Abstract
In the cloud computing environment with massive information services and decision-making resources, the accuracy and reliability of information are more important than previous single closed systems. Therefore, ensuring the reliability of information and the stable operation of the system are the core problems in the research fields such as the Internet Plus and the Internet of Things. The connectivity and diagnosability are two important measures for the fault tolerance of multiprocessor systems. The -good-neighbor conditional connectivity (-connectivity) is the minimum number of nodes that make the graph disconnected, and each node has at least neighbors in every remaining component. The -good-neighbor conditional diagnosability (-GNCD) is the maximum number of faulty processors that has been correctly identified in a system, and any fault-free processor has no less than fault-free neighbors. Exchanged -cubes are a class of irregular networks, obtained by deleting links from hypercubes and some variant networks of hypercubes (-cubes). They not only combine the advantages of -cubes but also reduce the interconnection complexity. Exchanged -cubes classify its nodes into two different classes clusters with a unique connecting rule. In this paper, we propose the generalized exchanged -cubes framework so that architecture can be constructed by different connecting rules. Furthermore, we study the -connectivity and -GNCD of generalized exchanged -cubes under the PMC and MM models. As applications, the -connectivity and -GNCD of generalized exchanged hypercubes, dual-cube-like networks, generalized exchanged crossed cubes, and locally generalized exchanged twisted cubes are determined, respectively.
1. Introduction
With the expansion of network scale and the improvement of complexity, the reliability and stability of the system become more and more important. How to ensure the correct and efficient operation of the system is an important research topic for wireless sensor networks and distributed systems. The distributed system disperses the computing tasks which are originally collected on one computer to polymorphic computer for parallel processing. It has many advantages, such as resource sharing, openness, concurrency, scalability, and fault tolerance. In the operation of the multiprocessor system, processor failure is inevitable. It may slow down the communication of information or even lead to paralyze of the system, thereby affecting the normal operation of the multiprocessor system and bringing huge losses. For example, on Google and Amazon systems, the failure of processors (servers) for several hours can bring millions of dollars of losses. Therefore, fault tolerance is very important for the construction and maintenance of systems [1, 2]. In fact, a multiprocessor system can be usually enlightened as a simple connected graph, where each processor represents a node of the graph, and each link between two processors represents an edge between two nodes in the graph. The graph is called the interconnection network of this multiprocessor system. Thus, some parameters of a graph as an interconnection network can be used to measure the reliability of a multiprocessor system. In the following, we do not distinguish among multiprocessor systems, interconnection networks, and graphs.
An important evaluating parameter for the fault tolerance of a system (modeled by graph ), the connectivity, is denoted by , which is the minimum number of nodes that make the graph disconnected. So far, connectivities for many famous networks have been proven. Nevertheless, there is a shortcoming for using traditional connectivity as a parameter of fault tolerance, which is considered a highly unlikely phenomenon in reality that all nodes adjacent to a node have failed simultaneously. Therefore, Esfahanian and Hakimi [3] proposed a new measure to overcome this shortcoming, the restricted connectivity, which limits that all adjacent nodes of any node cannot fail at the same time. Later, a generalized restricted connectivity concept, the -restricted connectivity (-connectivity), was proposed by Latifi et al. [4], which defines that each node of any remaining component after deleting all faulty nodes has degree at least . In recent years, they have attracted much interest of theoretical computer scientists and mathematicians. Xu et al. [5] determined the -connectivity of hierarchical cubic networks and complete cubic networks. Ning [6] studied the -connectivity of exchanged crossed cubes. Yuan et al. [7] explored the -connectivity of -ary -cube networks. Lin et al. [8] obtained the -connectivity of ()-arrangement graphs.
Identifying all faulty processors in a multiprocessor system (in brief, system) is called system-level diagnosis. A system is -diagnosable when all faulty processors can be detected, provided that the number of faulty processors in it does not exceed . The maximum number of faulty processors that the system can precisely point out is as known as the diagnosability of the system. In system-level diagnosis, there are several well-known models.
