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Local diagnosability of generic star-pyramid graph
Information Processing Letters 109 (2009) 695–699
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
Local diagnosability of generic star-pyramid graph ✩ Qi-yan Zhou ∗ , San-yang Liu, Qiang Zhu Department of Mathematics, University of Xidian of China, Xi’an, Shanxi 710071, China
a r t i c l e
i n f o
Article history: Received 15 April 2008 Available online 18 March 2009 Communicated by F. Dehne Keywords: Interconnection networks PMC model Strong local diagnosability Generic star-pyramid graph
a b s t r a c t The problem of fault diagnosis in network has been discussed widely. In this paper, we study the local diagnosability of a generic star-pyramid graph. We prove that under the PMC model the local diagnosability of each vertex in a generic star-pyramid graph is equal to its degree and the generic star-pyramid has the strong local diagnosability property. Then we study the local diagnosability of a faulty graph. After showing some properties of the graph, we prove that a generic star-pyramid graph keeps the strong property no matter how many edges are faulty under the condition that each vertex is incident with at least four fault-free edges. Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved.
1. Introduction The rapid development in technology has resulted in development of systems incorporating a very large number of processors. As the number of processors in systems increases, the diagnosis of such systems is faced with new challenges. One of the most important problem concerning multiprocessor systems is to locate and replace the faulty processors to maintain high reliability of the system. The process of identifying faulty processors in a system by analyzing the outcomes of available inter-processor tests is system-level diagnosis. The foundation of system diagnosis and an original diagnostic model is the PMC model [7]. Under the PMC model, all tests are performed between two adjacent processors. In the study of multiprocessor systems, the topology of a system is often represented by a graph G ( V , E ), where each node u ∈ V denotes a processor and each edge (u , v ) ∈ E denotes a link between nodes u and v. Throughout this paper we use a graph G ( V , E ) to represent a self-diagnosable system in which an edge directed from vertex u to vertex v means that u can test v. In this situation, u is called the tester and v is called the tested vertex. The outcome of a test result is 1 (respec✩ The work was supported by NNSF of China (Nos. 60674108 and 60574075). Corresponding author. E-mail address: [email protected] (Q.-Y. Zhou).
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0020-0190/$ – see front matter Crown Copyright doi:10.1016/j.ipl.2009.03.010
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tively, 0) if u evaluates v as faulty (respectively, fault-free), and it is assumed that the test is reliable (respectively, unreliable) if the processor that initiates the test is faultfree (respectively, fault). The collection of all testing results is called a syndrome. Formally, a syndrome is a function: σ : E → {0, 1}. For any subset S ⊂ V , the notation G − S represents the graph obtained by removing the vertices in S from G and deleting those edges with at least one end vertex in S simultaneously. Let S 1 , S 2 ⊆ V (G ) be two distinct sets. The symmetric difference of the two sets is defined as the set S 1 S 2 = ( S 1 − S 2 ) ∪ ( S 2 − S 1 ). G is called t-diagnosable if, given the test outcomes obtained by the testing link, all the faulty vertices can be uniquely identified without replacement, provided that the number of faulty vertices does not exceed t. The diagnosability of a system is defined as the maximum number t such that the system is diagnosable. Many topologies have been proposed to interconnect processors in a multiprocessor system. Among them, the hypercube and the star have drawn the greatest attention, since they possess many attractive properties and the two system have been widely studied in literatures [1–4,6,8]. In [6], Lai et al. introduced a measure of diagnosability called conditional diagnosability by restricting that a faulty set cannot contain all the neighbors of any vertex. Based on this restriction, the conditional diagnosability of the n-dimensional hypercube is shown to be 4(n − 2) + 1. Observing discussions in previous literature
by Elsevier B.V. All rights reserved.
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Q.-Y. Zhou et al. / Information Processing Letters 109 (2009) 695–699
about diagnosability of a system consider the global sense but ignore some local information, G.H. Hsu and Jimmy J.M. Tan proposed a new measure of diagnosability, called local diagnosability, and two useful structures for determining whether a system is locally t-diagnosable at a given node were proposed. They also introduced a concept called the strong local diagnosability property. A system has the strong local diagnosability property if the local diagnosability of every vertex is equal to its degree. This measure of diagnosability leads us to study the local diagnosability instead of the whole system. It has been shown that both hypercube and star have the strong local diagnosability property [3,4]. The algorithm and the time complexities of the new local diagnosability measure are clear from the discussion in paper [4]. In this paper, we study the diagnosability of the generic star-pyramid graph which was first introduced in [5]. The new topology has some good properties compared with the pyramid graph, such as better scalability, higher faulttolerance, and lower diameter. Based on the useful structures introduced in [3], the local diagnosability of each vertex in a generic star-pyramid graph is shown to be equal to its own degree and the graph has the strong local diagnosability property. Then we study the local diagnosability of an incomplete generic star-pyramid graph. Assuming that each vertex of the graph is incident with at least four faultfree edges, we show that a generic star-pyramid graph keeps this strong property no matter how many edges are faulty. Finally, we give a corollary. 2. Preliminaries and previous results A star graph S n of n-dimension is defined to be a symmetric graph G ( V , E ) with v being the set of all permutations of {1, 2, 3, . . . , n} and E consisting of the symmetric edges (u , v ) such that two permutations u and v are connected by an edge if one can be reached from the other by interchanging its first symbol with the ith symbol. So an n-star is an n-regular graph. The degree of each vertex in S n is n − 1. Each star graph S n can be decomposed into n (n − 1)-substar graphs, each is isomorphic to S n−1 . An n-star-pyramid, SPn , is constructed by piling up the star graphs of dimensions 1 to n in a hierarchy ( S 1 , S 2 , . . . , S n ) with each S i , 1 < i < n, connected to S i −1 from the top and to S i +1 from the bottom using some extra links. Each vertex v i of SPn in level i , 1 i n, can be distinctly labeled by a binary n-bit string. Say an arbitrary node v i = (a1 a2 . . . ai ) in level i, a node v j of level i + 1 is a neighbor of v i if and if only the index of v j is derived by stuffing of i + 1 symbol in any of the i + 1 possible positions in the v i index. A generic star-pyramid (m, n) is modeled as a graph GSPm,n which is obtained by removing the upper SPm−1 component from SPn , m n. It thus consists of levels m to n of SPn . Definition 1. Let G ( V , E ) be a graph, v ∈ V be a vertex and k be an integer, k 1, a Type I structure T 1 ( v ; k) of order k at vertex v is defined to be the following graph: T 1 ( v ; k) = [ V ( v ; k), E ( v ; k)], which is composed of 2k + 1 vertices and of 2k edges as illustrated in Fig. 1, where
Fig. 1. A Type I structure T 1 ( v ; k).
