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The diagnosability and 1-good-neighbor conditional diagnosability of hypercubes with missing links and broken-down nodes
Information Processing Letters 146 (2019) 20–26
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Information Processing Letters www.elsevier.com/locate/ipl
The diagnosability and 1-good-neighbor conditional diagnosability of hypercubes with missing links and broken-down nodes Xiaoyan Li a , Yuan-Hsiang Teng b , Tzu-Liang Kung c , Qi Chen a , Cheng-Kuan Lin d,∗ a
School of Computer Science and Technology, Soochow University, Suzhou 215006, China Department of Computer Science and Information Engineering, Providence University, Taichung 433, Taiwan c Department of Computer Science and Information Engineering, Asia University, Taichung 413, Taiwan d College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China b
a r t i c l e
i n f o
Article history: Received 23 February 2017 Received in revised form 7 January 2019 Accepted 11 January 2019 Available online 7 February 2019 Communicated by Łukasz Kowalik Keywords: Hypercube Diagnosability 1-Good-neighbor conditional diagnosability PMC model Fault tolerance
a b s t r a c t In system-level diagnosis, we propose to further classify faulty nodes into two categories. One category is the “ordinary” faulty nodes – they are malfunctioning, but they still participate in the diagnosis, rendering unreliable test results. The other category contains nodes that are completely broken down so that they cannot test other nodes, and they cannot be tested by other nodes either. In this paper, we study the diagnosability and 1-good-neighbor conditional diagnosability of hypercubes with both ordinary faulty nodes and broken-down nodes. Let S be a set of missing links and broken-down nodes in a hypercube Q n with | S | ≤ n − 1. We prove that the diagnosability of Q n − S is δ( Q n − S ) for n ≥ 3. Furthermore, we show that the 1-good-neighbor conditional diagnosability of Q n − S is δ( E ( Q n − S )) + 1 for n ≥ 4, which is the maximum number of faulty nodes can guarantee to identify, under the condition that every fault-free node has at least a fault-free neighbor. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Many multiprocessor systems take interconnection networks as underlying topology. The underlying topology of a system is usually modeled as a graph topology, where every node corresponds to a processor, and every edge corresponds to a communication link. Even slight malfunctions could make the service not work, and thus, the reliability of a system is the key issue. To ensure the reliability of the system, the devices found fault should be replaced with fault-free ones instantly. Therefore, it is important for the ability to identify any faulty devices, as known as system
*
Corresponding author. E-mail address: [email protected] (C.-K. Lin).
https://doi.org/10.1016/j.ipl.2019.01.007 0020-0190/© 2019 Elsevier B.V. All rights reserved.
diagnosis. The maximum number of faulty devices which can be identified correctly is diagnosability. If all the faulty devices in a system can be pointed out precisely with the faulty devices being t at most, the system is t-diagnosable. Many results about the diagnosis and the diagnosability have been proposed [11,17,19]. Since there is no restrictive condition on the distribution pattern of faulty processors, the classical diagnosability of a system is quite small. In order to increase the diagnosability, Lai et al. [12] proposed a more realistic parameter of diagnosability, conditional diagnosability, which limited that all the neighbors of any processor can not be faulty at the same time in a system. Recently, Peng et al. [2] proposed the notion of g-goodneighbor conditional diagnosability, which is the maximum number of faulty processors that can be identified under the condition that every fault-free processor has no
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less than g fault-free neighbors. The g-good-neighbor conditional diagnosability of various classes of systems have been studied in recent years [2,13,14,18]. In all the works to date on system-level diagnosability, a default assumption is that all faulty nodes are participating in the mutual test process. That is, all faulty nodes test other nodes, although they will render unreliable results, and all faulty nodes are tested by other nodes. The collective test results will be analyzed to determine which nodes are truly faulty. However, in reality this assumption may not be always true. It is likely that some nodes might be “totally disappearing” from the system. This can be caused by a complete, severe breakdown of the faulty node’s hardware, or by the depletion of battery in the case of wireless nodes. Preparata, Metze, and Chien have proposed the PMC diagnosis model in [15]. The PMC model is the tested-based diagnosis. Under the PMC model, a processor performs the diagnosis by testing the neighboring processors via the links between them. Hakimi and Amin have showed that, a system G is t-diagnosable if G is t-connected with at least 2t + 1 nodes under the PMC model [4]. Furthermore, they gave a necessary and sufficient condition for verifying whether or not a system is t-diagnosable under the PMC model. Some related studies have appeared in the literature [1,5,16]. In this paper, we propose to further classify faulty nodes into two categories. The first category is the “ordinary” faulty nodes – they are malfunctioning, but they still participate in the diagnosis, rendering unreliable test results. The second category contains nodes that are completely broken down so that they cannot test other nodes, and they cannot be tested by other nodes either. That is, these faulty nodes act like they have totally disappeared from the system. We call these nodes broken-down nodes. So the self-diagnosis systems contain three types of nodes – fault-free nodes, faulty nodes, broken-down nodes. In practice, the self-diagnosis systems still contain some missing links. The past literature in this field has seen studies of diagnosability for hypercubes with faulty (fault-free) nodes or regularly enhanced links. In this work, we will show that the diagnosability of a faulty hypercube Q n with these three types of nodes is δ( Q n − S ) for n ≥ 3, where S is the set of missing links and broken-down nodes, and δ( Q n − S ) is the minimum degree of Q n − S. Furthermore, we will show that the 1-good-neighbor conditional diagnosability of Q n − S is δ( E ( Q n − S )) + 1 for n ≥ 4, under the condition that every fault-free node has at least a fault-free neighbor. The remainder of this paper is organized as follows. Section 2 introduces the terminologies and notations used throughout the paper; Section 3 establishes the diagnosability of a faulty hypercube Q n − S; Section 4 establishes the 1-good-neighbor conditional diagnosability of a faulty hypercube Q n − S; Finally, we present our conclusion in Section 5. 2. Preliminaries The layout of processors and links in a multiprocessor system is usually represented by an undirected graph.
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For graph definitions and notations, we follow [6]. Let G = ( V (G ), E (G )) be a graph. The node set V (G ) is a finite set, and the edge set E (G ) is a subset of {(u , v )|(u , v ) is an unordered pair of V (G )}. Two nodes u and v are adjacent if (u , v ) ∈ E (G ). The neighborhood of a node u in G is the set of all nodes adjacent to u in G, denoted by N G (u ), N G (u ) = { v ∈ V (G ) | (u , v ) ∈ E (G )}. We define N G ( S ) = { v ∈ V (G ) − S | u ∈ S , (u , v ) ∈ E (G )} = ( N (u )) − S being the set of neighborhood of S in G. u∈ S
The degree of u in G is denoted by deg G (u ) = | N G (u )|. The graph G is k-regular if every node has degree k. We use δ(G ) = min{deg G (u ) : u ∈ V (G )}. A path P is a sequence of nodes in G of the form P = u 0 , u 1 , . . . , ul , such that u i and u i +1 are adjacent for i = 0, . . . , l − 1. A graph G is connected if for any two nodes, there is a path joining them, otherwise it is disconnected. The degree of edge (u , v ) in G is denoted by deg G ((u , v )) = | N G (u )| + | N G ( v )| − 2. We use δ( E (G )) = min{deg G ((u , v )) : (u , v ) ∈ E (G )}. For a set F of V (G ) ∪ E (G ), the notation G − F represents the graph obtained by removing the nodes in F from G and deleting those edges with at least one end node in F . If G − F is disconnected, then F is called a separating set or cut. The connectivity κ (G ) of a graph G is the minimum number of nodes whose removal results in a disconnected graph or only one node left. A graph G is k-connected if its connectivity is at least k. A graph H is a subgraph of G if V ( H ) ⊆ V (G ) and E ( H ) ⊆ E (G ). Under the PMC model, we assume that the adjacent processors can perform tests on each other. Let G = ( V (G ), E (G )) represent the underlying topology of a multiprocessor system. For any two adjacent nodes u and v in G, the ordered pair (u , v ) represents a test that u diagnoses v. In this situation, u is a tester, and v is a testee. If u evaluates v to be faulty (respectively, fault-free), the outcome of the test (u , v ) is 1 (respectively, 0). The faults considered here are permanent, hence the outcome is reliable if and only if the tester is fault-free. A faulty set F is the set of all faulty nodes in G. A test assignment for a system G is a collection of tests, which can be modeled as a directed graph T = ( V (G ), L ). The nodes u and v are adjacent in G, and the test (u , v ) ∈ L. The collection of all test results from the test assignment T is termed a syndrome. A syndrome of T is a mapping σ : L → {0, 1}. Let T = ( V (G ), L ) be a test assignment, and F be a subset of V (G ). For any given syndrome σ resulting from T , F is said to be consistent with σ if for a test (u , v ) ∈ L such that u ∈ V (G ) − F , then σ (u , v ) = 1 if and only if v ∈ F . Thus the fault-free testers always give correct test results, while the faulty testers give rise to unreliable results. Therefore, a faulty set F may be consistent with different syndromes. We use σ ( F ) to represent the set of all possible syndromes with which the faulty set F can be consistent. Let F 1 and F 2 be two distinct faulty sets of V (G ). Then F 1 and F 2 are distinguishable if σ ( F 1 ) ∩ σ ( F 2 ) = ∅; otherwise, F 1 and F 2 are indistinguishable. In other words, ( F 1 , F 2 ) is a distinguishable pair if σ ( F 1 ) ∩ σ ( F 2 ) = ∅; otherwise, ( F 1 , F 2 ) is an indistinguishable pair. Suppose that | F 1 | ≤ t and | F 2 | ≤ t in a system G. G is t-diagnosable if and only if ( F 1 , F 2 ) is a distinguishable pair. We use F 1 F 2 to denote the symmetric difference ( F 1 − F 2 ) ∪ ( F 2 − F 1 ) between F 1
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Fig. 1. An illustration for Theorem 1.