The PMC model is the first model, proposed by Preparate et al. [9], which is a test-based model, assumes that the adjacent processors can perform tests on each other. For any adjacent processors in a system, the ordered pair is called a test that diagnoses its neighbor , where is a tester and is a testee. In case diagnoses to be faulty (resp., fault-free), the outcome of the test is 1 (resp., 0). Moreover, the outcome is reliable in the present of the tester is fault-free. Another model, the MM model, was proposed by Maeng and Malek [10], which is a comparison-based model. In MM model, a comparator processor sends the same test to its two neighbors (i.e., comparison nodes) and then compares their responses. Let a labeled edge be a comparison performed that two processors and are compared by a processor , where is adjacent to and is also adjacent to . If the comparator processor is fault-free, and the responses of and are identical, then both comparison processors and are fault-free; on the other hand, if the responses of and are different, then at least one of is faulty. Furthermore, if both comparison processors and are faulty, the responses of and are distinct. In addition, the comparison is unreliable in the present if the comparator node is faulty. The MMmodel (proposed by Sengupta and Dahbura) [11] is a special MM model, which is assumed that each processor must compare each pair of its adjacent processors.
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 cannot be faulty at the same time in a system. Recently, Peng et al. [13] proposed the notion of -good-neighbor conditional diagnosability (-GNCD), which is the maximum number of faulty processors that can be identified under the condition that every fault-free processor has no less than fault-free neighbors. Peng et al. [13] (resp., Wang et al. [14]) established the -GNCD of hypercubes under the PMC model (resp., MM model). Li et al. [15] introduced the diagnosability and 1-good-neighbor conditional diagnosability of hypercubes with missing links and broken-down nodes under the PMC model. Yuan et al. [7] studied the -good-neighbor conditional diagnosabilities of -ary -cube networks under the PMC model and the MM model. Xu et al. [5] established the -good-neighbor conditional diagnosabilities of complete cubic networks under the PMC model and the MM model. Lin et al. [16] evaluated the -good-neighbor conditional diagnosabilities of -arrangement graphs under the PMC model and the MM model. Guo et al. [17] studied the -good-neighbor conditional diagnosability of the crossed cubes under the PMC model and the MM model. Li et al. [18] introduced this concept into a family of data center networks—DCell—and determined the -good-neighbor conditional diagnosabilities of DCell under the PMC model and the MM model.
The -connectivity (or -GNCD) of different networks are usually determined independently. It is a very worthwhile topic to explore a unified method to get them in different networks. A family of exchanged networks (i.e., exchanged -cubes) have some common properties, so that their -connectivity (or -GNCD) can be studied by a uniform method. The family of exchanged -cubes not only combine the advantages of hypercubes and some variant networks of hypercubes (-cubes) but also reduce the interconnection complexity. Exchanged -cubes classify its nodes into two different classes clusters with a unique connecting rule. In this paper, we propose the generalized exchanged -cubes framework so that architecture can be constructed by different connecting rules. There are some of the better properties in generalized exchanged -cubes, such as smaller diameter, fewer edges, lower cost factor, and low latency. Based on the fine properties, the network’s hardware and communication costs are reduced, and a greater balance between performance and cost can be achieved. Due to the excellent properties of the generalized exchanged -cubes, they can be used as the logical topologies in the peer-to-peer environment [19].
In recent years, the research on the relationship between the -connectivity and the -GNCD of regular networks under certain conditions has been widely developed [20–24], while this paper will study the -connectivity and the -GNCD of a class of irregular networks (i.e., generalized exchanged -cubes). We first establish the -connectivity of generalized exchanged -cubes. Next, we evaluate the -GNCD of generalized exchanged -cubes. As applications, we obtain the -connectivity and -GNCD of generalized exchanged hypercubes, dual-cube-like networks, generalized exchanged crossed cubes, and locally generalized exchanged twisted cubes.
The remainder of this paper is organized as follows. Section 2 provides the terms and notations used throughout the paper. Section 3 evaluates the -connectivity of generalized exchanged -cubes. Section 4 establishes the -GNCD of generalized exchanged -cubes. Section 5 gives some applications based on the results in Section 3 and Section 4. In Section 6, we illustrate the advantages of -connectivity and -GNCD compared to traditional connectivity and traditional diagnosability, respectively. Finally, we finish the whole paper by concluding in Section 7.
2. Preliminaries
2.1. Terminology and Notations
In this paper, a multiprocessor system is usually represented by a simple undirected graph (in brief, a graph). For terminology and notations not defined in this paper, we follow the reference [25]. We use to represent a graph, where representing a nonempty and finite node set and is an unordered pair of representing an edge set. Two nodes and are adjacent, denoted by . The set of neighbors of node in is denoted by . If , let denote the subgraph of induced by the node subset in . And we denote as . We set and . Two binary strings and are pair related, denoted by , if and only if . The case that and are not pair related is denoted by [26].