Fig. 2. A Type II structure T 2 ( v ; k, 2).
V ( v ; k) = { v } ∪ {xi , y i | 1 i k},
E ( v ; k) = ( v , xi ), (xi , y i ) | 1 i k . Definition 2. Let G ( V , E ) be a graph, v ∈ V be a vertex and k be an integer, k 1, a Type II structure T 2 ( v ; k, 2) of order k + 2 at vertex v is defined to be the following graph: T 2 ( v ; k, 2) = [ V ( v ; k, 2), E ( v ; k, 2)], which is composed of 2k + 5 vertices and of 2k + 5 edges as illustrated in Fig. 2, where V ( v ; k, 2) = { v } ∪ {xi , y i | 1 i k} ∪ { z1 , z2 , z3 , z4 },
E ( v ; k, 2) = ( v , xi ), (xi , y i ) | 1 i k
∪ ( v , z1 ), ( v , z2 ), (z1 , z3 ), (z2 , z3 ), (z3 , z4 ) .
Lemma 1. Let G ( V , E ) be a graph. For any two distinct sets F 1 , F 2 ⊂ V (G ), ( F 1 , F 2 ) is a distinguishable pair if and only if there exists a vertex u ∈ V − F 1 ∪ F 2 and a vertex v ∈ F 1 F 2 such that (u , v ) ∈ E. Lemma 2. Let G ( V , E ) be a graph, v ∈ V be a vertex. G is locally t-diagnosable at vertex v if and only if for any two distinct sets of vertices F 1 , F 2 ⊂ V (G ), | F 1 | t , | F 2 | t , v ∈ F 1 F 2 and ( F 1 , F 2 ) is a distinguishable pair. The local diagnosability of vertex v, written as tl (G ), is defined to be the maximum value of t such that G is locally t-diagnosable at vertex v. Lemma 3. Let G ( V , E ) be a graph, v ∈ V be a vertex with deg( v ) = t. The local diagnosability of vertex v is t if G contains a Type I structure T 1 ( v ; t ) of order t or a Type II structure T 2 ( v ; t − 2, 2) of order t at vertex v as a subgraph. 3. Strong local diagnosability property
n
There are i =m i ! vertices in a in a generic star-pyramid (m, n), GSPm,n . Each vertices v i in an intermediate level i, m < i < n, has degree 2i + 1. A node v m in the top level m has degree 2m and a node v n in the bottom level n has degree n.
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Definition 3. Let G ( V , E ) be a graph, v ∈ V be a vertex. Vertex v has the strong local diagnosability property if the local diagnosability of vertex v is equal to its degree. G has the strong local diagnosability property if every vertex v has the strong local diagnosability property. Lemma 4. Let S n be an n-dimensional star graph, n 3. S n has the strong local diagnosability property. Fig. 3. An indistinguishable pair ( F 1 , F 2 ).
Theorem 1. Let GSPm,n be a generic star-pyramid graph, 3 m < n, GSPm,n has the strong local diagnosability property. Proof. We use Lemma 3 to prove this result. Since an nstar is vertex-symmetric and owing to the regular links between levels, we can concentrate on the construction of a Type I structure at an arbitrary given node v i of level i, m i n. It is easy to see that if we find a private neighbor for each linking node of v i , then a Type I structure is constructed. In the following proof, we consider three cases. Case 1. For a node v i in intermediate level i , m < i < n. We know that GSPm,n is constructed by stuffing of i + 1 symbol in any of i + 1 possible positions in the v i index. Say a node v i = (a1 a2 a3 . . . ai −1 ai ), the nodes of level i + 1 which is connected to v i have the following permutations: (a1 a2 a3 . . . ai i + 1), (a1 a2 a3 . . . i + 1ai ), . . . , (a1 a2 i + 1 . . . ai −1 ai ), (a1 i + 1a2 . . . ai −1 ai ), (i + 1a1 a2 . . . ai −1 ai ). Let a = (a1 a2 i + 1 . . . ai −1 ai ), b = (a1 i + 1a2 . . . ai −1 ai ), c = (i + 1a1 a2 . . . ai −1 ai ). Among these neighbors, it is easy to see that only the two nodes b and c are neighbors and the two nodes c and a have a common neighbor d = (a2 a1 i + 1a3 . . . ai −1 ai ). Since the degree of each node is greater than three, we can find a different neighbor for b except c, and a different neighbor for a except the common neighbor d with c, and we take d as the private neighbor of c. Now we find a private neighbor for all the i + 1 linking nodes of v i in level i + 1. We have known that S n has the strong local diagnosability property. Now for a node v i in level i , v i has already had a Type I structure T 1 ( v ; ki ) of order ki , ki = 2i. Removing the i symbol in v i index, we then obtain the only neighbor of v i in level i − 1, let it be v i −1 . Since v i −1 has degree greater than three, we can find another neighbor for v i −1 except v i . By the above discussions, we find a Type I structure T 1 ( v ; k) of order k, k = deg( v i ) = 2i + 1 for each node in all intermediate levels. Case 2. For any node v m = (a1 a2 a3 . . . am−1 am ) in the top level m. Since the m-star S m has the strong local diagnosability property, v m has already had a Type I structure T 1 ( v ; m − 1) of order m − 1. By similar discussions as Case 1, we know that the m + 1 neighbors of v m in level m + 1 all have their private neighbor. So we find a Type I structure T 1 ( v ; 2m) of order 2m for v m . Case 3. For any node v n in the bottom level n. Similarly, we can find a Type I structure T 1 ( v ; n) of order n for v n . Now for each node in GSPm,n , we have found a Type I structure of T 1 ( v ; k) of order k, where k = deg( v ), by
Fig. 4. An example of SP3 .