and F 2 . In [3], Dahbura and Masson proposed a sufficient and necessary characterization of t-diagnosable systems. A faulty set F is called a g-good-neighbor conditional faulty set if | N G − F ( v )| ≥ g for every node v in V (G ) − F . The graph G is g-good-neighbor conditional t-diagnosable if each distinct pair of g-good-neighbor conditional faulty sets F 1 and F 2 of V (G ) with | F 1 | ≤ t and | F 2 | ≤ t are distinguishable. The g-good-neighbor conditional diagnosability t g (G ) of a graph G is the maximum value of t such that G is g-good-neighbor conditionally t-diagnosable. Theorem 1. [3] Let G = ( V (G ), E (G )) be a system. For any two distinct sets F 1 , F 2 ⊂ V (G ), F 1 and F 2 are distinguishable if and only if there exists at least one test from V (G ) − ( F 1 ∪ F 2 ) to F 1 F 2 . See Fig. 1 for an illustration. Lemma 1. [2] Let G be a system. Then t g (G ) ≥ t (G ). The hypercube is used as fundamental models for computer networks, in which the properties have been studied for many years. The hypercube is still eye-catching in recent years [7,9,10]. An n-dimensional hypercube Q n is an undirected graph with 2n nodes and n2n−1 edges. Each node in Q n can be represented as an n-bit binary string. For any two distinct nodes u and v in Q n , u is adjacent to v if and only if their binary string representation differs in only one bit position. For any node u = u 0 u 1 . . . un−1 in Q n , we set (u )i = u 0i u 1i . . . uni −1 being the neighbor of u
in dimension i of Q n , where u ij = u j for every i = j and
u ii = 1 − u i . Similarly, ((u )i ) j is the neighbor of (u )i in dimension j. We use Q ni to denote an (n − 1)-dimensional subgraph of Q n , where V ( Q ni ) = {u | u = u 0 u 1 . . . un−1 ∈ V ( Q n ) and un−1 = i }. Obviously, Q ni is isomorphic to Q n−1 for each i. Fig. 2 shows example of Q 3 and Q 4 with missing links and three types of nodes – fault-free nodes, faulty nodes, broken-down nodes. Suppose that n ≥ 3, and F is a set of missing links such that δ( Q n − F ) ≥ 3. Wang [17] proved that Q n − F is δ( Q n − F )-diagnosable. Peng et al. [2] show that t g ( Q n ) = 2 g (n − g ) + 2 g − 1 for 0 ≤ g ≤ n − 3. In the following sections, we will prove the diagnosability and 1-good-neighbor conditional diagnosability of hypercubes with missing links and broken-down nodes. 3. The diagnosability of Q n with missing links and broken-down nodes In this section, we will determine the diagnosability of Q n − S under the PMC model. Lemma 2. [8] The hypercube Q n is n-diagnosable for n ≥ 3.
Fig. 2. A 3-cube and a 4-cube with missing links and three types of nodes.
Lemma 3. Let w, x and y be any three distinct nodes of Q n with {( w , x), (x, y )} ⊂ E ( Q n ). Then | N Q n ( w ) ∪ N Q n (x) ∪ N Q n ( y )| = 3n − 2. Proof. Since ( w , x) ∈ E ( Q n ) and (x, y ) ∈ E ( Q n ), we have | N Q n ( w ) ∩ N Q n (x)| = 0 and | N Q n (x) ∩ N Q n ( y )| = 0. Since x ∈ N Q n ( w ) ∩ N Q n ( y ), and Q n does not contain a cycle of length 3, we have d Q n ( w , y ) = 2. By the definition of Q n , | N Q n ( w ) ∩ N Q n ( y )| = 2. Then, | N Q n ( w ) ∪ N Q n (x) ∪ N Q n ( y )| = | N Q n ( w )| + | N Q n (x)| + | N Q n ( y )| − | N Q n ( w ) ∩ N Q n (x)| − | N Q n (x) ∩ N Q n ( y )| − | N Q n ( w ) ∩ N Q n ( y )| + | N Q n ( w ) ∩ N Q n (x) ∩ N Q n ( y )| = n + n + n − 0 − 0 − 2 + 0 = 3n − 2. 2 Lemma 4. Suppose that n ≥ 3. Let S be a set of missing links and broken-down nodes in Q n with | S | ≤ n − 1, and let F 1 and F 2 be any two distinct node subsets of Q n with | F 1 | ≤ δ( Q n − S ) and | F 2 | ≤ δ( Q n − S ). Then F 1 and F 2 are distinguishable if there exist two distinct nodes x and y in F 1 F 2 with (x, y ) ∈ E ( Q n ).
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Proof. By Lemma 2, F 1 and F 2 are distinguishable if | S | = 0. Thus, we consider that | S | ≥ 1. Since (x, y ) ∈ E ( Q n ), we have N Q n − S (x) ∩ N Q n − S ( y ) = ∅. Suppose that F 1 and F 2 are indistinguishable. By Theorem 1, for every node z ∈ F 1 F 2 , we have N Q n − S ( z) ⊆ F 1 ∪ F 2 . Consider the following cases. Case 1. (x, y ) ∈ S. Since (x, y ) ∈ E ( Q n ), N Q n (x) ∩ N Q n ( y ) = ∅. Since (x, y ) ∈ S, x ∈ / N Q n − S ( y ) and y ∈ / N Q n − S (x). Thus | F 1 | + | F 2 | ≥ | N Q n − S (x) ∪ N Q n − S ( y ) ∪ {x, y }| = | N Q n − S (x)| + | N Q n − S ( y )| + |{x, y }| ≥ δ( Q n − S ) + δ( Q n − S ) + 2. We obtain a contradiction since | F 1 | + | F 2 | ≤ 2δ( Q n − S ).
/ S. Case 2. (x, y ) ∈ Since (x, y ) ∈ E ( Q n ) − S and N Q n (x) ∩ N Q n ( y ) = ∅, we have |( N Q n − S (x) ∪ N Q n − S ( y )) − {x, y }| = deg Q n − S (x) + deg Q n − S ( y ) − 2 ≥ 2n − | S | − 2. Since | F 1 ∩ F 2 | ≤ δ( Q n − S ) − 1, |( N Q n − S (x) ∪ N Q n − S ( y )) − {x, y }| − | F 1 ∩ F 2 | ≥ 2n − | S | − 2 − (δ( Q n − S ) − 1) = (n − | S |) + (n − δ( Q n − S )) − 1 ≥ 1 + 1 − 1 > 0. Thus, there exists a node w in ( N Q n − S (x) ∪ N Q n − S ( y )) − ({x, y } ∪ ( F 1 ∩ F 2 )). By Lemma 3, | F 1 | + | F 2 | ≥ | N Q n − S (x) ∪ N Q n − S ( y ) ∪ N Q n − S ( w )| − | S | ≥ (3n − 2) − (n − 1) = 2n − 1 > 2(n − 1) ≥ 2δ( Q n − S ). We obtain a contradiction since | F 1 | + | F 2 | ≤ 2δ( Q n − S ). 2 Theorem 2. Let S be a set of missing links and broken-down nodes in Q n with | S | ≤ n − 1. Then Q n − S is δ( Q n − S )-diagnosable if n ≥ 3. Proof. Suppose that there exists an indistinguishable pair F 1 and F 2 in Q n − S with max{| F 1 |, | F 2 |} ≤ δ( Q n − S ). Since F 1 and F 2 are indistinguishable, we have N Q n − S ( z) ⊆ F 1 ∪ F 2 for every node z ∈ ( F 1 − F 2 ) ∪ ( F 2 − F 1 ). Without loss of generality, we assume that | F 1 | ≥ | F 2 |. Let x be a node in F 1 − F 2 . Since | F 1 | ≤ δ( Q n − S ), deg Q n − S (x) ≥ δ( Q n − S ), N Q n − S (x) ⊆ F 1 ∪ F 2 and x ∈ F 1 − F 2 , there exists a node y ∈ ( N Q n − S (x) ∩ F 2 ) − F 1 . By Lemma 4, F 1 and F 2 are distinguishable, which contradicts the assumption that F 1 and F 2 are indistinguishable. Hence, Q n − S is δ( Q n − S )-diagnosable, the theorem holds. 2 Theorem 3. Let S be a set of missing links and broken-down nodes in Q n with | S | ≤ n − 1. Then the diagnosability of Q n − S under the PMC model is δ( Q n − S ) if n ≥ 3. Proof. By Theorem 2, Q n − S is δ( Q n − S )-diagnosable. Thus, we only need to show that there exists an indistinguishable pair F 1 and F 2 of Q n − S with | F 1 | ≤ δ( Q n − S ) + 1 and | F 2 | ≤ δ( Q n − S ) + 1. Let u be a node in Q n − S such that deg Q n − S (u ) = δ( Q n − S ). Suppose that F 1 = N Q n − S (u ) ∪ {u } and F 2 = N Q n − S (u ). Then we have | F 1 | = δ( Q n − S ) + 1 and | F 2 | = δ( Q n − S ). By Theorem 1, F 1 and F 2 are indistinguishable. Therefore, the diagnosability of Q n − S under the PMC model is δ( Q n − S ), the theorem holds. 2