The degree of in is denoted by . Let , . is defined as a complete graph with nodes. A path is a sequence of distinct nodes with any two consecutive nodes in that are adjacent. We use to represent the graph is isomorphic to the graph . A component is defined as a maximally connected subgraph of a graph.
Definition 1 (see [27]). Let . is called a node-cut if is disconnected. If there exists a node-cut with , then is called a -node-cut. The connectivity of is defined as the minimum such that has a -node-cut.
Definition 2 (see [4]). Let be a positive integer and . If is disconnected and each remaining component has minimum degree at least , then is called an -cut.
Definition 3 (see [4]). The -connectivity of , denoted by , is the minimum cardinality over all -cuts of .
2.2. The -Good-Neighbor Conditional Diagnosability
Under the PMC model and MM model, we call the notation as the syndrome of the system, which is defined as the set of all test (comparison) results in a system , where test results are based on the PMC model and comparison results are based on the MM model. Define a faulty set , where , is a faulty processor. Let be the set of test (comparison) results which could be produced if is the faulty node set. We use and to represent two distinct faulty sets of . In case , we call these two distinct faulty sets and distinguishable, and a distinguishable pair; otherwise, and are indistinguishable, and is an indistinguishable pair. Let be the symmetric difference between and . In [28], under the PMC model, the sufficient and necessary condition for two different subsets and is a distinguishable pair proposed by Dahbura and Masson. Moreover, under the MM model, the sufficient and necessary condition for two different subsets and is a distinguishable pair proposed by Sengupta and Dahbura [11].
Lemma 4 (see [28]). Let be a multiprocessor system. For any two distinct sets , and are distinguishable under the PMC model if and only if there exists at least one test from to (see Figure 1(a)).
Lemma 5 (see [11]). Let be a multiprocessor system. For any two distinct sets , and are distinguishable under the MM model if and only if there is a node such that one of the following conditions holds (see Figure 1(b)): (1) and ,(2),(3).
(a)
(b)
The concept of -GNCD of a system was proposed in the literature [13].
Definition 6 (see [13]). (1)Let and be a fault-set. If any node of has at least neighbors in , then is called a -good-neighbor conditional fault-set.(2)A system is -good-neighbor conditional -diagnosable if each distinct pair of -good-neighbor conditional faulty (-GNCF) sets and of with and are distinguishable.(3)The -GNCD, denoted by , is defined as the maximum value of such that is -good-neighbor conditionally -diagnosable. Let and be the -GNCD of under the PMC model and MM model, respectively.
2.3. Generalized Exchanged -Cubes
In this subsection, we give the definition of the family of generalized exchanged networks, denoted by generalized exchanged-cubes, which have some common properties, so that the their -connectivity (or -GNCD) can be studied by a uniform method. Since generalized exchanged -cubes are derived by BC networks (bijective connection networks), we first review the definition of the BC network.
Definition 7 (see [29]). The one-dimensional BC network contains only two nodes which forms an edge. We use to represent the family of the one-dimensional BC network with . A graph belongs to the family of -dimensional BC networks if and only if there exists such that the following two conditions hold: is a perfect matching between and in
For any , by Definition 7, there exist satisfying the conditions. We use , to denote the induced subgraph , , respectively. Clearly, they are both -dimensional BC networks, and , , is a decomposition of . We define the decomposition as .
BC networks are a class of networks containing a number of famous networks such as hypercubes [13], the Möbius cubes [30], crossed cubes [31], and locally twisted cubes [32] as members. An -dimensional BC network is -regular and consisting of nodes. Figure 2 shows two three-dimensional BC networks.
Lemma 8 (see [33]). For and , if, then .
Lemma 9 (see [34]). (1)For and , if , then .(2)For , .
Lemma 10 (see [35]). For , there are at most two common neighbors between any two nodes of .
(a)
(b)
Next, we introduce the definition of generalized exchanged -cubes.
Definition 11. The -dimensional generalized exchanged -cubes is defined as a graph for and . consists of two disjoint subgraphs and . And consists of subgraphs, denoted by for . Similarly, consists of subgraphs, denoted by for . Moreover, satisfies the following conditions (see Figure 3): (a)For any integers and , and . Further, and (b)Each node in has a sole neighbor in and vice versa. In addition, for distinct nodes in each , their neighbors of lie in different (c)For any two different subgraphs and with , there exists no edge between them. Similar for and with .