Lemma 3 and Definition 3, GSPm,n has the strong local diagnosability property. 2 In the following, we give an example to show that the condition m 3 is tight. We show that a node v in level 2 in a star-pyramid SPn does not have the strong local diagnosability property since tl ( v ) = deg( v ), so the graph does not have the strong property. As shown in Figs. 3 and 4, let F 1 = { v , b, c , e , h}, F 2 = {a, b, c , e , h} with | F 1 | 5, | F 2 | 5. Since there is no edge between V (G ) − F 1 ∪ F 2 and F 1 F 2 , by Lemma 1, ( F 1 , F 2 ) is an indistinguishable pair. Therefore, the local diagnosability of vertex v is at most 4 which is smaller than its degree. 4. Conditional fault local diagnosability In [4], G.H. Hsu and Jimmy J.M. Tan showed that an incomplete star preserved the strong local diagnosability property when removing all the edges in S from S n , 0 | S | n − 3. And if there are n − 2 faulty edges, all these faulty edges are incident with a single vertex and this vertex is incident with only one fault-free edge, we know that S n does not have the strong local diagnosability property. Then we are led to the following questions: How many edges can be removed from GSPm,n such that GSPm,n keeps the strong local diagnosability property? First let us explore some properties of GSPm,n . Proposition 1. Let GSPm,n be a generic star-pyramid graph, 3 m < n, for any two nodes v i and u i in level i , m i n, they have no common neighbor in level i + 1. It is easy to see this from the fact that GSPm,n is constructed by stuffing of i + 1 symbol in any of i + 1 possible positions in the v i index and each v i has a different permutation. Proposition 2. Let GSPm,n be a generic star-pyramid graph, 3 m < n, a given node v i is contained in at most two cycles of length three, m i n.
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Proof. Given an arbitrary node v i = (a1 a2 a3 . . . aα . . . ai −1 ai ) in level i, neighbors of v i in level i have the permutations: (aα a2 a3 . . . a1 . . . ai −1 ai ), 2 α i and neighbors of v i in level i + 1 are shown in Case 1 of Theorem 1. We know that there is no cycle of length less than six in a star graph, therefore we shall find cycle of length three formed by nodes of adjacent levels. By Proposition 1, any two nodes v i and u i in level i has no common neighbor in level i + 1, so there is no cycle of length three formed by two nodes in level i and a node in level i + 1. Of the nodes connected to v i in level i + 1, only nodes b and c are neighbors, so v i bc is the only cycle of length three that v i is contained in between adjacent level i and level i + 1, where b = (a1 i + 1a2 . . . ai −1 ai ), c = (i + 1a1 a2 . . . ai −1 ai ). Now we will find another cycle of length three formed by nodes of adjacent level i and level i − 1. We have known that node v i has only one neighbor in level i − 1, let it be z, then the situation a cycle of length three exists would be the case: a neighbor of v i in level i also has linking with z. Now by analyzing the position of the i symbol in v i , we show that v i is contained in at most two cycles of length three. If a2 is the i symbol in v i index, z1 = (a1 a3 . . . ai −1 ai ) is the neighbor of v i in level i − 1, among neighbors of v i in level i , x1 = (ia1 . . . ai −1 ai ) is the only node that also has linking with z. Then the three nodes form a cycle v i z1 x1 of length three. So v i is in the two cycles v i z1 x1 and v i bc. Similarly, if a1 is the i symbol in v i index, we can find v i is in the two cycles v i z2 x1 and v i bc, where z2 = (a2 a3 . . . ai −1 ai ). In these two cases, v i is contained in two cycles of length three. If the i symbol is not in the first and the second position in v i index, there is no neighbor of v i in level i that is also connected to node z, then v i bc is the only cycle of length three that v i is contained in. So any node v i can be contained in at most two cycles of length three. 2 We can conclude that there exists a node in the bottom level which is not contained in a cycle of length three, any node in the top level must be contained in a cycle of length three and a node that can be contained in two cycles of length three is in intermediate levels. By simply search, we can also find that the neighbor (aα a2 a3 . . . a1 . . . ai −1 ai ) of v i in level i and a neighbor of v i in level i + 1 can have a common neighbor in level i + 1 which has one of the following permutations: (aα a2 a3 . . . a1 . . . i + 1ai ), (aα a2 a3 . . . a1 . . . i + 1ai ), . . . , (aα a2 i + 1 . . . a1 . . . ai −1 ai ), (aα i + 1a2 . . . a1 . . . ai −1 ai ). Then there exist cycles of length four in GSPm,n . So any two nodes have at most two common neighbors. Lemma 5. Let G ( V , E ) be a bipartite graph. The maximum size of a matching in G equals the minimum size of a vertex cover of G. Let G ( V , E ) be a graph with and F ⊂ E (G ) be a set of edges, removing edges in F from G, we define BG( v ) = ( L 1 ( v )∪ L 2 ( v ), E ) to be the bipartite graph under v, where
L 1 ( v ) = x ∈ V (G ) | vertex x is adjacent to v in G − F ,
Fig. 5. A Type II structure T 2 ( v ; k − 2, 2).