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4. The 1-good-neighbor conditional diagnosability of Q n with missing links and broken-down nodes In this section, we will determine the 1-good-neighbor conditional diagnosability of Q n − S under the PMC model. Lemma 5. Let F 1 and F 2 be any two distinct node sets of graph G where | V (G ) −( F 1 ∪ F 2 )| ≥ 1 and G −( F 1 ∩ F 2 ) is connected. Then F 1 and F 2 are distinguishable. Proof. Let y be a node of G − ( F 1 ∪ F 2 ). Since G − ( F 1 ∩ F 2 ) is connected, there is a path R of G − ( F 1 ∩ F 2 ) joining y to F 1 F 2 . That is, there is an edge ( p , q) ∈ E (G − ( F 1 ∩ F 2 )) where p ∈ V (G − ( F 1 ∩ F 2 )) and q ∈ F 1 F 2 . By Theorem 1, F 1 and F 2 are distinguishable. 2 Lemma 6. Let S be a set of missing links and broken-down nodes in Q n with | S | ≤ n − 1, and let F 1 and F 2 be any two distinct 1-good-neighbor conditional faulty sets of Q n − S. Then A = ( F 1 ∩ F 2 ) ∪ S is a 1-good-neighbor conditional faulty set. Proof. Let x be a node of Q n − A. We have following cases. Case 1. x ∈ Q n − S and x ∈ / F1. Since F 1 is a 1-good-neighbor conditional faulty set of Q n − S, thus, there is a node y ∈ V ( Q n − S ) − F 1 such that (x, y ) ∈ E ( Q n − S ). Case 2. x ∈ F 1 − F 2 . Since F 2 is a 1-good-neighbor conditional faulty set of Q n − S, thus, there is a node y ∈ V ( Q n − S ) − F 2 such that (x, y ) ∈ E ( Q n − S ). Based on above two cases, A is a 1-good-neighbor conditional faulty set. 2 Lemma 7. Let F be a set of V ( Q n ) ∪ E ( Q n ) with | F | ≤ 2n − 3 such that every node in V ( Q n ) − F has a neighbor in V ( Q n ) − F . Then Q n − F is connected for n ≥ 2. Proof. Since the connectivity of Q 2 is 2, Q 2 − F is connected. Suppose that this lemma holds on Q k for every 2 ≤ k ≤ n − 1. We set F 0 = ( V ( Q n0 ) ∪ E ( Q n0 )) ∩ F and F 1 = ( V ( Q n1 ) ∪ E ( Q n1 )) ∩ F . Since | F | ≤ 2n − 3, min{| F 0 |, | F 1 |} ≤ n − 2. Without loss of generality, we assume that | F 1 | ≤ n − 2. Thus, Q n1 − F 1 is connected because the connectivity of Q n1 is n − 1. Let x be a node in Q n0 − F 0 . If (x)n−1 ∈ V ( Q n1 ) does not in F 1 , x is connecting to Q n1 − F 1 . Suppose that (x)n−1 be a node in F 1 . Since every node in V ( Q n ) − F has a neighbor in V ( Q n ) − F and (x)n−1 ∈ F , there is a node y ∈ V ( Q n0 ) − F such that (x, y ) ∈ E ( Q n ) − F . Let {x1 , x2 , · · · , xn−2 } be the set of neighbor of x in V ( Q n0 ) − { y } and { y 1 , y 2 , · · · , yn−2 } be the set of neighbor of y in V ( Q n0 ) − {x}. Since Q n does not contain a cycle of length 3, {x1 , x2 , · · · , xn−2 } ∩ { y 1 , y 2 , · · · , yn−2 } = ∅. We set P i = x, xi , (xi )n−1 for every 1 ≤ i ≤ n − 2 and P j = x, y , y j −(n−2) , ( y j −(n−2) )n−1 for every n − 1 ≤ j ≤
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Fig. 3. An illustration for Lemma 9.
2n − 2. Since | F | ≤ 2n − 3, there is a path P t such that ( V ( P t ) ∪ E ( P t )) ∩ F = ∅. That is, x is connecting to a node in Q n1 − F 1 . Since every node in Q n0 − F 0 is connecting to a node in Q n1 − F 1 and Q n1 − F 1 is connected, Q n − F is connected. 2 Theorem 4. For n ≥ 3, suppose that F is a minimum cardinality cut of Q n such that every node in V ( Q n ) − F has a neighbor in V ( Q n ) − F . Then | F | = 2n − 2. Proof. We set (u , v ) ∈ E ( Q n ), let F be a set of V ( Q n ) ∪ E ( Q n ) in which each node adjacent to (u or v) and each edge end in (u or v). It is clear that F is a desired cut of Q n such that every node in V ( Q n ) − F has a neighbor in V ( Q n ) − F . Moreover, by Lemma 7, we have F is a minimum cardinality cut of Q n with | F | = 2n − 2 such that every node in V ( Q n ) − F has a node in V ( Q n ) − F . 2 Lemma 8. For n ≥ 4, let S be a set of missing links and brokendown nodes in Q n with | S | ≤ n − 1, and let F 1 and F 2 be any two distinct 1-good-neighbor conditional faulty sets of Q n − S with | F 1 | ≤ δ( E ( Q n − S )) + 1 and | F 2 | ≤ δ( E ( Q n − S )) + 1. If F 1 and F 2 are two indistinguishable faulty sets, then | F 1 ∩ F 2 | ≥ 2n − 2 − | S |. Proof. Suppose that | F 1 ∩ F 2 | ≤ 2n − 3 − | S |. By Lemma 6, we have A = ( F 1 ∩ F 2 ) ∪ S is a 1-good-neighbor conditional faulty set. We have following cases.
Lemma 9. For n ≥ 4, let S be a set of missing links and brokendown nodes in Q n with | S | ≤ n − 1, and let F 1 and F 2 be any two distinct 1-good-neighbor conditional faulty sets of Q n − S with | F 1 | ≤ δ( E ( Q n − S )) + 1 and | F 2 | ≤ δ( E ( Q n − S )) + 1. Then t 1 ( Q n − S ) ≥ δ( E ( Q n − S )) + 1. Proof. Suppose that F 1 and F 2 are indistinguishable 1-good-neighbor conditional faulty sets. Since F 1 = F 2 , there is a node x1 ∈ F 1 F 2 . Without loss of generality, we assume that x1 ∈ F 1 − F 2 . Since F 1 and F 2 are indistinguishable and F 2 is a 1-good-neighbor conditional faulty set, there is a node x2 ∈ F 1 − F 2 with (x1 , x2 ) ∈ E ( Q n − S ). If N Q n − S (x1 ) ∪ N Q n − S (x2 ) ⊆ F 1 , then | F 1 | ≥ | N Q n − S (x1 )| + | N Q n − S (x2 )| ≥ δ( E ( Q n − S )) + 2. Since | F 1 | ≤ δ( E ( Q n − S )) + 1, thus, there is a node x3 ∈ F 2 − F 1 with x3 ∈ N Q n − S (x1 ) ∪ N Q n − S (x2 ), without loss of generality, we assume that x3 ∈ N Q n − S (x2 ). Similarly, there exists a node x4 ∈ F 2 − F 1 with (x3 , x4 ) ∈ E ( Q n − S ). See Fig. 3 for an illustration. By the definition of Q n , we have: (1) | N Q n − S (xi ) ∩ N Q n − S (xi +1 )| = 0 for 1 ≤ i ≤ 3; (2) | N Q n − S (x1 ) ∩ N Q n − S (x4 )| = 0; (3) | N Q n − S (x j ) ∩ N Q n − S (x j +2 )| ≤ 2 for 1 ≤ j ≤ 2. Let H = {x1 , x2 , x3 , x4 }. Then
| N Q n − S ( H )| =
4
Case 1. | S | = 0.
i =1
Since | A | = | F 1 ∩ F 2 | ≤ 2n − 3 and A is also a 1-goodneighbor conditional faulty set. By Lemma 7, we have Q n − A is connected. Moreover, V ( Q n ) − (| F 1 ∪ F 2 |) = 2n − ( F 1 ∪ F 2 ) ≥ 2n − 2(2(n − 1) + 1) = 2n − 4n + 2 > 0 if n ≥ 4. That is there is a node y ∈ Q n − ( F 1 ∪ F 2 ). By Lemma 5, F 1 and F 2 are distinguishable.
−
Case 2. | S | ≥ 1. Since | A | = |( F 1 ∩ F 2 ) ∪ S | ≤ 2n − 3 and A is also a 1-good-neighbor conditional faulty set. By Lemma 7, we have Q n − A is connected. Moreover, we have δ( E ( Q n − S )) ≤ 2(n − 1) − 1 = 2n − 3. Thus, | F 1 | ≤ 2n − 2 and | F 2 | ≤ 2n − 2. Then V ( Q n ) − (| F 1 ∪ F 2 | + | S |) = 2n − (| F 1 ∪ F 2 | + | S |) ≥ 2n − (| F 1 | + | F 2 | + | S |) ≥ 2n − (2(2n − 2)) + (n − 1)) = 2n − 5n + 5 > 0. That is there is a node y ∈ Q n − S − ( F 1 ∪ F 2 ). By Lemma 5, F 1 and F 2 are distinguishable. Based on above two cases, we have | F 1 ∩ F 2 | ≥ 2n − 2 − | S | if F 1 and F 2 are two indistinguishable faulty sets. 2
| N Q n − S (xi )|
2
| N Q n − S (x j ) ∩ N Q n − S (x j +2 )| − | H |
j =1
≥
4
| N Q n − S (xi )| − 8
i =1
= deg Q n − S ((x1 , x2 )) + deg Q n − S ((x3 , x4 )) − 4 ≥ 2δ( E ( Q n − S )) − 4. By Lemma 8, | F 1 ∩ F 2 | ≥ 2n − 2 − | S | ≥ n − 1 ≥ 3 if n ≥ 4. We have
| N Q n − S ( H )| − |( F 1 ∪ F 2 ) − H | ≥ 2δ( E ( Q n − S )) − 4 − | F 1 ∪ F 2 | + 4 = 2δ( E ( Q n − S )) − (| F 1 | + | F 2 | − | F 1 ∩ F 2 |) ≥ 2δ( E ( Q n − S )) − (2(δ( E ( Q n − S )) + 1) − 3) > 0.