(a)
(b)
By Definition 11, we can deduce that . Let each of and be a cluster of . Obviously, consists of clusters. If we contract each cluster as a node, then is contracted into a complete bipartite graph . The edges that connect different clusters are called cross edges. In the following discussion, we consider , and thus, , .
3. The -Connectivity of
In this section, we establish the -connectivity of with . In what follows, we exploit some useful lemmas for our further investigation.
Lemma 12. For any integers and , let be a subgraph of with , and let be a subgraph of such that . Then .
Proof. We conduct induction on .
If , by fixing , the lemma holds obviously. Suppose that the lemma holds for , let be a subgraph of with and be a subgraph of such that , then for and . In the following, we will prove that the lemma holds for . Since can be merged through a perfect matching by two , namely and , we discuss the two cases below.
Case 1. and .
Let and . Then and . Let and be two subgraphs of and with and , respectively. Thus, by the induction hypothesis, we have
Then, for , the lemma holds.
Case 2. or .
Without loss of generality, we suppose that . By Lemma 8, we have . Further, by the induction hypothesis, . Then
Hence, the lemma holds.
Lemma 13. For any integers and , .
Proof. By Definition 11, can be decomposed into two disjoint subgraphs and , where can be partitioned into subgraphs (clusters) and can be partitioned into subgraphs (clusters). Without loss of generality, let such that . Clearly, . By Definition 7 and Lemma 9, . Further, by Definition 11, each node in has a sole neighbor in . And, for distinct nodes in each , their neighbors of lie in different . In addition, for any two different subgraphs and with , the edge between them is nonexistent. Thus, each node in has exactly one neighbor in . Then . Thus, we have
Since and , is disconnected. Then is a node-cut of .
In what follows, as an -cut of will be proved. That is, .
Since , . By Lemma 9, is an -cut of , where . As a result, with . Moreover, by Definition 11, each node in has exactly one neighbor in , and for distinct nodes in , their neighbors in lie in different . Since , for any node .
Summary of the above discussion, we have . Then is a -good-neighbor cut of . Hence, with and , the lemma holds.
Lemma 16. For any integers and , .
Proof. We assume as a minimum -cut of . Let and , where . Then we will show that with and . We consider three cases as follows.
Case 1. and are connected for each , where .
We prove this case by contradiction. Suppose that . In the following, we will prove that is not an -cut of .
Since is a minimum -cut of , is disconnected. In addition, there must exist a component with traverses clusters, where . Let , where be one of these clusters with . As a result, . By Definition 11, for any node in , it has a sole neighbor in . And, for distinct nodes in each , their neighbors of lie in different . In addition, for any two different subgraphs and with , there exists no edge between them. Then there exist at most cross edges between and , where . Moreover, there are at least cross edges between and with . Clearly, and . Since there is no edge between and , . Then, we have
Let with . We obtain . Thus, is an increasing function. Therefore, and . In addition, let with . We obtain that . Thus, is a decreasing function. Therefore, . Then , which results in a contradiction with .
Case 2. Only one of and is disconnected, where .
Without loss generality, assume that is disconnected. Since is an -cut of , . Then is a -good-neighbor cut of . By Lemma 9, . By contradiction, suppose that with and . Let and . Then .
Assume that is disconnected. Then there must exist a component such that traverses clusters, where . Let be one of these clusters for and . As a result, . By Definition 11, for any node in , it has a sole neighbor in . And for distinct nodes in each , their neighbors of lie in different . In addition, for any two different subgraphs and with , the edge between them is nonexistent. Then there exist at most cross edges between and , where . Moreover, there are at least cross edges between and with . Clearly, and . Since there is no edge between and , . Figure 4 shows an illustration for this case. Then, we have
Let with . We obtain that . Thus, is an increasing function. Therefore, . And . In addition, let with and . We get that . Thus, is a decreasing function. Therefore, . Then , which results in a contradiction with .
Thus, is connected. Since is disconnected, there must exists a component in such that there is no edge between and . Then . Since , . Moreover, since is an -cut of , . By Lemma 8 and , we have , which results in a contradiction with .
Case 3. For any integers , , there are at least two of and that are disconnected.