L 2 ( v ) = y ∈ V (G ) | there exists a vertex x ∈ L 1 ( v )
such that (x, y ) ∈ E (G ) in G − F − { v },
E BG( v ) = (x, y ) ∈ E (G ) | x ∈ L 1 ( v ), y ∈ L 2 ( v ) . L 1 ( v ), L 2 ( v ) is called the level one and level two vertex under v respectively. Theorem 2. Let GSPm,n be a generic star-pyramid graph with 3 m < n, and F ⊂ E (GSPm,n ) be a set of edges. Assume that each vertex of GSPm,n − F is incident with at least four fault-free edges, removing all the edges in F from GSPm,n , GSPm,n has still the strong local diagnosability property. Proof. According to Lemma 3, we can concentrate on the construction of Type I structure or Type II structure at each vertex. Consider a vertex v in GSPm,n − F , deg( v ) = k. Let BG( v ) = ( L 1 ( v ) ∪ L 2 ( v ), E ) to be the bipartite graph under v, then | L 1 ( v )| = k. Let M ⊂ E (BG( v )) be a maximum matching from L 1 ( v ) to L 2 ( v ). In the following proof, we consider three cases by the size of M. Case 1. | M | = k. Since | M | = k and | L 1 ( v )| = k, there exists a Type I structure T 1 ( v ; k) of order k at vertex v. By Lemma 3, the local diagnosability of vertex v is equal to k. Case 2. | M | = k − 1. As shown in Fig. 5, since each node is incident with at least four fault-free edges. By simple discussion, we can find a Type II structure T 2 ( v ; k − 2, 2) of order k at vertex v. By Lemma 3, the local diagnosability of vertex v is equal to k. Case 3. | M | k − 2. We will see that this is an impossible case. By Lemma 5, the minimum size of a vertex cover of the bipartite graph BG( v ) is no greater than k − 2. However, we claim that any k − 2 vertices of BG( v ) can not cover all the edges of BG( v ). Suppose we take a vertex cover with the minimum size and let VL1 ( v ) ⊂ L 1 ( v ), VL2 ( v ) ⊂ L 2 ( v ) and VL1 ( v ) ∪ VL2 ( v ) be the vertex cover. VL1 ( v ) and VL2 ( v ) can cover all the edges of BG( v ). Since |VL1 ( v )| + |VL2 ( v )| k − 2. We rewrite this inequality into the equivalent form: 2(k − |VL1 ( v )|) 2(VL2 ( v ) + 2). Let NVL1 ( v ) = L 1 ( v ) − VL1 ( v ). Since each vertex of GSPm,n − F is incident with at least four fault-free edges and by Proposition 2, vertex v is contained in at most two cycles of length three and the two
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By Lemma 3 and the same matching method in Theorem 3, we have the following result. Corollary 1. Let G ( V , E ) be a graph, v ∈ V be a vertex. S ⊂ E ( S n ) be a set of edges and G has no cycle of length less than five. Assume that each vertex of G − S is incident with at least two fault-free edges, removing all the edges in S from G, the local diagnosability of each vertex is equal to its remaining degree, and G − S has the strong local diagnosability property. References Fig. 6. An illustration for Case 3 of Theorem 3.
cycles have no common edge, then for a node x ∈ NVL1 ( v ), if x is contained in a cycles of length three, it has at least two connecting edge in VL2 ( v ) aside from the edge (x, v ) as shown in Fig. 6 and there are at most four such nodes. Each of the another NVL1 ( v ) − 4 nodes has at least three edges connecting with the vertices in VL2 ( v ). So the total number of edges incident with the vertices in VL2 ( v ) is at least 3(NVL1 ( v ) − 4) + 8. We have stated that any two nodes have at most two common neighbors, for each vertex y ∈ VL2 ( v ), at most two edges connecting y are incident with the vertices in NVL1 ( v ). So the total number of edges incident with the vertices in NVL1 ( v ) is at most 2|VL2 ( v )|. We have the following inequality:
3NVL1 ( v ) − 4 + 8 = 3 k − VL1 ( v ) − 4 + 8
3 VL2 ( v ) + 2 − 4 + 8 > 2VL2 ( v ).
The lower bound 3(NVL2 ( v ) − 4) + 8 is greater than the upper bound 2|VL2 ( v )|. It means that some edges are not covered by VL1 ( v ) or VL2 ( v ). Thus, our claim follows. 2
[1] Toru Araki, Yukio Shibata, Optimal design of diagnosable system on network constructed by graph operations, IEEE Transactions on Computers 85 (2002) 509–518. [2] Arun Somani, Ofer Peleg, On diagnosability of large fault sets in regular topology-based computer system, IEEE Transactions on Computers 45 (1996) 892–902. [3] Guo-Huang Hua, Jimmy J.M. Tan, A local diagnosability measure for multiprocessor systems, IEEE Transactions on Computers 18 (2007) 176–184. [4] Guo-Huang Hsu, Jimmy J.M. Tan, Local diagnosability under the PMC model with application to star graph, IEEE Transactions on Parallel and Distributed Systems (2007) 159–164. [5] N. Imani, H. Sarbazi-Azad, The Star-Pyramid Graph: An Attractive Alternative to the Pyramid, in: Lecture Notes in Computer Science, Springer, 2005, pp. 509–519. [6] P.L. Lai, J.J.M. Tan, C.P. Chang, L.H. Hsu, Conditional diagnosability measures for large multiprocessor systems, IEEE Transactions on Computers 54 (2005) 165–175. [7] F.P. Preparata, G. Metze, R.T. Chien, On the connection assignment problem of diagnosable systems, IEEE Trans. Electronic Computers 16 (1967) 848–854. [8] S. Zheng, S. Zhou, Diagnosability of the incomplete star graph, Tsinghua Science and Technology 12 (2007) 105–109.
