollary 1.For t1; t2> 2, let G1 and G2 be two t1-connected
and t2-connectednetworks, with regularity t1and t2,
respec-tively. Let G ¼ ðV ; EÞ be the product network of G1and G2.
Then, the product network G ¼ G1 G2 is ðt1þ
t2Þ-diagnosable with regularity t1þ t2.
Proof. G1 is t1-regular and t1-connected, so at least t1þ 1
nodes exist in G1. Similarly, the number of nodes in G2
is at least t2þ 1. Therefore, G contains at least ðt1þ
1Þðt2þ 1Þ nodes. Moreover, by Lemma 1, the degree of
every node in G is t1þ t2 (regularity t1þ t2). ðGÞ is
used to denote the minimum degree of G. That [7] ðGÞ ðGÞ is well-known. However, by Lemma 1,
ðGÞ ðG1Þ þ ðG2Þ ¼ t1þ t2. Since
t1þ t2 ðGÞ ðGÞ ¼ t1þ t2;
ðGÞ ¼ t1þ t2. Since ðt1þ 1Þðt2þ 1Þ > 2ðt1þ t2Þ þ 3 for
t1; t2> 2, Theorem 1 implies that G is ðt1þ t2Þ-diagnosable.
Therefore, the corollary follows. tu
Notice that the number Ni of nodes is greater than or
equal to niþ 1 for ni-connected i¼ 1; 2 in Corollary 1. The
following corollary is immediately obtained from Corollary 1 and by induction.
Corollary 2.Let G be a product network of G1; G2; . . ., and Gk.
Each Gi is ti-regular and ti-connected and ti> 2 for
1 i k, where k > 2. Then, the product network G is ðt1þ
t2þ þ tkÞ-regular and ðt1þ t2þ þ tkÞ-diagnosable.
Theorem 1 indicates that a t-connected network with
N 2t þ 3 nodes is also diagnosable. However, a
t-diagnosable network is not necessarily a t-connected net-work (as depicted in Fig. 5). The example shown in Fig. 5 is 4-regular and 4-diagnosable, but not 4-connected. The t-diagnosability and t-connectivity are not equivalent terms, Fig. 4. An example of 4-connected and 3-diagnosable.
but these two concepts are closely related; Theorem 1 provides an example.
4
D
IAGNOSABILITY OFP
RODUCTN
ETWORKSThe product networks are distinguished into homogeneous product networks and heterogeneous product networks. Homogeneous product networks refer to every factor network of the product that is t-diagnosable and t-regular (or being t-connected and t-regular, respectively), while heterogeneous products are of factor networks one of which is t-diagnosable and the other is t-connected. Section 4.1 addresses the diagnosability of homogeneous product networks. Section 4.2 presents the diagnosability of hetero-geneous product networks.
4.1 Diagnosability of Homogeneous Product
Networks
By Corollary 1, the homogeneous product network G1 G2
is ðt1þ t2Þ-diagnosable, where Gi is ti-connected and
ti-regular, ti> 2 i¼ 1; 2. The homogeneous product
net-work G1 G2 is also ðt1þ t2Þ-diagnosable, where Gi is
ti-diagnosableand ti-regular, ti > 2 i¼ 1; 2. Several lemmas
must be proven first.
Lemma 8. Let G ¼ ðV ; EÞ be a t-regular network with N
2tþ 1 nodes. Suppose each node of G has order t, t > 2. If
V0 V and jV V0j t, then T ðG; V0Þ ¼ V V0.
Proof.Let v be an arbitrary node in V V0, and let GðvÞ be
the order graph of v in G. The following two cases are considered:
Case 1. jV V0j < t. For jV V0j < t, the degree of
each node is t, so each node in V0 has at least one
neighbor in V0. Therefore, no isolated node exists in V0.
Similarly, every node in V V0has at least one neighbor
in V0. Hence, T ðG; V0Þ ¼ V V0.