Without loss of generality, suppose that and are disconnected. Since is an -cut of and by Definition 11, we have and . Then and are two -good-neighbor cuts of . By Lemma 9, and . Then
Thus, .
Hence, the lemma holds.
Combining Lemma 13 and Lemma 16, the following theorem holds.
Theorem 14. For any integersand, .
4. The -Good-Neighbor Conditional Diagnosability of
In this section, we will determine the -GNCD of under the PMC model and MM model, respectively, where .
Theorem 15. For any integers and , .
Proof. First, we show that with and . Let with such that . Clearly, . Suppose that and . By Lemma 13, we have , and , where . Since and , and are two -GNCF sets of with and . On the other hand, since and , there is no edge between and . By Lemma 4, and are indistinguishable under the PMC model. By Definition 6 (2), is not -good-neighbor conditional -diagnosable under the PMC model. That is, for .
Next, we prove that with and . We suppose, to the contrary, that for . And assume that there are two indistinguishable -GNCF sets and with and . In what follows, we consider two cases.
Case 1. .
Since , by Definition 11, we have
Since , we have
Let with and . We obtain that . Thus, is a decreasing function. Therefore, for and ,
which induces a contradiction since .
Case 2. .
Since , we may assume that . There exists no edge between and because and are indistinguishable. Moreover, since is a -good-neighbor conditional faulty set, it is easy to verify that . By Lemma 8, . On the other hand, since both and are -GNCF sets, is also a -good-neighbor conditional faulty set. Moreover, there is no edge between and ; thus, is disconnected. Then is an -cut of . By Theorem 14, with and . Hence,
which results in a contradiction since .
To sum up, we can conclude that for any integers and .
Hence, the theorem holds.
Theorem 24. For any integers and , .
Proof. The proof of with and is similar to Theorem 15, so it is omitted.
Next, we prove that with and . We suppose, to the contrary, that with and . Moreover, we assume that there are two indistinguishable -GNCF sets and , where and .
Since , by Definition 11, we have
Furthermore, it is easy to get that . Then, we have
Let with and . We obtain that . Thus, is a decreasing function. Therefore, for and ,
which results in a contradiction since .
Thus, . In addition, an important claim is given as follows.
Claim 25. has no isolated node.
By contradiction, suppose that has at least one isolated node. Then, we prove that the two cases both contradict the supposition.
Case 1. .
Since , without loss of generality, we suppose that . When , since is a -GNCF set, has no isolated node. Now, we consider . The given is the set of all isolated nodes and . Since is a -GNCF set, for any .
Since and are indistinguishable, there exists at most one node with is adjacent to by Lemma 5. Thereby, there exists only one node with adjacent to . It is easy to see that there is only one node with adjacent to . Since , there are at most neighbors of in with any isolated node . Since and , . Hence,
It follows that . Thus,
Let . We can deduce that . Then is an increasing function. Therefore, , a contradiction. Thus, .
Since the fault-pair does not satisfy Lemma 5 and any node in is not isolated, there exists no edge between and . Moreover, is also a -GNCF set. Thus, is an -cut of . By Theorem 14, . Since , , and , . Let and . Then . Hence, we have and for any isolated node . By Lemma 10, there are at most two common neighbors between any two nodes in . In addition, by Definition 11, any two cross edges have no common end node. Then we deduce that any two nodes in have at most two common neighbors. Thus, . Since there is no common node between any two cross edges and is triangle-free, is triangle-free. Thereby,
Therefor, for , we have
which results in a contradiction since .
Case 2. .
Without loss of generality, we suppose that . Since is a -GNCF set of , with any node . For any , there exists at most one neighbor in because the fault-pair is indistinguishable by Lemma 5. Therefore, . Since is arbitrary, every node of is not an isolated one.
To sum up, Claim 25 holds.
Since there exists no isolated node in by Claim 25 we have, for any , there exists some node such that . If for any , satisfies condition in Lemma 5. Therefore, the -GNCF sets and are distinguishable, which results in a contradiction. By the arbitrariness of , there exists no edge between and .
Since is a -GNCF set and , . Thus, . Since and are both -GNCF sets and there exists no edge between and , is also an -cut of . By Theorem 14, . Then , which contradicts with .
Thus, for any integers and .
Hence, the proof of theorem is completed.