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
Local diagnosability of generic star-pyramid graph ✩ Qi-yan Zhou ∗ , San-yang Liu, Qiang Zhu Department of Mathematics, University of Xidian of China, Xi’an, Shanxi 710071, China
a r t i c l e
i n f o
Article history: Received 15 April 2008 Available online 18 March 2009 Communicated by F. Dehne Keywords: Interconnection networks PMC model Strong local diagnosability Generic star-pyramid graph
a b s t r a c t The problem of fault diagnosis in network has been discussed widely. In this paper, we study the local diagnosability of a generic star-pyramid graph. We prove that under the PMC model the local diagnosability of each vertex in a generic star-pyramid graph is equal to its degree and the generic star-pyramid has the strong local diagnosability property. Then we study the local diagnosability of a faulty graph. After showing some properties of the graph, we prove that a generic star-pyramid graph keeps the strong property no matter how many edges are faulty under the condition that each vertex is incident with at least four fault-free edges. Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved.
1. Introduction The rapid development in technology has resulted in development of systems incorporating a very large number of processors. As the number of processors in systems increases, the diagnosis of such systems is faced with new challenges. One of the most important problem concerning multiprocessor systems is to locate and replace the faulty processors to maintain high reliability of the system. The process of identifying faulty processors in a system by analyzing the outcomes of available inter-processor tests is system-level diagnosis. The foundation of system diagnosis and an original diagnostic model is the PMC model [7]. Under the PMC model, all tests are performed between two adjacent processors. In the study of multiprocessor systems, the topology of a system is often represented by a graph G ( V , E ), where each node u ∈ V denotes a processor and each edge (u , v ) ∈ E denotes a link between nodes u and v. Throughout this paper we use a graph G ( V , E ) to represent a self-diagnosable system in which an edge directed from vertex u to vertex v means that u can test v. In this situation, u is called the tester and v is called the tested vertex. The outcome of a test result is 1 (respec✩ The work was supported by NNSF of China (Nos. 60674108 and 60574075). Corresponding author. E-mail address: [email protected] (Q.-Y. Zhou).
*
0020-0190/$ – see front matter Crown Copyright doi:10.1016/j.ipl.2009.03.010
© 2009 Published
tively, 0) if u evaluates v as faulty (respectively, fault-free), and it is assumed that the test is reliable (respectively, unreliable) if the processor that initiates the test is faultfree (respectively, fault). The collection of all testing results is called a syndrome. Formally, a syndrome is a function: σ : E → {0, 1}. For any subset S ⊂ V , the notation G − S represents the graph obtained by removing the vertices in S from G and deleting those edges with at least one end vertex in S simultaneously. Let S 1 , S 2 ⊆ V (G ) be two distinct sets. The symmetric difference of the two sets is defined as the set S 1 S 2 = ( S 1 − S 2 ) ∪ ( S 2 − S 1 ). G is called t-diagnosable if, given the test outcomes obtained by the testing link, all the faulty vertices can be uniquely identified without replacement, provided that the number of faulty vertices does not exceed t. The diagnosability of a system is defined as the maximum number t such that the system is diagnosable. Many topologies have been proposed to interconnect processors in a multiprocessor system. Among them, the hypercube and the star have drawn the greatest attention, since they possess many attractive properties and the two system have been widely studied in literatures [1–4,6,8]. In [6], Lai et al. introduced a measure of diagnosability called conditional diagnosability by restricting that a faulty set cannot contain all the neighbors of any vertex. Based on this restriction, the conditional diagnosability of the n-dimensional hypercube is shown to be 4(n − 2) + 1. Observing discussions in previous literature
by Elsevier B.V. All rights reserved.