Case 2. jV V0j ¼ t. For jV V0j ¼ t, each node in
V V0has at least one neighbor in V0. NðV0; vÞ is used to
denote the neighbor set of v in V0. Assume that no node
in NðV0; vÞ is adjacent to any other node in V0. Then,
every node in NðV0; vÞ is adjacent only to V V0 (as
shown in Fig. 6). Thus, V V0 fvg is a node cover of
GðvÞ because every node in NðV0; vÞ is an isolated node
in V0.The cardinality of a minimum node cover of the
order graph GðvÞ can be easily determined to be at most
t 1. However, this contradicts the hypothesis that each
node has order t. Therefore, NðV0; vÞ contains at least one
neighbor u of v such that the node u is adjacent to
another node w in V0. Hence, T ðG; V0Þ ¼ V V0. tu
Lemma 9.Let H be a t-regular network, t > 2, and let K2be the
complete network with two nodes. Suppose that the order of each node in H is t. Then, each node of the product network
G¼ H K2 has order t þ 1.
Proof.Let G0 and G1be two copies of H in G. M ¼ ðV ; CÞ
represents the comparison scheme of G. Let v be a node of G and let GðvÞ be the order graph of v in G. Without
loss of generality, assume that v is a node in G0and that
uis a neighbor of v in G1. There exists at least one node w
in G1 such that ðv; wÞ
u2 C. Then, let G0ðvÞ be the order
graph of v in G0. Since G0ðvÞ is a proper subgraph of
GðvÞ, every node cover of GðvÞ must contain a node
cover of G0ðvÞ. However, ðw; uÞ is an edge in GðvÞ rather
than in G0ðvÞ. Therefore, a node cover of GðvÞ must
include at least either u or w. The order of v in G
therefore exceeds that of v in G0by one. Thus, the lemma
is proven. tu
Theorem 2. For t > 2, let H be a t-regular and t-diagnosable
network with N nodes. Then, the product network G ¼
H K2is ðt þ 1Þ-diagnosable.
Proof.Let G0¼ ðV0; E0Þ and G1¼ ðV1; E1Þ be two copies
of H in G ¼ ðV ; EÞ. Let S1 and S2 be two distinct
s u b s e t s o f V a n d l e t V00¼ S1[ S2 w i t h
jS1j ¼ jS2j ¼ t þ 1, jS1\ S2j ¼ p, and 0 p t. Then,
let V0¼ V V00 with jV0j ¼ 2N 2ðt þ 1Þ þ p. Since G
has 2N nodes, 2N 2ð2t þ 1Þ > 2ðt þ 1Þ þ 1. Lemma 9 implies that each node of G has order t þ 1. Hence, the theorem is proven if one of the conditions of Lemma 2
is satisfied. Now, let V00
¼ V0\ V0 and V10
¼ V0\ V1.
G0 and G1 are isomorphic to H, so, without loss of
generality, assume that jV00
j jV10 j. Let jV0 V00 j ¼ k and jV1 V10 j ¼ 2ðt þ 1Þ p k. Since jV00 j jV10 j,
k 2ðt þ 1Þ p k. Thus, the proof is divided into
the following cases:
Case 1. 2ðt þ 1Þ p k tand k < t. From Lemma 8,
jT ðG; V0Þj jT ðG0; V00
Þj þ jT ðG1; V10
Þj ¼ k þ 2ðt þ 1Þ p k ¼ 2ðt þ 1Þ p:
Since p t, jT ðG; V0Þj 2ðt þ 1Þ p > p. By Lemma 6,
this case holds.
C a s e 2 . 1 . 2ðt þ 1Þ p k > t and k < t. F r om
Lemma 8, jT ðG0; V00
Þj ¼ k. Since V0 V00
contains k <
tnodes, each node in V00has at least one neighbor in V00.
Therefore, no isolated node is present in V00
. Notably, at least 2ðt þ 1Þ p 2k nodes in V1 V10 are adjacent to some 2ðt þ 1Þ p 2k nodes in V00 . Thus, jT ðG; V0Þj jT ðG0; V00 Þj þ NðV1 V10 ; V00Þ k þ 2ðt þ 1Þ p 2k ¼ 2ðt þ 1Þ p k: Since 2ðt þ 1Þ p k > t p, by Lemma 6, the case holds.