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Fig. 4. The illustration that ( F 1 , F 2 ) is an indistinguishable 1-goodneighbor conditional pair of Q n − S.
That is, there is a node y ∈ Q n − S such that y ∈ / F1 ∪ F2. By Lemma 5, F 1 and F 2 are distinguishable. We obtain a contradiction since F 1 and F 2 are indistinguishable. Thus, t 1 ( Q n − S ) ≥ δ( E ( Q n − S )) + 1. 2 Lemma 10. For n ≥ 4, let S be a set of missing links and broken-down nodes in Q n with | S | ≤ n − 1, then t 1 ( Q n − S ) ≤ δ( E ( Q n − S )) + 1. Proof. We give an example to show that t 1 ( Q n − S ) is less than δ( E ( Q n − S )) + 1. Let u and v be two adjacent fault-free nodes of Q n − S with deg Q n − S ((u , v )) = δ( E ( Q n − S )). We set F 1 = N Q n − S ((u , v )) and F 2 = {u , v } ∪ F 1 . Then | F 1 | = | N Q n − S ((u , v ))| = | N Q n − S (u )| + | N Q n − S (u )| − 2 = deg Q n − S ((u , v )) = δ( E ( Q n − S )) and | F 2 | = | F 1 | + 2 = δ( E ( Q n − S )) + 2. Since {u , v } = F 1 F 2 and N Q n − S ((u , v )) = F 1 ⊂ F 2 , there is no edge between V ( Q n − S ) − ( F 1 ∪ F 2 ) and F 1 F 2 . By Theorem 1, F 1 and F 2 are indistinguishable under the PMC model. See Fig. 4 for an illustration. Now we verify that F 1 and F 2 are two 1-good-neighbor conditional faulty sets of Q n − S. Let X be the set V ( Q n − S ) − ( F 1 ∪ F 2 ). Since F 1 ⊂ F 2 , X = V ( Q n − S ) − F 2 . We consider the nodes in X. By the definition of X , for every x in X , there exists at least a fault-free neighbor in X . Then F 2 is a 1-good-neighbor conditional faulty set of Q n − S. Since v is the fault-free neighbor of u in {u , v } (u is the fault-free neighbor of v in {u , v }), then F 2 is a 1-good-neighbor conditional faulty set of Q n − S. Thus, Q n − S is not 1-good-neighbor conditionally (δ( E ( Q n − S )) + 2)-diagnosable under the PMC model since F 1 and F 2 are two 1-good-neighbor conditional faulty sets of Q n − S with | F 1 | ≤ δ( E ( Q n − S )) + 2 and | F 2 | ≤ δ( E ( Q n − S )) + 2. Then the upper bound of 1-good-neighbor conditional diagnosability of Q n − S is t 1 ( Q n − S ) ≤ δ( E ( Q n − S )) + 1. 2 Lemma 11. Let S be a set of missing links and broken-down nodes in Q 3 with | S | ≤ 2, Q 3 − S is not 1-good-neighbor conditionally (δ( E ( Q 3 − S )) + 1)-diagnosable under the PMC model. Proof. We will distinguish between the following three cases. Case 1. | S | = 0.
Fig. 5. An illustration for Lemma 11.
As shown in Fig. 5(a), F 1 and F 2 are two 1-goodneighbor conditional faulty sets of Q n with | F 1 | = 4 and | F 2 | = 4. Then Q 3 is not 1-good-neighbor conditionally 4-diagnosable, so t 1 ( Q 3 ) ≤ 3. It follows from Lemma 1 and Lemma 2 that t 1 ( Q 3 ) ≥ 3. Thus, t 1 ( Q 3 ) = 3 < 5 = δ( E ( Q 3 )) + 1. Case 2. | S | = 1. As shown in Fig. 5(b), F 1 and F 2 are two 1-goodneighbor conditional faulty sets of Q n − S with | F 1 | = 4 and | F 2 | = 3. Then Q 3 − S is not 1-good-neighbor conditionally 4-diagnosable, so t 1 ( Q 3 − S ) ≤ 3. It follows from Lemma 1 and Theorem 3 that t 1 ( Q 3 − S ) ≥ δ( Q 3 − S ) = 2. Thus, t 1 ( Q 3 − S ) = 4 = δ( E ( Q 3 − S )) + 1. Case 3. | S | = 2. We have following two subcases. As shown in Fig. 5(c), F 1 and F 2 are two 1-goodneighbor conditional faulty sets of Q n − S with | F 1 | = 3 and | F 2 | = 3. Then Q 3 − S is not 1-good-neighbor conditionally 3-diagnosable, so t 1 ( Q 3 − S ) ≤ 2. It follows from Lemma 1 and Theorem 3 that t 1 ( Q 3 − S ) ≥ δ( Q 3 − S ) = 2. Thus, t 1 ( Q 3 − S ) = 2 < 3 = δ( E ( Q 3 − S )) + 1. As shown in Fig. 5(d), F 1 and F 2 are two 1-goodneighbor conditional faulty sets of Q n − S with | F 1 | = 3 and | F 2 | = 3. Then Q 3 − S is not 1-good-neighbor conditionally 3-diagnosable, so t 1 ( Q 3 − S ) ≤ 2. It follows from Lemma 1 and Theorem 3 that t 1 ( Q 3 − S ) ≥ δ( Q 3 − S ) = 1. Thus, t 1 ( Q 3 − S ) = 3 = δ( E ( Q 3 − S )) + 1. Based on above discussion, we conclude that Q 3 − S is not 1-good-neighbor conditionally (δ( E ( Q 3 − S )) + 1)-diagnosable under the PMC model. 2 By Lemma 9 and Lemma 10, we have t 1 ( Q n − S ) = δ( E ( Q n − S )) + 1 for n ≥ 4. Furthermore, by Lemma 11,
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Q 3 − S is not 1-good-neighbor conditionally (δ( E ( Q 3 − S )) + 1)-diagnosable under the PMC model. Finally, the 1-good-neighbor conditional diagnosability of Q n − S under the PMC model is stated as follows. Theorem 5. Let S be a set of missing links and broken-down nodes in Q n with | S | ≤ n − 1. The 1-good-neighbor conditional diagnosability of Q n − S under the PMC model is δ( E ( Q n − S )) + 1 for n ≥ 4. 5. Conclusion In particular, some processors could be broken-down and some links could be missing in a system. A system with these processors and links is called an incomplete participant system. In this paper, we propose the concept of the diagnosability with missing links and broken-down nodes. To grant more accurate measurement of diagnosability for a large-scale processing system, we introduce the diagnosability and 1-good-neighbor conditional diagnosability of a system with missing links and broken-down nodes under the PMC model. It is an attractive work to develop more different measures of diagnosability with missing links and broken-down nodes based on application environment, network topology, network reliability, and statistics related to fault patterns. Acknowledgement This work was supported by the National Natural Science Foundation of China (No. 61602333, No. 61872257), Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX17_2005), Application Foundation Research of Suzhou of China (SYG201653). References [1] N.-W. Chang, S.-Y. Hsieh, Structural properties and conditional diagnosability of star graphs by using the PMC model, IEEE Trans. Parallel Distrib. Syst. 25 (2014) 3002–3011.