5. Applications to a Family of Famous Networks
In Section 2, the definition of the generalized exchanged -cube has been given. Furthermore, we determine the -connectivity and -GNCD of in Section 3 and Section 4, respectively. Applying the theorems of Section 3 and Section 4, we can directly establish the -connectivity and -GNCD of some generalized exchanged -cubes, including generalized exchanged hypercubes, dual-cube-like networks, generalized exchanged crossed cubes, and locally generalized exchanged twisted cubes. In this section, we will give the applications to these networks.
5.1. The Generalized Exchanged Hypercube
In 2005, Loh et al. [36] proposed the exchanged hypercube, which obtained by removing edges from a hypercube . We denote , where is a given position integer. For each , the sequence is a binary string of length if . The definition of exchanged hypercubes is presented as follows.
Definition 16 (see [36]). Let , the exchanged hypercube consists of the node set and the edge set , two nodes and are linked by an edge, called -dimensional edge, if and only if the following conditions are satisfied: (i) and differ exactly in one bit on the -th bit or on the last bit(ii)if , then (iii), then
The generalized exchange hypercube was proposed by Cheng et al. [37]. Let , the generalized exchanged hypercube consists of two classes of hypercubes: one class contains ’, referred to as the Class-0 clusters; and the other contains ’, referred to as the Class-1 clusters. Class-0 and Class-1 clusters will be referred to as clusters of opposite class of each other, same class otherwise. The function is a bijection between nodes of Class-0 clusters and those of Class-1 clusters; for two nodes in the same cluster, and are in two different clusters, and the edge is a cross edge. The bijection ensures the existence of a perfect matching between nodes of Class-0 clusters and those in the Class-1 clusters but ignores the specifics of the perfect matching. Hence, we present the following proposition.
Proposition 17. can be decomposed into two subgraphs and . Further, can be partitioned into subgraphs, denoted by for . Similarly, can be partitioned into subgraphs, denoted by for . And satisfies the following conditions (Figure 5 shows the and ): (a)For any , and . Further, and (b)Each node in has a sole neighbor in and vice versa. In addition, for distinct nodes in each , their neighbors of lie in different (c)For any two different subgraphs and with , there exists no edge between them. Similar for and with .
(a)
(b)
The dual-cube is a special case of the exchanged hypercube when , proposed by Li and Peng [38]. That is, . The dual-cube-like network [39], which is a generalization of dual-cubes, is isomorphic to , a special case of (see in Figure 6).
By Proposition 17, the generalized exchanged hypercube is the member of generalized exchanged -cubes, where the -cube is a hypercube. Then, the following theorems hold obviously.
Theorem 18. (1)For any integers and , (2)For any integers and , .
Theorem 19. (1)For any integers and , (2)For any integers and , .
Theorem 20. (1)For any integers and , (2)For any integers and , .
5.2. The Generalized Exchanged Crossed Cube
Li et al. [26] give the definition of , which is obtained by removing edges from a crossed cube . In what follows, we review the definition of exchanged crossed cubes.
Definition 21 (see [26]). The -dimensional exchanged crossed cube is defined as a graph for , The node set , where and . The edge set consisting of three types of disjoint sets , , and is shown as follows.
, where is the exclusive-OR operator.
, is denoted byandis denoted by. Andandare adjacent by the following rule: for any integer, if and only if there is anwith= ; ifis even;, where
, is denoted byandis denoted by. Andandare adjacent by the following rule: for any integer, if and only if there is anwith= ifis even;, where.
is the bit pattern offrom dimensionto dimension.
Let , the generalized crossed cube comprises two classes of crossed cubes, referred to as the Class-0 clusters and the Class-1 clusters, respectively. The Class-0 clusters contain ’ and the Class-1 clusters contain ’. They will be referred to as clusters of opposite class of each other, same class otherwise. The function is a bijection between nodes of Class-0 clusters and those of Class-1 clusters such that, for , two nodes of the same cluster, and , are in two different clusters, and the edge is a cross edge. The bijection ensures the existence of a perfect matching between two nodes in different clusters, but there is no requirement for the specifics of the perfect matching. Therefore, we have the following proposition.
Proposition 22. can be decomposed into two disjoint subgraphs and . And and are the subgraphs induced by and , respectively, where with and . with and .
By Definition 21, can be partitioned into subgraphs, denoted by such that for , , , where . Similarly, can be partitioned into subgraphs, denoted by such that , , , for . And satisfies the following conditions (see in Figure 7): (1)For any , and . Further, and (2)Each node in has a sole neighbor in and vice versa. In addition, for distinct nodes in each , their neighbors of lie in different (3)For any two different subgraphs and with , there exists no edge between them. Similar for and with .