Case 2.2. 2ðt þ 1Þ p k > t and k ¼ t. Since 2ðt þ
1Þ p k > t and k ¼ t, ðt þ 2Þ p > t, implying p < 2. From Lemma 8,
jT ðG; V0Þj jT ðG0; V00
Þj ¼ t > 2 > p: Then, the case follows.
Case 2.3. 2ðt þ 1Þ p k > t and k > t. Since 2ðt þ
1Þ p k > t and k > t, the number of nodes in V V0is
2ðt þ 1Þ, indicating p ¼ 0. Condition 1 in Lemma 2 is first
supposed to be satisfied in G0. Then, the subgraph
induced by V00
includes at least one connected compo-nent R with a cardinality of at least 2. Given
jV0 V00
j ¼ t þ 1, Lemma 6 implies jT ðG0; V00
Þj t > 2
since G0is t-diagnosable. Therefore,
jT ðG; V0Þj jT ðG0; V00
Þj > 2 > p:
This result implies that condition 1 in Lemma 2 is also satisfied in G.
Next, consider that condition 1 in Lemma 2 is violated
in G0. Then, either condition 2 or condition 3 in Lemma 2
is satisfied in G0. Since G0 is t-regular and t > 2, one
node v in V00
is adjacent to at least three nodes in
V0 V00
. Now, let u, w, and x be three nodes in V0 V00
such that u; w 2 S1, and x 2 S2. Since u; w 2 S1 S2,
v2 V S1 S2, and p ¼ 0, condition 2 in Lemma 2 is
also satisfied in G. The theorem follows. tu
Let Gi be a ti-regular interconnection network i ¼ 1; 2
and let G ¼ G1 G2be the product network of G1and G2.
Then, the order of each node v in G is estimated from the following lemma.
Lemma 10. Let Gi¼ ðVi; EiÞ be a ti-regular network with
ti> 2. Suppose each node of Gi has order at least ti, i ¼ 1; 2.
Then, each node of the product network G ¼ G1 G2 has
ordert t1þ t2.
Proof.Let v ¼ hx; yi be an arbitrary node of G and let GðvÞ
be the order graph of v in G. According to the definition
of product networks, x is a node of V1and y is a node of
V2. Therefore, the order of x is at least t1and the order of
yis at least t2. Let G1ðxÞ be the order graph of x in G1and
let G2ðyÞ be the order graph of y in G2. NðxÞ is a node
cover of G1ðxÞ, so the order of node x is exactly t1.
Similarly, the order of node y is t2. Let Gy1ðvÞ be the order
graph of v in the subgraph Gy1of G and let Gx
2ðvÞ be the
order graph of v in the subgraph Gx
2 of G. Since V1y\ Vx 2 ¼ v, V ðG y 1ðvÞÞ \ V ðGx2ðvÞÞ ¼ ;, where V ðG y 1ðvÞÞ and V ðGx
2ðvÞÞ are the node sets of G
y
1ðvÞ and Gx2ðvÞ,
respectively. Gy1ðvÞ and Gx2ðvÞ are observed to be
subgraphs of GðvÞ. Thus, every node cover of GðvÞ must
contain a node cover of both Gy1ðvÞ and Gx2ðvÞ. Since the
subgraphs Gy1and Gx2of G are isomorphic to G1and G2,
respectively, Gy1ðvÞ is isomorphic to G1ðxÞ and Gx2ðvÞ is
isomorphic to G2ðyÞ. Therefore, the order of v in Gy1ðvÞ is
t1 a n d t h e o r d e r o f v i n Gx2ðvÞ i s t2. S i n c e
VðGy1ðvÞÞ \ V ðGx2ðvÞÞ ¼ ;, the order of v in GðvÞ is
t1þ t2. Hence, the lemma follows. tu
Corollary 1 was proven; it states that the product
network G1 G2 is ðt1þ t2Þ-diagnosable, in which Gi is
ti-connectedfor ti> 2 i¼ 1; 2. The previous section also
established that a ti-diagnosablenetwork is not equivalent
to a ti-connectednetwork. The following theorem states that
the product network G1 G2is ðt1þ t2Þ-diagnosable, where
Giis ti-diagnosablefor ti > 2 i¼ 1; 2. Theorem 3 is proven in
Appendix A.