[2] S.-L. Peng, C.-K. Lin, J.J.M. Tan, L.-H. Hsu, The g-good-neighbor conditional diagnosability of hypercube under PMC model, Appl. Math. Comput. 218 (2012) 10406–10412. [3] A.T. Dahbura, G.M. Masson, An O (n2.5 ) fault identification algorithm for diagnosable systems, IEEE Trans. Comput. 33 (1984) 486–492. [4] S.L. Hakimi, A.T. Amin, Characterization of connection assignment of diagnosable systems, IEEE Trans. Comput. 23 (1974) 86–88. [5] S.-Y. Hsieh, T.-Y. Chuang, The strong diagnosability of regular networks and product networks under the PMC model, IEEE Trans. Parallel Distrib. Syst. 20 (2009) 367–378. [6] L.-H. Hsu, C.-K. Lin, Graph Theory and Interconnection Networks, CRC Press, 2008. [7] P.K. Jha, 1-Perfect codes over dual-cubes vis-à-vis hamming codes over hypercubes, IEEE Trans. Inf. Theory 61 (2015) 4259–4268. [8] A. Kavianpour, K.H. Kim, Diagnosability of hypercube under the pessimistic one-step diagnosis strategy, IEEE Trans. Comput. 40 (1991) 232–237. [9] T.-L. Kueng, C.-K. Lin, L.-H. Hsu, On the maximum number of faultfree mutually independent hamiltonian cycles in the faulty hypercube, J. Comb. Optim. 27 (2014) 328–344. [10] S. Lati, M. Hegde, M.N. Pour, Conditional connectivity measures for large multiprocessor systems, IEEE Trans. Comput. 43 (1994) 218–222. [11] L. Lin, S. Zhou, L. Xu, D. Wang, The extra connectivity and conditional diagnosability of alternating group networks, IEEE Trans. Parallel Distrib. Syst. 26 (2015) 2352–2362. [12] P.-L. Lai, J.J.M. Tan, C.-P. Chang, L.-H. Hsu, Conditional diagnosability measure for large multiprocessors systems, IEEE Trans. Comput. 54 (2005) 165–175. [13] L. Lin, L. Xu, D. Wang, S. Zhou, The g-good-neighbor conditional diagnosability of arrangement graphs, IEEE Trans. Dependable Secure Comput. 15 (2018) 542–548. [14] X. Xu, S. Zhou, J. Li, The reliability of complete cubic networks under the condition of g-good-neighbor, Comput. J. 26 (2016) 1165–1177. [15] F.P. Preparata, G. Metze, R.T. Chien, On the connection assignment problem of diagnosis systems, IEEE Trans. Electron. Comput. 16 (1967) 848–854. [16] Y.-H. Teng, C.-K. Lin, A test round controllable local diagnosis algorithm under the PMC diagnosis model, Appl. Math. Comput. 244 (2014) 613–623. [17] D. Wang, The diagnosability of hypercubes with arbitrarily missing links, J. Syst. Archit. 46 (2000) 519–527. [18] J. Yuan, A. Liu, X. Ma, X. Liu, The g-good-neighbor conditional diagnosability of k-ary n-cubes under the PMC model and MM model, IEEE Trans. Parallel Distrib. Syst. 26 (2015) 1165–1177. [19] S. Zhou, L. Lin, L. Xu, D. Wang, The t /k-diagnosability of star graph networks, IEEE Trans. Comput. 64 (2015) 547–555.
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
The diagnosability and 1-good-neighbor conditional diagnosability of hypercubes with missing links and broken-down nodes Xiaoyan Li a , Yuan-Hsiang Teng b , Tzu-Liang Kung c , Qi Chen a , Cheng-Kuan Lin d,∗ a
School of Computer Science and Technology, Soochow University, Suzhou 215006, China Department of Computer Science and Information Engineering, Providence University, Taichung 433, Taiwan c Department of Computer Science and Information Engineering, Asia University, Taichung 413, Taiwan d College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China b
a r t i c l e
i n f o
Article history: Received 23 February 2017 Received in revised form 7 January 2019 Accepted 11 January 2019 Available online 7 February 2019 Communicated by Łukasz Kowalik Keywords: Hypercube Diagnosability 1-Good-neighbor conditional diagnosability PMC model Fault tolerance
a b s t r a c t In system-level diagnosis, we propose to further classify faulty nodes into two categories. One category is the “ordinary” faulty nodes – they are malfunctioning, but they still participate in the diagnosis, rendering unreliable test results. The other category contains nodes that are completely broken down so that they cannot test other nodes, and they cannot be tested by other nodes either. In this paper, we study the diagnosability and 1-good-neighbor conditional diagnosability of hypercubes with both ordinary faulty nodes and broken-down nodes. Let S be a set of missing links and broken-down nodes in a hypercube Q n with | S | ≤ n − 1. We prove that the diagnosability of Q n − S is δ( Q n − S ) for n ≥ 3. Furthermore, we show that the 1-good-neighbor conditional diagnosability of Q n − S is δ( E ( Q n − S )) + 1 for n ≥ 4, which is the maximum number of faulty nodes can guarantee to identify, under the condition that every fault-free node has at least a fault-free neighbor. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Many multiprocessor systems take interconnection networks as underlying topology. The underlying topology of a system is usually modeled as a graph topology, where every node corresponds to a processor, and every edge corresponds to a communication link. Even slight malfunctions could make the service not work, and thus, the reliability of a system is the key issue. To ensure the reliability of the system, the devices found fault should be replaced with fault-free ones instantly. Therefore, it is important for the ability to identify any faulty devices, as known as system
*
Corresponding author. E-mail address: [email protected] (C.-K. Lin).
https://doi.org/10.1016/j.ipl.2019.01.007 0020-0190/© 2019 Elsevier B.V. All rights reserved.
diagnosis. The maximum number of faulty devices which can be identified correctly is diagnosability. If all the faulty devices in a system can be pointed out precisely with the faulty devices being t at most, the system is t-diagnosable. Many results about the diagnosis and the diagnosability have been proposed [11,17,19]. Since there is no restrictive condition on the distribution pattern of faulty processors, the classical diagnosability of a system is quite small. In order to increase the diagnosability, Lai et al. [12] proposed a more realistic parameter of diagnosability, conditional diagnosability, which limited that all the neighbors of any processor can not be faulty at the same time in a system. Recently, Peng et al. [2] proposed the notion of g-goodneighbor conditional diagnosability, which is the maximum number of faulty processors that can be identified under the condition that every fault-free processor has no
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less than g fault-free neighbors. The g-good-neighbor conditional diagnosability of various classes of systems have been studied in recent years [2,13,14,18]. In all the works to date on system-level diagnosability, a default assumption is that all faulty nodes are participating in the mutual test process. That is, all faulty nodes test other nodes, although they will render unreliable results, and all faulty nodes are tested by other nodes. The collective test results will be analyzed to determine which nodes are truly faulty. However, in reality this assumption may not be always true. It is likely that some nodes might be “totally disappearing” from the system. This can be caused by a complete, severe breakdown of the faulty node’s hardware, or by the depletion of battery in the case of wireless nodes. Preparata, Metze, and Chien have proposed the PMC diagnosis model in [15]. The PMC model is the tested-based diagnosis. Under the PMC model, a processor performs the diagnosis by testing the neighboring processors via the links between them. Hakimi and Amin have showed that, a system G is t-diagnosable if G is t-connected with at least 2t + 1 nodes under the PMC model [4]. Furthermore, they gave a necessary and sufficient condition for verifying whether or not a system is t-diagnosable under the PMC model. Some related studies have appeared in the literature [1,5,16]. In this paper, we propose to further classify faulty nodes into two categories. The first category is the “ordinary” faulty nodes – they are malfunctioning, but they still participate in the diagnosis, rendering unreliable test results. The second category contains nodes that are completely broken down so that they cannot test other nodes, and they cannot be tested by other nodes either. That is, these faulty nodes act like they have totally disappeared from the system. We call these nodes broken-down nodes. So the self-diagnosis systems contain three types of nodes – fault-free nodes, faulty nodes, broken-down nodes. In practice, the self-diagnosis systems still contain some missing links. The past literature in this field has seen studies of diagnosability for hypercubes with faulty (fault-free) nodes or regularly enhanced links. In this work, we will show that the diagnosability of a faulty hypercube Q n with these three types of nodes is δ( Q n − S ) for n ≥ 3, where S is the set of missing links and broken-down nodes, and δ( Q n − S ) is the minimum degree of Q n − S. Furthermore, we will show that the 1-good-neighbor conditional diagnosability of Q n − S is δ( E ( Q n − S )) + 1 for n ≥ 4, under the condition that every fault-free node has at least a fault-free neighbor. The remainder of this paper is organized as follows. Section 2 introduces the terminologies and notations used throughout the paper; Section 3 establishes the diagnosability of a faulty hypercube Q n − S; Section 4 establishes the 1-good-neighbor conditional diagnosability of a faulty hypercube Q n − S; Finally, we present our conclusion in Section 5. 2. Preliminaries The layout of processors and links in a multiprocessor system is usually represented by an undirected graph.
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For graph definitions and notations, we follow [6]. Let G = ( V (G ), E (G )) be a graph. The node set V (G ) is a finite set, and the edge set E (G ) is a subset of {(u , v )|(u , v ) is an unordered pair of V (G )}. Two nodes u and v are adjacent if (u , v ) ∈ E (G ). The neighborhood of a node u in G is the set of all nodes adjacent to u in G, denoted by N G (u ), N G (u ) = { v ∈ V (G ) | (u , v ) ∈ E (G )}. We define N G ( S ) = { v ∈ V (G ) − S | u ∈ S , (u , v ) ∈ E (G )} = ( N (u )) − S being the set of neighborhood of S in G. u∈ S
The degree of u in G is denoted by deg G (u ) = | N G (u )|. The graph G is k-regular if every node has degree k. We use δ(G ) = min{deg G (u ) : u ∈ V (G )}. A path P is a sequence of nodes in G of the form P = u 0 , u 1 , . . . , ul , such that u i and u i +1 are adjacent for i = 0, . . . , l − 1. A graph G is connected if for any two nodes, there is a path joining them, otherwise it is disconnected. The degree of edge (u , v ) in G is denoted by deg G ((u , v )) = | N G (u )| + | N G ( v )| − 2. We use δ( E (G )) = min{deg G ((u , v )) : (u , v ) ∈ E (G )}. For a set F of V (G ) ∪ E (G ), the notation G − F represents the graph obtained by removing the nodes in F from G and deleting those edges with at least one end node in F . If G − F is disconnected, then F is called a separating set or cut. The connectivity κ (G ) of a graph G is the minimum number of nodes whose removal results in a disconnected graph or only one node left. A graph G is k-connected if its connectivity is at least k. A graph H is a subgraph of G if V ( H ) ⊆ V (G ) and E ( H ) ⊆ E (G ). Under the PMC model, we assume that the adjacent processors can perform tests on each other. Let G = ( V (G ), E (G )) represent the underlying topology of a multiprocessor system. For any two adjacent nodes u and v in G, the ordered pair (u , v ) represents a test that u diagnoses v. In this situation, u is a tester, and v is a testee. If u evaluates v to be faulty (respectively, fault-free), the outcome of the test (u , v ) is 1 (respectively, 0). The faults considered here are permanent, hence the outcome is reliable if and only if the tester is fault-free. A faulty set F is the set of all faulty nodes in G. A test assignment for a system G is a collection of tests, which can be modeled as a directed graph T = ( V (G ), L ). The nodes u and v are adjacent in G, and the test (u , v ) ∈ L. The collection of all test results from the test assignment T is termed a syndrome. A syndrome of T is a mapping σ : L → {0, 1}. Let T = ( V (G ), L ) be a test assignment, and F be a subset of V (G ). For any given syndrome σ resulting from T , F is said to be consistent with σ if for a test (u , v ) ∈ L such that u ∈ V (G ) − F , then σ (u , v ) = 1 if and only if v ∈ F . Thus the fault-free testers always give correct test results, while the faulty testers give rise to unreliable results. Therefore, a faulty set F may be consistent with different syndromes. We use σ ( F ) to represent the set of all possible syndromes with which the faulty set F can be consistent. Let F 1 and F 2 be two distinct faulty sets of V (G ). Then F 1 and F 2 are distinguishable if σ ( F 1 ) ∩ σ ( F 2 ) = ∅; otherwise, F 1 and F 2 are indistinguishable. In other words, ( F 1 , F 2 ) is a distinguishable pair if σ ( F 1 ) ∩ σ ( F 2 ) = ∅; otherwise, ( F 1 , F 2 ) is an indistinguishable pair. Suppose that | F 1 | ≤ t and | F 2 | ≤ t in a system G. G is t-diagnosable if and only if ( F 1 , F 2 ) is a distinguishable pair. We use F 1 F 2 to denote the symmetric difference ( F 1 − F 2 ) ∪ ( F 2 − F 1 ) between F 1
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Fig. 1. An illustration for Theorem 1.