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Q.-Y. Zhou et al. / Information Processing Letters 109 (2009) 695–699
about diagnosability of a system consider the global sense but ignore some local information, G.H. Hsu and Jimmy J.M. Tan proposed a new measure of diagnosability, called local diagnosability, and two useful structures for determining whether a system is locally t-diagnosable at a given node were proposed. They also introduced a concept called the strong local diagnosability property. A system has the strong local diagnosability property if the local diagnosability of every vertex is equal to its degree. This measure of diagnosability leads us to study the local diagnosability instead of the whole system. It has been shown that both hypercube and star have the strong local diagnosability property [3,4]. The algorithm and the time complexities of the new local diagnosability measure are clear from the discussion in paper [4]. In this paper, we study the diagnosability of the generic star-pyramid graph which was first introduced in [5]. The new topology has some good properties compared with the pyramid graph, such as better scalability, higher faulttolerance, and lower diameter. Based on the useful structures introduced in [3], the local diagnosability of each vertex in a generic star-pyramid graph is shown to be equal to its own degree and the graph has the strong local diagnosability property. Then we study the local diagnosability of an incomplete generic star-pyramid graph. Assuming that each vertex of the graph is incident with at least four faultfree edges, we show that a generic star-pyramid graph keeps this strong property no matter how many edges are faulty. Finally, we give a corollary. 2. Preliminaries and previous results A star graph S n of n-dimension is defined to be a symmetric graph G ( V , E ) with v being the set of all permutations of {1, 2, 3, . . . , n} and E consisting of the symmetric edges (u , v ) such that two permutations u and v are connected by an edge if one can be reached from the other by interchanging its first symbol with the ith symbol. So an n-star is an n-regular graph. The degree of each vertex in S n is n − 1. Each star graph S n can be decomposed into n (n − 1)-substar graphs, each is isomorphic to S n−1 . An n-star-pyramid, SPn , is constructed by piling up the star graphs of dimensions 1 to n in a hierarchy ( S 1 , S 2 , . . . , S n ) with each S i , 1 < i < n, connected to S i −1 from the top and to S i +1 from the bottom using some extra links. Each vertex v i of SPn in level i , 1 i n, can be distinctly labeled by a binary n-bit string. Say an arbitrary node v i = (a1 a2 . . . ai ) in level i, a node v j of level i + 1 is a neighbor of v i if and if only the index of v j is derived by stuffing of i + 1 symbol in any of the i + 1 possible positions in the v i index. A generic star-pyramid (m, n) is modeled as a graph GSPm,n which is obtained by removing the upper SPm−1 component from SPn , m n. It thus consists of levels m to n of SPn . Definition 1. Let G ( V , E ) be a graph, v ∈ V be a vertex and k be an integer, k 1, a Type I structure T 1 ( v ; k) of order k at vertex v is defined to be the following graph: T 1 ( v ; k) = [ V ( v ; k), E ( v ; k)], which is composed of 2k + 1 vertices and of 2k edges as illustrated in Fig. 1, where
Fig. 1. A Type I structure T 1 ( v ; k).
Fig. 2. A Type II structure T 2 ( v ; k, 2).
V ( v ; k) = { v } ∪ {xi , y i | 1 i k},
E ( v ; k) = ( v , xi ), (xi , y i ) | 1 i k . Definition 2. Let G ( V , E ) be a graph, v ∈ V be a vertex and k be an integer, k 1, a Type II structure T 2 ( v ; k, 2) of order k + 2 at vertex v is defined to be the following graph: T 2 ( v ; k, 2) = [ V ( v ; k, 2), E ( v ; k, 2)], which is composed of 2k + 5 vertices and of 2k + 5 edges as illustrated in Fig. 2, where V ( v ; k, 2) = { v } ∪ {xi , y i | 1 i k} ∪ { z1 , z2 , z3 , z4 },
E ( v ; k, 2) = ( v , xi ), (xi , y i ) | 1 i k
∪ ( v , z1 ), ( v , z2 ), (z1 , z3 ), (z2 , z3 ), (z3 , z4 ) .
Lemma 1. Let G ( V , E ) be a graph. For any two distinct sets F 1 , F 2 ⊂ V (G ), ( F 1 , F 2 ) is a distinguishable pair if and only if there exists a vertex u ∈ V − F 1 ∪ F 2 and a vertex v ∈ F 1 F 2 such that (u , v ) ∈ E. Lemma 2. Let G ( V , E ) be a graph, v ∈ V be a vertex. G is locally t-diagnosable at vertex v if and only if for any two distinct sets of vertices F 1 , F 2 ⊂ V (G ), | F 1 | t , | F 2 | t , v ∈ F 1 F 2 and ( F 1 , F 2 ) is a distinguishable pair. The local diagnosability of vertex v, written as tl (G ), is defined to be the maximum value of t such that G is locally t-diagnosable at vertex v. Lemma 3. Let G ( V , E ) be a graph, v ∈ V be a vertex with deg( v ) = t. The local diagnosability of vertex v is t if G contains a Type I structure T 1 ( v ; t ) of order t or a Type II structure T 2 ( v ; t − 2, 2) of order t at vertex v as a subgraph. 3. Strong local diagnosability property
n
There are i =m i ! vertices in a in a generic star-pyramid (m, n), GSPm,n . Each vertices v i in an intermediate level i, m < i < n, has degree 2i + 1. A node v m in the top level m has degree 2m and a node v n in the bottom level n has degree n.
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Definition 3. Let G ( V , E ) be a graph, v ∈ V be a vertex. Vertex v has the strong local diagnosability property if the local diagnosability of vertex v is equal to its degree. G has the strong local diagnosability property if every vertex v has the strong local diagnosability property. Lemma 4. Let S n be an n-dimensional star graph, n 3. S n has the strong local diagnosability property. Fig. 3. An indistinguishable pair ( F 1 , F 2 ).