Theorem 3.For ti> 2, let Gi¼ ðVi; EiÞ be a ti-diagnosableand
ti-regularnetwork with Ninodes i ¼ 1; 2. Let G ¼ ðV ; EÞ be
the product network of G1and G2. Then, the product network
G¼ G1 G2 is ðt1þ t2Þ-diagnosable with regularity
t1þ t2.
Notice that, in Theorem 3, the number of nodes Ni is
greater than or equal to 2tiþ 1 for ti-diagnosable i¼ 1; 2.
From Theorem 3 and by induction, the following corollary is obtained.
Corollary 3.Let G be the product network of G1; G2; . . ., and Gk,
where each Giis ti-diagnosablewith regularity tiand ti> 2for
1 i k. Then, the product network G is ðt1þ t2þ þ
tkÞ-diagnosable with regularity ðt1þ t2þ þ tkÞ.
4.2 Diagnosability of Heterogeneous Product
Networks
This section considers different combinations of
ti-diagnosability and ti-connectivity to study the
diagnosa-bility of the product networks. The diagnosadiagnosa-bility of the
heterogeneous product network G of G1 and G2 is
considered in which G1 is t1-diagnosable and G2 is
t2-connected. Although the heterogeneous product network
differs from the homogeneous product network, a similar result is obtained as that obtained for the homogeneous product network. Lemmas 7 and 10 immediately yield the following lemma.
Lemma 11.Let G1be a t1-regularand t1-diagnosablenetwork
with t1> 2 and let G2 be a t2-regular and t2-connected
network with N2 2t2þ 1 nodes and t2> 2. Then, each node
of the product network G ¼ G1 G2 has order t1þ t2.
Section 3 presents some examples to show that a t-diagnosable network is not equivalent to a t-connected network. Therefore, the following theorem is not implied by Theorem 3, but it can be proven by a similar technique. Theorem 4 is proven in Appendix B.
Theorem 4.For t1; t2> 2, let G1¼ ðV1; E1Þ be a t1-regularand
t1-diagnosablenetwork with N1 nodes and let G2¼ ðV2; E2Þ
be a t2-regularand t2-connectednetwork with N2 2t2þ 1
nodes. Then, the product network G ¼ G1 G2 is ðt1þ
t2Þ-diagnosable with regularity t1þ t2.
In the above theorem, the factor network G2must have at
least 2t2þ 1 nodes. Therefore, by Corollary 3 and
Theorem 4, the following corollary holds.
Corollary 4. Let G be the product network of G1; G2; . . ., and
Gk. Suppose that G1 is t1-regular and t1-connected with
N1 2t1þ 1 nodes and suppose that Gi is ti-regular and
ti-diagnosable, ti> 2 for 2 i k. Then, the product
network G is ðt1þ t2þ þ tkÞ-diagnosable with regularity
ðt1þ t2þ þ tkÞ.
However, Corollaries 2 and 3 yield the following corollary.
Corollary 5. Let G be the product network of G1; G2; . . ., and
Gk. Suppose that Giis ti-regularand ti-connected, ti> 2for
1 i m, where m > 2, and suppose that Gj is tj-regular
product network G is ðt1þ t2þ þ tkÞ-diagnosable with
regularity ðt1þ t2þ þ tkÞ.