and F 2 . In [3], Dahbura and Masson proposed a sufficient and necessary characterization of t-diagnosable systems. A faulty set F is called a g-good-neighbor conditional faulty set if | N G − F ( v )| ≥ g for every node v in V (G ) − F . The graph G is g-good-neighbor conditional t-diagnosable if each distinct pair of g-good-neighbor conditional faulty sets F 1 and F 2 of V (G ) with | F 1 | ≤ t and | F 2 | ≤ t are distinguishable. The g-good-neighbor conditional diagnosability t g (G ) of a graph G is the maximum value of t such that G is g-good-neighbor conditionally t-diagnosable. Theorem 1. [3] Let G = ( V (G ), E (G )) be a system. For any two distinct sets F 1 , F 2 ⊂ V (G ), F 1 and F 2 are distinguishable if and only if there exists at least one test from V (G ) − ( F 1 ∪ F 2 ) to F 1 F 2 . See Fig. 1 for an illustration. Lemma 1. [2] Let G be a system. Then t g (G ) ≥ t (G ). The hypercube is used as fundamental models for computer networks, in which the properties have been studied for many years. The hypercube is still eye-catching in recent years [7,9,10]. An n-dimensional hypercube Q n is an undirected graph with 2n nodes and n2n−1 edges. Each node in Q n can be represented as an n-bit binary string. For any two distinct nodes u and v in Q n , u is adjacent to v if and only if their binary string representation differs in only one bit position. For any node u = u 0 u 1 . . . un−1 in Q n , we set (u )i = u 0i u 1i . . . uni −1 being the neighbor of u
in dimension i of Q n , where u ij = u j for every i = j and
u ii = 1 − u i . Similarly, ((u )i ) j is the neighbor of (u )i in dimension j. We use Q ni to denote an (n − 1)-dimensional subgraph of Q n , where V ( Q ni ) = {u | u = u 0 u 1 . . . un−1 ∈ V ( Q n ) and un−1 = i }. Obviously, Q ni is isomorphic to Q n−1 for each i. Fig. 2 shows example of Q 3 and Q 4 with missing links and three types of nodes – fault-free nodes, faulty nodes, broken-down nodes. Suppose that n ≥ 3, and F is a set of missing links such that δ( Q n − F ) ≥ 3. Wang [17] proved that Q n − F is δ( Q n − F )-diagnosable. Peng et al. [2] show that t g ( Q n ) = 2 g (n − g ) + 2 g − 1 for 0 ≤ g ≤ n − 3. In the following sections, we will prove the diagnosability and 1-good-neighbor conditional diagnosability of hypercubes with missing links and broken-down nodes. 3. The diagnosability of Q n with missing links and broken-down nodes In this section, we will determine the diagnosability of Q n − S under the PMC model. Lemma 2. [8] The hypercube Q n is n-diagnosable for n ≥ 3.
Fig. 2. A 3-cube and a 4-cube with missing links and three types of nodes.
Lemma 3. Let w, x and y be any three distinct nodes of Q n with {( w , x), (x, y )} ⊂ E ( Q n ). Then | N Q n ( w ) ∪ N Q n (x) ∪ N Q n ( y )| = 3n − 2. Proof. Since ( w , x) ∈ E ( Q n ) and (x, y ) ∈ E ( Q n ), we have | N Q n ( w ) ∩ N Q n (x)| = 0 and | N Q n (x) ∩ N Q n ( y )| = 0. Since x ∈ N Q n ( w ) ∩ N Q n ( y ), and Q n does not contain a cycle of length 3, we have d Q n ( w , y ) = 2. By the definition of Q n , | N Q n ( w ) ∩ N Q n ( y )| = 2. Then, | N Q n ( w ) ∪ N Q n (x) ∪ N Q n ( y )| = | N Q n ( w )| + | N Q n (x)| + | N Q n ( y )| − | N Q n ( w ) ∩ N Q n (x)| − | N Q n (x) ∩ N Q n ( y )| − | N Q n ( w ) ∩ N Q n ( y )| + | N Q n ( w ) ∩ N Q n (x) ∩ N Q n ( y )| = n + n + n − 0 − 0 − 2 + 0 = 3n − 2. 2 Lemma 4. Suppose that n ≥ 3. Let S be a set of missing links and broken-down nodes in Q n with | S | ≤ n − 1, and let F 1 and F 2 be any two distinct node subsets of Q n with | F 1 | ≤ δ( Q n − S ) and | F 2 | ≤ δ( Q n − S ). Then F 1 and F 2 are distinguishable if there exist two distinct nodes x and y in F 1 F 2 with (x, y ) ∈ E ( Q n ).
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Proof. By Lemma 2, F 1 and F 2 are distinguishable if | S | = 0. Thus, we consider that | S | ≥ 1. Since (x, y ) ∈ E ( Q n ), we have N Q n − S (x) ∩ N Q n − S ( y ) = ∅. Suppose that F 1 and F 2 are indistinguishable. By Theorem 1, for every node z ∈ F 1 F 2 , we have N Q n − S ( z) ⊆ F 1 ∪ F 2 . Consider the following cases. Case 1. (x, y ) ∈ S. Since (x, y ) ∈ E ( Q n ), N Q n (x) ∩ N Q n ( y ) = ∅. Since (x, y ) ∈ S, x ∈ / N Q n − S ( y ) and y ∈ / N Q n − S (x). Thus | F 1 | + | F 2 | ≥ | N Q n − S (x) ∪ N Q n − S ( y ) ∪ {x, y }| = | N Q n − S (x)| + | N Q n − S ( y )| + |{x, y }| ≥ δ( Q n − S ) + δ( Q n − S ) + 2. We obtain a contradiction since | F 1 | + | F 2 | ≤ 2δ( Q n − S ).