Theorem 1. Let GSPm,n be a generic star-pyramid graph, 3 m < n, GSPm,n has the strong local diagnosability property. Proof. We use Lemma 3 to prove this result. Since an nstar is vertex-symmetric and owing to the regular links between levels, we can concentrate on the construction of a Type I structure at an arbitrary given node v i of level i, m i n. It is easy to see that if we find a private neighbor for each linking node of v i , then a Type I structure is constructed. In the following proof, we consider three cases. Case 1. For a node v i in intermediate level i , m < i < n. We know that GSPm,n is constructed by stuffing of i + 1 symbol in any of i + 1 possible positions in the v i index. Say a node v i = (a1 a2 a3 . . . ai −1 ai ), the nodes of level i + 1 which is connected to v i have the following permutations: (a1 a2 a3 . . . ai i + 1), (a1 a2 a3 . . . i + 1ai ), . . . , (a1 a2 i + 1 . . . ai −1 ai ), (a1 i + 1a2 . . . ai −1 ai ), (i + 1a1 a2 . . . ai −1 ai ). Let a = (a1 a2 i + 1 . . . ai −1 ai ), b = (a1 i + 1a2 . . . ai −1 ai ), c = (i + 1a1 a2 . . . ai −1 ai ). Among these neighbors, it is easy to see that only the two nodes b and c are neighbors and the two nodes c and a have a common neighbor d = (a2 a1 i + 1a3 . . . ai −1 ai ). Since the degree of each node is greater than three, we can find a different neighbor for b except c, and a different neighbor for a except the common neighbor d with c, and we take d as the private neighbor of c. Now we find a private neighbor for all the i + 1 linking nodes of v i in level i + 1. We have known that S n has the strong local diagnosability property. Now for a node v i in level i , v i has already had a Type I structure T 1 ( v ; ki ) of order ki , ki = 2i. Removing the i symbol in v i index, we then obtain the only neighbor of v i in level i − 1, let it be v i −1 . Since v i −1 has degree greater than three, we can find another neighbor for v i −1 except v i . By the above discussions, we find a Type I structure T 1 ( v ; k) of order k, k = deg( v i ) = 2i + 1 for each node in all intermediate levels. Case 2. For any node v m = (a1 a2 a3 . . . am−1 am ) in the top level m. Since the m-star S m has the strong local diagnosability property, v m has already had a Type I structure T 1 ( v ; m − 1) of order m − 1. By similar discussions as Case 1, we know that the m + 1 neighbors of v m in level m + 1 all have their private neighbor. So we find a Type I structure T 1 ( v ; 2m) of order 2m for v m . Case 3. For any node v n in the bottom level n. Similarly, we can find a Type I structure T 1 ( v ; n) of order n for v n . Now for each node in GSPm,n , we have found a Type I structure of T 1 ( v ; k) of order k, where k = deg( v ), by
Fig. 4. An example of SP3 .
Lemma 3 and Definition 3, GSPm,n has the strong local diagnosability property. 2 In the following, we give an example to show that the condition m 3 is tight. We show that a node v in level 2 in a star-pyramid SPn does not have the strong local diagnosability property since tl ( v ) = deg( v ), so the graph does not have the strong property. As shown in Figs. 3 and 4, let F 1 = { v , b, c , e , h}, F 2 = {a, b, c , e , h} with | F 1 | 5, | F 2 | 5. Since there is no edge between V (G ) − F 1 ∪ F 2 and F 1 F 2 , by Lemma 1, ( F 1 , F 2 ) is an indistinguishable pair. Therefore, the local diagnosability of vertex v is at most 4 which is smaller than its degree. 4. Conditional fault local diagnosability In [4], G.H. Hsu and Jimmy J.M. Tan showed that an incomplete star preserved the strong local diagnosability property when removing all the edges in S from S n , 0 | S | n − 3. And if there are n − 2 faulty edges, all these faulty edges are incident with a single vertex and this vertex is incident with only one fault-free edge, we know that S n does not have the strong local diagnosability property. Then we are led to the following questions: How many edges can be removed from GSPm,n such that GSPm,n keeps the strong local diagnosability property? First let us explore some properties of GSPm,n . Proposition 1. Let GSPm,n be a generic star-pyramid graph, 3 m < n, for any two nodes v i and u i in level i , m i n, they have no common neighbor in level i + 1. It is easy to see this from the fact that GSPm,n is constructed by stuffing of i + 1 symbol in any of i + 1 possible positions in the v i index and each v i has a different permutation. Proposition 2. Let GSPm,n be a generic star-pyramid graph, 3 m < n, a given node v i is contained in at most two cycles of length three, m i n.
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Proof. Given an arbitrary node v i = (a1 a2 a3 . . . aα . . . ai −1 ai ) in level i, neighbors of v i in level i have the permutations: (aα a2 a3 . . . a1 . . . ai −1 ai ), 2 α i and neighbors of v i in level i + 1 are shown in Case 1 of Theorem 1. We know that there is no cycle of length less than six in a star graph, therefore we shall find cycle of length three formed by nodes of adjacent levels. By Proposition 1, any two nodes v i and u i in level i has no common neighbor in level i + 1, so there is no cycle of length three formed by two nodes in level i and a node in level i + 1. Of the nodes connected to v i in level i + 1, only nodes b and c are neighbors, so v i bc is the only cycle of length three that v i is contained in between adjacent level i and level i + 1, where b = (a1 i + 1a2 . . . ai −1 ai ), c = (i + 1a1 a2 . . . ai −1 ai ). Now we will find another cycle of length three formed by nodes of adjacent level i and level i − 1. We have known that node v i has only one neighbor in level i − 1, let it be z, then the situation a cycle of length three exists would be the case: a neighbor of v i in level i also has linking with z. Now by analyzing the position of the i symbol in v i , we show that v i is contained in at most two cycles of length three. If a2 is the i symbol in v i index, z1 = (a1 a3 . . . ai −1 ai ) is the neighbor of v i in level i − 1, among neighbors of v i in level i , x1 = (ia1 . . . ai −1 ai ) is the only node that also has linking with z. Then the three nodes form a cycle v i z1 x1 of length three. So v i is in the two cycles v i z1 x1 and v i bc. Similarly, if a1 is the i symbol in v i index, we can find v i is in the two cycles v i z2 x1 and v i bc, where z2 = (a2 a3 . . . ai −1 ai ). In these two cases, v i is contained in two cycles of length three. If the i symbol is not in the first and the second position in v i index, there is no neighbor of v i in level i that is also connected to node z, then v i bc is the only cycle of length three that v i is contained in. So any node v i can be contained in at most two cycles of length three. 2 We can conclude that there exists a node in the bottom level which is not contained in a cycle of length three, any node in the top level must be contained in a cycle of length three and a node that can be contained in two cycles of length three is in intermediate levels. By simply search, we can also find that the neighbor (aα a2 a3 . . . a1 . . . ai −1 ai ) of v i in level i and a neighbor of v i in level i + 1 can have a common neighbor in level i + 1 which has one of the following permutations: (aα a2 a3 . . . a1 . . . i + 1ai ), (aα a2 a3 . . . a1 . . . i + 1ai ), . . . , (aα a2 i + 1 . . . a1 . . . ai −1 ai ), (aα i + 1a2 . . . a1 . . . ai −1 ai ). Then there exist cycles of length four in GSPm,n . So any two nodes have at most two common neighbors. Lemma 5. Let G ( V , E ) be a bipartite graph. The maximum size of a matching in G equals the minimum size of a vertex cover of G. Let G ( V , E ) be a graph with and F ⊂ E (G ) be a set of edges, removing edges in F from G, we define BG( v ) = ( L 1 ( v )∪ L 2 ( v ), E ) to be the bipartite graph under v, where
L 1 ( v ) = x ∈ V (G ) | vertex x is adjacent to v in G − F ,
Fig. 5. A Type II structure T 2 ( v ; k − 2, 2).