5
C
ONCLUSIONSThe reliability of an interconnection network is an im-portant issue. The diagnosability is also an imim-portant factor in measuring the reliability of an interconnection network. The connectivity is used to measure the fault tolerance of an interconnection network. This study addresses how the connectivity and the diagnosability of an interconnection network are related. Given an n-regular and n-connected interconnection network with at least 2n þ 3 nodes, the interconnection network is n-diagnosable. Illustrative ex-amples reveal that the result may not hold if the condition of more than 2n þ 3 nodes is replaced by 2n þ 1 or 2n þ 2. This finding suggests that fault diagnosis improves with fault tolerance. Among the many well-known interconnec-tion networks that meet this condiinterconnec-tion are hypercubes, tori, crossed cubes, k-ary n-cubes, and generalized hypercubes, among others, and such interconnection networks are n-diagnosable. Besides, the Cartesian product has defined several interconnection networks and the product networks construct important classes of interconnection networks. This work also investigates the diagnosability of product
networks. The homogeneous product network G of G1and
G2 is proven to be ðn1þ n2Þ-diagnosable, given that Gi is
either ni-diagnosableor ni-connected with regularity ni for
i¼ 1; 2. Furthermore, different co mbinat ions of
ni-diagnosabilityand ni-connectivityare considered to study
the diagnosability of the product networks. The
hetero-geneous product network G of G1 and G2 is shown to be
ðn1þ n2Þ-diagnosable, given that G1 is n1-diagnosable with
regularity n1, and G2 is n2-regular and n2-connected with
2n2þ 1 nodes. Similarly, the product network G is also
generalized in terms of the k factor networks, G1; G2; . . .,
and Gk, all with regularity ni, such that each Gi is either
ni-diagnosable or ni-connected for 1 i k. The product
network G is shown to be ðn1þ n2þ þ nkÞ-diagnosable.
A
PPENDIXA
Proof. Let S1 and S2 be two distinct subsets of V and let
V00¼ S1[ S2 with jS1j ¼ jS2j ¼ ðt1þ t2Þ, jS1\ S2j ¼ p,
and 0 p ðt1þ t2Þ 1. Then, let V0¼ V V00 with
jV0j ¼ N
1N2 2ðt1þ t2Þ þ p. Since G1 is t1-diagnosable
and G2 is t2-diagnosable,
jV j ¼ N1N2 ð2t1þ 1Þð2t2þ 1Þ 2ðt1þ t2Þ þ 1:
By Lemma 10, each node of the product network G has
order t1þ t2. Therefore, the proof is complete if Lemma 6
can be shown to be satisfied. G1and G2are t1-diagnosable
and t2-diagnosable, respectively, so, without loss of
generality, assume that t1 t2. Hence, Gy1¼ ðV
y 1; E
y 1Þ is
isomorphic to G1 for all 1 y N2. Let V10y¼ V0\ V
y 1,
V100y¼ V00\ Vy
1 for all 1 y N2. A set K is defined as
K¼ fy : jV100yj > 0g. Let S be a subset of K with jV100yj > t1
and let J be V2 K. The following cases are discussed:
Case 1. jKj t2. For all 1 y N2, since Gy1 is
isomorphic to G1, Gy1 must also be t1-diagnosable. Two
cases for V100y are thus distinguished.
Case 1.1. jV100yj t1for all y 2 K. In this case, the set S
is empty. From Lemma 8,
jT ðG; V0Þj X
y2k
jT ðGy1; V10yÞj ¼X
y2K
jV100yj ¼ 2ðt1þ t2Þ p > p:
By Lemma 6, the case holds.
Case 1.2.At least one V100y exists for y 2 K such that
jV100yj > t1. G2is t2-diagnosableand, in this case, jKj t2,
so Lemma 8 implies that T ðG2; JÞ ¼ K. This lemma also
implies that each y 2 K is adjacent to at least one connected component in J. Hence, for y 2 K and r 2 J,
each V1yis adjacent to at least one V1r. Since r is in J and
r62 K, jV00r
1 j ¼ 0 such that V1r is also V10r. Gs1 V10s
represents the subgraph of G induced by Vs
1 V10s for
all s 2 S. Let Vs
1 V10s be V100s. For r 2 J, s 2 S, one such
V10ris always adjacent to one specific Vs
1 with jV100sj > t1.
jT ðGs
1 V10s; V10rÞj ¼ NðV100s; V10rÞ ¼ jV100sj is thus
ob-tained for such s and r. Therefore,
jT ðG; V0Þj X y2K;y62S jT ðGy1; V 0y 1 Þj þ X y2S;r2J jT ðGs1 V10s; V10rÞj ¼ X y2K;y62S jV100yj þX s2S jV100sj ¼ 2ðt1þ t2Þ p > p:
Case 2. jKj > t2. Similarly, two cases for V100y are
considered.