/ S. Case 2. (x, y ) ∈ Since (x, y ) ∈ E ( Q n ) − S and N Q n (x) ∩ N Q n ( y ) = ∅, we have |( N Q n − S (x) ∪ N Q n − S ( y )) − {x, y }| = deg Q n − S (x) + deg Q n − S ( y ) − 2 ≥ 2n − | S | − 2. Since | F 1 ∩ F 2 | ≤ δ( Q n − S ) − 1, |( N Q n − S (x) ∪ N Q n − S ( y )) − {x, y }| − | F 1 ∩ F 2 | ≥ 2n − | S | − 2 − (δ( Q n − S ) − 1) = (n − | S |) + (n − δ( Q n − S )) − 1 ≥ 1 + 1 − 1 > 0. Thus, there exists a node w in ( N Q n − S (x) ∪ N Q n − S ( y )) − ({x, y } ∪ ( F 1 ∩ F 2 )). By Lemma 3, | F 1 | + | F 2 | ≥ | N Q n − S (x) ∪ N Q n − S ( y ) ∪ N Q n − S ( w )| − | S | ≥ (3n − 2) − (n − 1) = 2n − 1 > 2(n − 1) ≥ 2δ( Q n − S ). We obtain a contradiction since | F 1 | + | F 2 | ≤ 2δ( Q n − S ). 2 Theorem 2. Let S be a set of missing links and broken-down nodes in Q n with | S | ≤ n − 1. Then Q n − S is δ( Q n − S )-diagnosable if n ≥ 3. Proof. Suppose that there exists an indistinguishable pair F 1 and F 2 in Q n − S with max{| F 1 |, | F 2 |} ≤ δ( Q n − S ). Since F 1 and F 2 are indistinguishable, we have N Q n − S ( z) ⊆ F 1 ∪ F 2 for every node z ∈ ( F 1 − F 2 ) ∪ ( F 2 − F 1 ). Without loss of generality, we assume that | F 1 | ≥ | F 2 |. Let x be a node in F 1 − F 2 . Since | F 1 | ≤ δ( Q n − S ), deg Q n − S (x) ≥ δ( Q n − S ), N Q n − S (x) ⊆ F 1 ∪ F 2 and x ∈ F 1 − F 2 , there exists a node y ∈ ( N Q n − S (x) ∩ F 2 ) − F 1 . By Lemma 4, F 1 and F 2 are distinguishable, which contradicts the assumption that F 1 and F 2 are indistinguishable. Hence, Q n − S is δ( Q n − S )-diagnosable, the theorem holds. 2 Theorem 3. Let S be a set of missing links and broken-down nodes in Q n with | S | ≤ n − 1. Then the diagnosability of Q n − S under the PMC model is δ( Q n − S ) if n ≥ 3. Proof. By Theorem 2, Q n − S is δ( Q n − S )-diagnosable. Thus, we only need to show that there exists an indistinguishable pair F 1 and F 2 of Q n − S with | F 1 | ≤ δ( Q n − S ) + 1 and | F 2 | ≤ δ( Q n − S ) + 1. Let u be a node in Q n − S such that deg Q n − S (u ) = δ( Q n − S ). Suppose that F 1 = N Q n − S (u ) ∪ {u } and F 2 = N Q n − S (u ). Then we have | F 1 | = δ( Q n − S ) + 1 and | F 2 | = δ( Q n − S ). By Theorem 1, F 1 and F 2 are indistinguishable. Therefore, the diagnosability of Q n − S under the PMC model is δ( Q n − S ), the theorem holds. 2
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4. The 1-good-neighbor conditional diagnosability of Q n with missing links and broken-down nodes In this section, we will determine the 1-good-neighbor conditional diagnosability of Q n − S under the PMC model. Lemma 5. Let F 1 and F 2 be any two distinct node sets of graph G where | V (G ) −( F 1 ∪ F 2 )| ≥ 1 and G −( F 1 ∩ F 2 ) is connected. Then F 1 and F 2 are distinguishable. Proof. Let y be a node of G − ( F 1 ∪ F 2 ). Since G − ( F 1 ∩ F 2 ) is connected, there is a path R of G − ( F 1 ∩ F 2 ) joining y to F 1 F 2 . That is, there is an edge ( p , q) ∈ E (G − ( F 1 ∩ F 2 )) where p ∈ V (G − ( F 1 ∩ F 2 )) and q ∈ F 1 F 2 . By Theorem 1, F 1 and F 2 are distinguishable. 2 Lemma 6. Let S be a set of missing links and broken-down nodes in Q n with | S | ≤ n − 1, and let F 1 and F 2 be any two distinct 1-good-neighbor conditional faulty sets of Q n − S. Then A = ( F 1 ∩ F 2 ) ∪ S is a 1-good-neighbor conditional faulty set. Proof. Let x be a node of Q n − A. We have following cases. Case 1. x ∈ Q n − S and x ∈ / F1. Since F 1 is a 1-good-neighbor conditional faulty set of Q n − S, thus, there is a node y ∈ V ( Q n − S ) − F 1 such that (x, y ) ∈ E ( Q n − S ). Case 2. x ∈ F 1 − F 2 . Since F 2 is a 1-good-neighbor conditional faulty set of Q n − S, thus, there is a node y ∈ V ( Q n − S ) − F 2 such that (x, y ) ∈ E ( Q n − S ). Based on above two cases, A is a 1-good-neighbor conditional faulty set. 2 Lemma 7. Let F be a set of V ( Q n ) ∪ E ( Q n ) with | F | ≤ 2n − 3 such that every node in V ( Q n ) − F has a neighbor in V ( Q n ) − F . Then Q n − F is connected for n ≥ 2. Proof. Since the connectivity of Q 2 is 2, Q 2 − F is connected. Suppose that this lemma holds on Q k for every 2 ≤ k ≤ n − 1. We set F 0 = ( V ( Q n0 ) ∪ E ( Q n0 )) ∩ F and F 1 = ( V ( Q n1 ) ∪ E ( Q n1 )) ∩ F . Since | F | ≤ 2n − 3, min{| F 0 |, | F 1 |} ≤ n − 2. Without loss of generality, we assume that | F 1 | ≤ n − 2. Thus, Q n1 − F 1 is connected because the connectivity of Q n1 is n − 1. Let x be a node in Q n0 − F 0 . If (x)n−1 ∈ V ( Q n1 ) does not in F 1 , x is connecting to Q n1 − F 1 . Suppose that (x)n−1 be a node in F 1 . Since every node in V ( Q n ) − F has a neighbor in V ( Q n ) − F and (x)n−1 ∈ F , there is a node y ∈ V ( Q n0 ) − F such that (x, y ) ∈ E ( Q n ) − F . Let {x1 , x2 , · · · , xn−2 } be the set of neighbor of x in V ( Q n0 ) − { y } and { y 1 , y 2 , · · · , yn−2 } be the set of neighbor of y in V ( Q n0 ) − {x}. Since Q n does not contain a cycle of length 3, {x1 , x2 , · · · , xn−2 } ∩ { y 1 , y 2 , · · · , yn−2 } = ∅. We set P i = x, xi , (xi )n−1 for every 1 ≤ i ≤ n − 2 and P j = x, y , y j −(n−2) , ( y j −(n−2) )n−1 for every n − 1 ≤ j ≤
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Fig. 3. An illustration for Lemma 9.
2n − 2. Since | F | ≤ 2n − 3, there is a path P t such that ( V ( P t ) ∪ E ( P t )) ∩ F = ∅. That is, x is connecting to a node in Q n1 − F 1 . Since every node in Q n0 − F 0 is connecting to a node in Q n1 − F 1 and Q n1 − F 1 is connected, Q n − F is connected. 2 Theorem 4. For n ≥ 3, suppose that F is a minimum cardinality cut of Q n such that every node in V ( Q n ) − F has a neighbor in V ( Q n ) − F . Then | F | = 2n − 2. Proof. We set (u , v ) ∈ E ( Q n ), let F be a set of V ( Q n ) ∪ E ( Q n ) in which each node adjacent to (u or v) and each edge end in (u or v). It is clear that F is a desired cut of Q n such that every node in V ( Q n ) − F has a neighbor in V ( Q n ) − F . Moreover, by Lemma 7, we have F is a minimum cardinality cut of Q n with | F | = 2n − 2 such that every node in V ( Q n ) − F has a node in V ( Q n ) − F . 2 Lemma 8. For n ≥ 4, let S be a set of missing links and brokendown nodes in Q n with | S | ≤ n − 1, and let F 1 and F 2 be any two distinct 1-good-neighbor conditional faulty sets of Q n − S with | F 1 | ≤ δ( E ( Q n − S )) + 1 and | F 2 | ≤ δ( E ( Q n − S )) + 1. If F 1 and F 2 are two indistinguishable faulty sets, then | F 1 ∩ F 2 | ≥ 2n − 2 − | S |. Proof. Suppose that | F 1 ∩ F 2 | ≤ 2n − 3 − | S |. By Lemma 6, we have A = ( F 1 ∩ F 2 ) ∪ S is a 1-good-neighbor conditional faulty set. We have following cases.
Lemma 9. For n ≥ 4, let S be a set of missing links and brokendown nodes in Q n with | S | ≤ n − 1, and let F 1 and F 2 be any two distinct 1-good-neighbor conditional faulty sets of Q n − S with | F 1 | ≤ δ( E ( Q n − S )) + 1 and | F 2 | ≤ δ( E ( Q n − S )) + 1. Then t 1 ( Q n − S ) ≥ δ( E ( Q n − S )) + 1. Proof. Suppose that F 1 and F 2 are indistinguishable 1-good-neighbor conditional faulty sets. Since F 1 = F 2 , there is a node x1 ∈ F 1 F 2 . Without loss of generality, we assume that x1 ∈ F 1 − F 2 . Since F 1 and F 2 are indistinguishable and F 2 is a 1-good-neighbor conditional faulty set, there is a node x2 ∈ F 1 − F 2 with (x1 , x2 ) ∈ E ( Q n − S ). If N Q n − S (x1 ) ∪ N Q n − S (x2 ) ⊆ F 1 , then | F 1 | ≥ | N Q n − S (x1 )| + | N Q n − S (x2 )| ≥ δ( E ( Q n − S )) + 2. Since | F 1 | ≤ δ( E ( Q n − S )) + 1, thus, there is a node x3 ∈ F 2 − F 1 with x3 ∈ N Q n − S (x1 ) ∪ N Q n − S (x2 ), without loss of generality, we assume that x3 ∈ N Q n − S (x2 ). Similarly, there exists a node x4 ∈ F 2 − F 1 with (x3 , x4 ) ∈ E ( Q n − S ). See Fig. 3 for an illustration. By the definition of Q n , we have: (1) | N Q n − S (xi ) ∩ N Q n − S (xi +1 )| = 0 for 1 ≤ i ≤ 3; (2) | N Q n − S (x1 ) ∩ N Q n − S (x4 )| = 0; (3) | N Q n − S (x j ) ∩ N Q n − S (x j +2 )| ≤ 2 for 1 ≤ j ≤ 2. Let H = {x1 , x2 , x3 , x4 }. Then
| N Q n − S ( H )| =
4
Case 1. | S | = 0.
i =1
Since | A | = | F 1 ∩ F 2 | ≤ 2n − 3 and A is also a 1-goodneighbor conditional faulty set. By Lemma 7, we have Q n − A is connected. Moreover, V ( Q n ) − (| F 1 ∪ F 2 |) = 2n − ( F 1 ∪ F 2 ) ≥ 2n − 2(2(n − 1) + 1) = 2n − 4n + 2 > 0 if n ≥ 4. That is there is a node y ∈ Q n − ( F 1 ∪ F 2 ). By Lemma 5, F 1 and F 2 are distinguishable.