By Proposition 22, the exchanged crossed cube is an exchanged -cube, where the -cube is a crossed cube. Then, the following theorems hold obviously.
Theorem 23. For any integers and , .
Theorem 24. (1)For any integers and , (2)For any integers and , .
5.3. The Locally Generalized Exchanged Twisted Cube
The locally exchanged twisted cube proposed by Chang et al. [29], obtained by removing edges from a locally twisted cube . The definition of locally exchanged twisted cube is introduced as follows.
Definition 25 (see [29]). The -dimensional locally exchanged twisted cube is defined as a graph for , The node set . is the edge set consisting of the following three types of disjoint sets , , and .
Let ; there are two classes of locally twisted cubes in the locally generalized exchanged twisted cube : one class, referred to as the Class-0 clusters, contains ’; and the other, referred to as the Class-1 clusters, contains ’. They will be referred to as clusters of opposite class of each other, same class otherwise. There exists a bijection function between nodes of Class-0 clusters and those of Class-1 clusters. For two nodes in the same cluster, and belong to two different ones, and the edge is a cross edge. The bijection ensures the existence of a perfect matching between nodes of Class-0 clusters and those in the Class-1 clusters, but the specifics of the perfect matching can be ignored. Further, we obtain the proposition as follows.
Proposition 26. can be decomposed into two disjoint subgraphs and . can be partitioned into subgraphs, denoted by for . Similarly, can be partitioned into subgraphs, denoted by for . And satisfies the following conditions (see in Figure 8): (a)For any , and . Further, and (b)Each node in has a sole neighbor in and vice versa. In addition, for distinct nodes in each , their neighbors of lie in different (c)For any two different subgraphs and with , there exits no edge connects them. Similar for and with .
By Proposition 26, the locally exchanged twisted cube is a member of generalized exchanged -cubes, where the -cube is a locally twisted cube. Then, we have the following theorems.
Theorem 27. (1)For any integers and , (2)For any integers and , (3)For any integers and , .
6. Compare Results
In this section, we will illustrate the advantages of -connectivity and -GNCD compared to traditional connectivity and traditional diagnosability, respectively. Let us review their definition. The connectivity, which is less than the minimum degree of graph, is the minimum number of nodes that make the graph disconnected. The maximum number of faulty processors that the system can precisely point out is as known as the diagnosability of the system, which is equal to the minimum degree of graph in most cases. The -good-neighbor conditional connectivity (-connectivity) is the minimum number of nodes that make the graph disconnected, and each node has at least neighbors in every remaining component. The -good-neighbor conditional diagnosability (-GNCD) is the maximum number of faulty processors that can be identified under the condition that every fault-free processor has no less than fault-free neighbors. We have determined that the -connectivity of is and the -GNCD of is . Figure 9 shows that -connectivity and -GNCD are both about times the minimum degree of graph. Therefore, we can speculate that -connectivity is about times traditional connectivity and -GNCD is about times traditional diagnosability, which means that -connectivity and -GNCD can better evaluate the fault tolerance of network.
(a)
(b)
7. Conclusion
The -connectivity and -GNCD are two significant metrics for reliability of multiprocessor systems. Exchanged -cubes are a class of irregular networks, obtained by deleting links from hypercubes and some variant networks of hypercubes (-cubes). They not only combine the advantages of -cubes but also reduce the interconnection complexity. Exchanged -cubes classify its nodes into two different classes clusters with a unique connecting rule. In this paper, we propose the generalized exchanged -cubes framework so that architecture can be constructed by different connecting rules. We first give the definition of a family of generalized exchanged -cubes, including generalized exchanged hypercubes, dual-cube-like networks, generalized exchanged crossed cubes, and locally exchanged twisted cubes as members. Then we determine the -connectivity and -GNCD of generalized exchanged -cubes. Finally, the -connectivity and -GNCD of generalized exchanged hypercubes, dual-cube-like networks, generalized exchanged crossed cubes, and locally exchanged twisted cubes are established directly. As a future research, we attempt to evaluate the -connectivity and -GNCD of other generalized exchanged -cubes using methods extended from the proposed method in this paper.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors have declared that no conflict of interest exists.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. 61872257 and No. 61977016) and the Joint Found of the National Natural Science Foundation of China (No. U1905211).