Case 2.1. jV100yj t1for all y 2 K. The proof is similar
to that of Case 1.1.
Case 2.2.At least one V100y exists for y 2 K such that
jV100yj > t1. First, 0 p < t1 is considered. Now, S is
nonempty and jV00j 2ðt
1þ t2Þ, so jV00 V100sj < t1þ 2t2
for s 2 S. Since G2is t2-regular, s has exactly t2neighbors
in V2. Let z1; z2; . . . ; zt2 be these neighbors of s. Thus, all
Vz1
1 ; V z2
1 ; . . ., and V
zt2
1 are adjacent to V1s. Since
jV00 V00s
1 j < t1þ 2t2, s has at least one neighbor zi for
1 i t2 s u c h t h a t jV100zij < ðt1þ 2t2Þ=t2 t1 f o r
t1 t2 2. For such zi and s, the subgraph induced by
Vzi
1 [ V1s is isomorphic to G1 K2. Since jV100sj > t1 and
jV00zi
1 j < t1, it follows from Case 2.1 of Theorem 2 that
jðT ðG; V0Þj jT ðGs 1[ G zi 1; V 0zi 1 Þj jV 00s 1 j > t1> p:
The other cases, with other relative positions of ziand s,
can be treated similarly.
Next, t1 p < t1þ t2 is considered. Then, jV00j
t1þ 2t2 and jV00 V100sj < 2t2 for s 2 S. If another V100t
exists with t 2 S, then jV00 V00s
1 V100tj 2t2 1 t1
1¼ 2t2 t1 2 t2 2 for t1 t2. Therefore, the
num-ber of V100y is at most t2 for y 2 K, violating the
assumption that jKj > t2. Hence, exactly one V100ssatisfies
jV00s
1 j > t1 and jKj 1 V100ys have jV
00y
1 j t1. Since G2 is
t2-regularand jV00 V100sj < 2t2, this situation is similar to
the case 0 p t1in which, for at least one neighbor z of
s, jV00z
1 j < 2t2=t2¼ 2. Similarly, the subgraph induced by
Vz
1 [ V1s is isomorphic to G1 K2. Following the same
argument as in case 0 p < t1 with jV100sj > t1 and
jV00z
jT ðG; V0Þj X y2K;y6¼s;z jT ðGy1; W10yÞj ! þ jT ðGs 1[ G z 1; V10zÞj X y2K;y6¼s;z jV100yj ! þ jV100sj ¼ ð2ðt1þ t2Þ p jV100sjÞ þ jV 00z 1 jÞ þ jV 00s 1 j ¼ 2ðt1þ t2Þ p jV100zj: Since jV100zj < 2, 2ðt1þ t2Þ p jV100zj 2ðt1þ t2Þ p 1 t1þ t2> p:
In other cases, jT ðG; V0Þj > p can be similarly proven.
Furthermore, by Lemma 1, the degree of every node in
G is t1þ t2 (regularity t1þ t2). Hence, the proof is
completed. tu
A
PPENDIXB
Proof.Let G ¼ ðV ; EÞ be the product network of G1and G2,
G¼ G1 G2. Now, G1is t1-diagnosableand the number
of nodes in G2is N2, so
jV j ¼ N1N2 ð2t1þ 1Þð2t2þ 1Þ > 2ðt1þ t2Þ þ 1:
By Lemma 11, each node of the product network G has
order t1þ t2. Thus, the proof is complete if we can show
Lemma 6 is satisfied. Let S1; S2be two distinct subsets of
V and let V00¼ S
1[ S2 with jS1j ¼ jS2j ¼ ðt1þ t2Þ,
jS1\ S2j ¼ p, 0 p ðt1þ t2Þ 1. Now, let V0¼ V V00
with jV0j ¼ N
1N2 2ðt1þ t2Þ þ p. For t1 t2, the proof is
similar to that of Theorem 3.