L 2 ( v ) = y ∈ V (G ) | there exists a vertex x ∈ L 1 ( v )
such that (x, y ) ∈ E (G ) in G − F − { v },
E BG( v ) = (x, y ) ∈ E (G ) | x ∈ L 1 ( v ), y ∈ L 2 ( v ) . L 1 ( v ), L 2 ( v ) is called the level one and level two vertex under v respectively. Theorem 2. Let GSPm,n be a generic star-pyramid graph with 3 m < n, and F ⊂ E (GSPm,n ) be a set of edges. Assume that each vertex of GSPm,n − F is incident with at least four fault-free edges, removing all the edges in F from GSPm,n , GSPm,n has still the strong local diagnosability property. Proof. According to Lemma 3, we can concentrate on the construction of Type I structure or Type II structure at each vertex. Consider a vertex v in GSPm,n − F , deg( v ) = k. Let BG( v ) = ( L 1 ( v ) ∪ L 2 ( v ), E ) to be the bipartite graph under v, then | L 1 ( v )| = k. Let M ⊂ E (BG( v )) be a maximum matching from L 1 ( v ) to L 2 ( v ). In the following proof, we consider three cases by the size of M. Case 1. | M | = k. Since | M | = k and | L 1 ( v )| = k, there exists a Type I structure T 1 ( v ; k) of order k at vertex v. By Lemma 3, the local diagnosability of vertex v is equal to k. Case 2. | M | = k − 1. As shown in Fig. 5, since each node is incident with at least four fault-free edges. By simple discussion, we can find a Type II structure T 2 ( v ; k − 2, 2) of order k at vertex v. By Lemma 3, the local diagnosability of vertex v is equal to k. Case 3. | M | k − 2. We will see that this is an impossible case. By Lemma 5, the minimum size of a vertex cover of the bipartite graph BG( v ) is no greater than k − 2. However, we claim that any k − 2 vertices of BG( v ) can not cover all the edges of BG( v ). Suppose we take a vertex cover with the minimum size and let VL1 ( v ) ⊂ L 1 ( v ), VL2 ( v ) ⊂ L 2 ( v ) and VL1 ( v ) ∪ VL2 ( v ) be the vertex cover. VL1 ( v ) and VL2 ( v ) can cover all the edges of BG( v ). Since |VL1 ( v )| + |VL2 ( v )| k − 2. We rewrite this inequality into the equivalent form: 2(k − |VL1 ( v )|) 2(VL2 ( v ) + 2). Let NVL1 ( v ) = L 1 ( v ) − VL1 ( v ). Since each vertex of GSPm,n − F is incident with at least four fault-free edges and by Proposition 2, vertex v is contained in at most two cycles of length three and the two
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By Lemma 3 and the same matching method in Theorem 3, we have the following result. Corollary 1. Let G ( V , E ) be a graph, v ∈ V be a vertex. S ⊂ E ( S n ) be a set of edges and G has no cycle of length less than five. Assume that each vertex of G − S is incident with at least two fault-free edges, removing all the edges in S from G, the local diagnosability of each vertex is equal to its remaining degree, and G − S has the strong local diagnosability property. References Fig. 6. An illustration for Case 3 of Theorem 3.
cycles have no common edge, then for a node x ∈ NVL1 ( v ), if x is contained in a cycles of length three, it has at least two connecting edge in VL2 ( v ) aside from the edge (x, v ) as shown in Fig. 6 and there are at most four such nodes. Each of the another NVL1 ( v ) − 4 nodes has at least three edges connecting with the vertices in VL2 ( v ). So the total number of edges incident with the vertices in VL2 ( v ) is at least 3(NVL1 ( v ) − 4) + 8. We have stated that any two nodes have at most two common neighbors, for each vertex y ∈ VL2 ( v ), at most two edges connecting y are incident with the vertices in NVL1 ( v ). So the total number of edges incident with the vertices in NVL1 ( v ) is at most 2|VL2 ( v )|. We have the following inequality:
3NVL1 ( v ) − 4 + 8 = 3 k − VL1 ( v ) − 4 + 8
3 VL2 ( v ) + 2 − 4 + 8 > 2VL2 ( v ).
The lower bound 3(NVL2 ( v ) − 4) + 8 is greater than the upper bound 2|VL2 ( v )|. It means that some edges are not covered by VL1 ( v ) or VL2 ( v ). Thus, our claim follows. 2
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