−
Case 2. | S | ≥ 1. Since | A | = |( F 1 ∩ F 2 ) ∪ S | ≤ 2n − 3 and A is also a 1-good-neighbor conditional faulty set. By Lemma 7, we have Q n − A is connected. Moreover, we have δ( E ( Q n − S )) ≤ 2(n − 1) − 1 = 2n − 3. Thus, | F 1 | ≤ 2n − 2 and | F 2 | ≤ 2n − 2. Then V ( Q n ) − (| F 1 ∪ F 2 | + | S |) = 2n − (| F 1 ∪ F 2 | + | S |) ≥ 2n − (| F 1 | + | F 2 | + | S |) ≥ 2n − (2(2n − 2)) + (n − 1)) = 2n − 5n + 5 > 0. That is there is a node y ∈ Q n − S − ( F 1 ∪ F 2 ). By Lemma 5, F 1 and F 2 are distinguishable. Based on above two cases, we have | F 1 ∩ F 2 | ≥ 2n − 2 − | S | if F 1 and F 2 are two indistinguishable faulty sets. 2
| N Q n − S (xi )|
2
| N Q n − S (x j ) ∩ N Q n − S (x j +2 )| − | H |
j =1
≥
4
| N Q n − S (xi )| − 8
i =1
= deg Q n − S ((x1 , x2 )) + deg Q n − S ((x3 , x4 )) − 4 ≥ 2δ( E ( Q n − S )) − 4. By Lemma 8, | F 1 ∩ F 2 | ≥ 2n − 2 − | S | ≥ n − 1 ≥ 3 if n ≥ 4. We have
| N Q n − S ( H )| − |( F 1 ∪ F 2 ) − H | ≥ 2δ( E ( Q n − S )) − 4 − | F 1 ∪ F 2 | + 4 = 2δ( E ( Q n − S )) − (| F 1 | + | F 2 | − | F 1 ∩ F 2 |) ≥ 2δ( E ( Q n − S )) − (2(δ( E ( Q n − S )) + 1) − 3) > 0.
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Fig. 4. The illustration that ( F 1 , F 2 ) is an indistinguishable 1-goodneighbor conditional pair of Q n − S.
That is, there is a node y ∈ Q n − S such that y ∈ / F1 ∪ F2. By Lemma 5, F 1 and F 2 are distinguishable. We obtain a contradiction since F 1 and F 2 are indistinguishable. Thus, t 1 ( Q n − S ) ≥ δ( E ( Q n − S )) + 1. 2 Lemma 10. For n ≥ 4, let S be a set of missing links and broken-down nodes in Q n with | S | ≤ n − 1, then t 1 ( Q n − S ) ≤ δ( E ( Q n − S )) + 1. Proof. We give an example to show that t 1 ( Q n − S ) is less than δ( E ( Q n − S )) + 1. Let u and v be two adjacent fault-free nodes of Q n − S with deg Q n − S ((u , v )) = δ( E ( Q n − S )). We set F 1 = N Q n − S ((u , v )) and F 2 = {u , v } ∪ F 1 . Then | F 1 | = | N Q n − S ((u , v ))| = | N Q n − S (u )| + | N Q n − S (u )| − 2 = deg Q n − S ((u , v )) = δ( E ( Q n − S )) and | F 2 | = | F 1 | + 2 = δ( E ( Q n − S )) + 2. Since {u , v } = F 1 F 2 and N Q n − S ((u , v )) = F 1 ⊂ F 2 , there is no edge between V ( Q n − S ) − ( F 1 ∪ F 2 ) and F 1 F 2 . By Theorem 1, F 1 and F 2 are indistinguishable under the PMC model. See Fig. 4 for an illustration. Now we verify that F 1 and F 2 are two 1-good-neighbor conditional faulty sets of Q n − S. Let X be the set V ( Q n − S ) − ( F 1 ∪ F 2 ). Since F 1 ⊂ F 2 , X = V ( Q n − S ) − F 2 . We consider the nodes in X. By the definition of X , for every x in X , there exists at least a fault-free neighbor in X . Then F 2 is a 1-good-neighbor conditional faulty set of Q n − S. Since v is the fault-free neighbor of u in {u , v } (u is the fault-free neighbor of v in {u , v }), then F 2 is a 1-good-neighbor conditional faulty set of Q n − S. Thus, Q n − S is not 1-good-neighbor conditionally (δ( E ( Q n − S )) + 2)-diagnosable under the PMC model since F 1 and F 2 are two 1-good-neighbor conditional faulty sets of Q n − S with | F 1 | ≤ δ( E ( Q n − S )) + 2 and | F 2 | ≤ δ( E ( Q n − S )) + 2. Then the upper bound of 1-good-neighbor conditional diagnosability of Q n − S is t 1 ( Q n − S ) ≤ δ( E ( Q n − S )) + 1. 2 Lemma 11. Let S be a set of missing links and broken-down nodes in Q 3 with | S | ≤ 2, Q 3 − S is not 1-good-neighbor conditionally (δ( E ( Q 3 − S )) + 1)-diagnosable under the PMC model. Proof. We will distinguish between the following three cases. Case 1. | S | = 0.
Fig. 5. An illustration for Lemma 11.
As shown in Fig. 5(a), F 1 and F 2 are two 1-goodneighbor conditional faulty sets of Q n with | F 1 | = 4 and | F 2 | = 4. Then Q 3 is not 1-good-neighbor conditionally 4-diagnosable, so t 1 ( Q 3 ) ≤ 3. It follows from Lemma 1 and Lemma 2 that t 1 ( Q 3 ) ≥ 3. Thus, t 1 ( Q 3 ) = 3 < 5 = δ( E ( Q 3 )) + 1. Case 2. | S | = 1. As shown in Fig. 5(b), F 1 and F 2 are two 1-goodneighbor conditional faulty sets of Q n − S with | F 1 | = 4 and | F 2 | = 3. Then Q 3 − S is not 1-good-neighbor conditionally 4-diagnosable, so t 1 ( Q 3 − S ) ≤ 3. It follows from Lemma 1 and Theorem 3 that t 1 ( Q 3 − S ) ≥ δ( Q 3 − S ) = 2. Thus, t 1 ( Q 3 − S ) = 4 = δ( E ( Q 3 − S )) + 1. Case 3. | S | = 2. We have following two subcases. As shown in Fig. 5(c), F 1 and F 2 are two 1-goodneighbor conditional faulty sets of Q n − S with | F 1 | = 3 and | F 2 | = 3. Then Q 3 − S is not 1-good-neighbor conditionally 3-diagnosable, so t 1 ( Q 3 − S ) ≤ 2. It follows from Lemma 1 and Theorem 3 that t 1 ( Q 3 − S ) ≥ δ( Q 3 − S ) = 2. Thus, t 1 ( Q 3 − S ) = 2 < 3 = δ( E ( Q 3 − S )) + 1. As shown in Fig. 5(d), F 1 and F 2 are two 1-goodneighbor conditional faulty sets of Q n − S with | F 1 | = 3 and | F 2 | = 3. Then Q 3 − S is not 1-good-neighbor conditionally 3-diagnosable, so t 1 ( Q 3 − S ) ≤ 2. It follows from Lemma 1 and Theorem 3 that t 1 ( Q 3 − S ) ≥ δ( Q 3 − S ) = 1. Thus, t 1 ( Q 3 − S ) = 3 = δ( E ( Q 3 − S )) + 1. Based on above discussion, we conclude that Q 3 − S is not 1-good-neighbor conditionally (δ( E ( Q 3 − S )) + 1)-diagnosable under the PMC model. 2 By Lemma 9 and Lemma 10, we have t 1 ( Q n − S ) = δ( E ( Q n − S )) + 1 for n ≥ 4. Furthermore, by Lemma 11,
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Q 3 − S is not 1-good-neighbor conditionally (δ( E ( Q 3 − S )) + 1)-diagnosable under the PMC model. Finally, the 1-good-neighbor conditional diagnosability of Q n − S under the PMC model is stated as follows. Theorem 5. Let S be a set of missing links and broken-down nodes in Q n with | S | ≤ n − 1. The 1-good-neighbor conditional diagnosability of Q n − S under the PMC model is δ( E ( Q n − S )) + 1 for n ≥ 4. 5. Conclusion In particular, some processors could be broken-down and some links could be missing in a system. A system with these processors and links is called an incomplete participant system. In this paper, we propose the concept of the diagnosability with missing links and broken-down nodes. To grant more accurate measurement of diagnosability for a large-scale processing system, we introduce the diagnosability and 1-good-neighbor conditional diagnosability of a system with missing links and broken-down nodes under the PMC model. It is an attractive work to develop more different measures of diagnosability with missing links and broken-down nodes based on application environment, network topology, network reliability, and statistics related to fault patterns. Acknowledgement This work was supported by the National Natural Science Foundation of China (No. 61602333, No. 61872257), Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX17_2005), Application Foundation Research of Suzhou of China (SYG201653). References [1] N.-W. Chang, S.-Y. Hsieh, Structural properties and conditional diagnosability of star graphs by using the PMC model, IEEE Trans. Parallel Distrib. Syst. 25 (2014) 3002–3011.
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