G2 is not t2-diagnosable, so considering only t1 t2
does not suffice. The case t1< t2 must therefore also be
considered in the following proof. By definition of the
product networks, Gx
2 ¼ ðV2x; Ex2Þ is isomorphic to G2for
all 1 x N1. Let V20x¼ V0\ V2x, V200x¼ V00\ V2x for all
1 x N1. A set K is defined as K ¼ fx : jV200xj > 0g. Let
S be a subset of K with jV00x
2 j > t2 and let J be V1 K.
The following cases are discussed:
Case 1. jKj t1. G1 is t1-diagnosable and
jV1 Jj ¼ jKj t1, s o L e m m a 8 i m p l i e s t h a t
TðG1; JÞ ¼ K. The lemma also implies that each x 2 K
is adjacent to at least one connected component in J.
Thus, for x 2 K and r 2 J, each Vx
2 is adjacent to at least
one Vr
2. Since r is in J and r 62 K, jV200rj ¼ 0 such that V2ris
also V0r
2 . For S ¼ ;, the proof is similar to that of Case 1.1
in Theorem 3. Therefore, the case S 6¼ ; is considered as
follows: Gs
2 V20s represents the subgraph of G induced
by Vs
2 V20s for all s 2 S. Let V2s V20s be V200s. For r 2 J,
s2 S, there always exists one such V0r
2 that is adjacent to
one specific Vs
2 with jV20sj > t2. The following are thus
obtained:
jðT ðGs
2 V20s; V20rÞj ¼ NðV200s; V20rÞ ¼ jV200sj
for such s and r. Therefore,
jT ðG; V0Þj X x2K;x62S jT ðGx 2; V 0x 2 Þj þ X s2S;r2J jT ðGs 2; V 0s 2 ; V 0r 2 Þj ¼ X x2K;x62S jV00x 2 j þ X s2S jV00s 2 j ¼ 2ðt1þ t2Þ p > p:
Case 2. jKj > t1. Since Gx2 is isomorphic to G2, it
follows that Gx
2 is t2-connected. Thus, two cases for V200x
are distinguished.
Case 2.1. jV200xj t2for all x 2 K. In this case, the set S
is empty. From Lemma 8,
jT ðG; V0Þj X x2K jT ðGx 2; V20xÞj ¼ X x2K jV200xj ¼ 2ðt1þ t2Þ p > p:
Lemma 6 implies that the case holds.
Case 2.2. At least one V00x
2 exists for x 2 K such
that jV200xj > t2. First, 0 p < t2 is considered. Since S
is nonempty and jV00j 2ðt1þ t2Þ, jV00 V200sj < 2t1þ t2
for s 2 S. G1 is t1-regular, so exactly t1 neighbors of s
are present in V1. Let z1; z2; . . . ; zt1 be such neighbors
of s. Hence, all Vz1 2 ; V z2 2 ; . . ., and V zt1 2 are adjacent to Vs
2. Since jV00 V200sj < 2t1þ t2, there exists at least
one neighbor zi of s for 1 i t1 such that jV200zij <
ð2t1þ t2Þ=t1< t2 for t2> t1. For such zi, the subgraph
induced by V0zi 2 is connected. Furthermore, TðGzi 2; V 0zi 2 Þ ¼ NðV 00zi 2 ; V 0zi 2 Þ ¼ V 00zi 2 . Notably, at least jV00s 2 j jV 00zi
2 j nodes in V200s that are adjacent to some
jV00s 2 j jV 00zi 2 j nodes in V 0zi 2 . Therefore, jT ðG; V0Þj jT ðGs 2; V 0zi 2 Þj þ jT ðG zi 2; V 0zi 2 Þj ðjV200sj jV 00zi 2 jÞ þ jV 00zi 2 j ¼ jV200sj > t2> p:
The other cases with different relative positions of ziand
sare similarly treated.
Now, consider t2 p < t1þ t2. Then, jV00j 2t1þ t2
and jV00 V00s
2 j < 2t1 for s 2 S. Another V200t exists with
t2 S, then jV00 V00s
2 V200tj 2t1 1 t2 1 < t1 2
for t2> t1. Thus, the number of V200x is less than t1 for
x2 K, contradicting the assumption that jKj > t1.
There-fore, exactly one V00s
2 has jV200sj > t2and jKj 1 V200xs have
jV00x
2 j t2. G1 is t1-regular and jV00 V200sj < 2t1, so this
case is similar to that of 0 p < t2, in which at least one
neighbor z of s has jV00z
2 j < 2t1=t1¼ 2. Obviously, the
subgraph induced by V20z is connected. For such z and s
with jV200sj > t2 and jV200zj < 2, jT ðG2s[ Gz2; V20zÞj jV200sj
can be obtained. By Lemma 8, jT ðGx
2; V20xÞj ¼ jV200xj for x2 K and x 6¼ s. Therefore, jT ðG; V0Þj X x2K;x6¼s;z jT ðGx 2; V20xÞj ! þ jT ðGs 2[ G z 2; V20zÞj X x2K;x6¼s;z jV200xj ! þ jV200sj ¼ ð2ðt2þ t2Þ p jV200sj jV 00z 2 jÞ þ jV 00s 2 j ¼ 2ðt1þ t2Þ p jV200zj: Since jV00z 2 j < 2, 2ðt1þ t2Þ p jV100zj 2ðt1þ t2Þ p 1 t1þ t2> p:
In other cases, jT ðG; V0Þj > p can be similarly proven.
Furthermore, Lemma 1 implies that the degree of every
node in G is t1þ t2 (regularity t1þ t2). Therefore, the
proof is completed. tu
A
CKNOWLEDGMENTSThe authors would like to thank the anonymous referees for their valuable comments which helped them improve the paper’s quality. This work was supported in part by the National Science Council of Republic of China under Contract NSC 91-2213-E-014-006.
R
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Chien-Ping Chang received the BS degree in electrical engineering from Chung Cheng Insti-tute of Technology in 1986 and the PhD degree in computer and information science from Na-tional Chiao Tung University, Taiwan, Republic of China, in 1998. He is currently an assistant professor in the Department of Electrical En-gineering, Chung Cheng Institute of Technology, Taiwan, Republic of China. His research inter-ests include parallel computing, interconnection networks, graph theory, image processing, and data hiding.
Pao-Lien Lai received the BS degree in electronic engineering from the Feng Chia University, Taiwan, Republic of China, in 1990 and the MS degree in electrical engineering from the National Tsing Hua University, Taiwan, Republic of China, in 1992, respectively. Cur-rently, she is pursuing a doctoral program in the Department of Computer and Information Science, National Chiao Tung University, Tai-wan, Republic of China. Her research interests include parallel computing, interconnection networks, and graph theory. Jimmy Jiann-Mean Tan received the BS and MS degrees in mathematics from the National Taiwan University, Taiwan, Republic of China, in 1970 and 1973, respectively, and the PhD degree from Carleton University, Ottawa, Cana-da, in 1981. He has been on the faculty of the Department of Computer and Information Science, National Chiao Tung University since 1983. His research interests include design and analysis of algorithms, combinatorial optimiza-tion, interconnection networks, and graph theory.
Lih-Hsing Hsu received the BS degree in mathematics from Chung Yuan Christian Uni-versity, Taiwan, Republic of China, in 1975, and the PhD degree in mathematics from the State University of New York at Stony Brook in 1981. He is currently a professor in the Department of Information Engineering, Ta Hwa Institute of Technology, Taiwan, Republic of China. His research interests include interconnection net-works, algorithms, graph theory, and VLSI layout.
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