k-diagnosability of the data center network DCell

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The extra connectivity, extra conditional diagnosability and t /k-diagnosability of the data center network DCell Xiaoyan Li a , Jianxi Fan a,b,∗ , Cheng-Kuan Lin a , Baolei Cheng a , Xiaohua Jia c a b c

School of Computer Science and Technology, Soochow University, Suzhou 215006, China Jiangsu High Technology Research Key Laboratory for Wireless Sensor Networks, Jiangsu Province, Nanjing 210003, China Department of Computer Science, City University of Hong Kong, Hong Kong

a r t i c l e

i n f o

Article history: Received 20 May 2017 Received in revised form 7 August 2018 Accepted 20 September 2018 Available online xxxx Communicated by S.-Y. Hsieh Keywords: Data center networks DCell PMC model Fault tolerance g-extra connectivity g-extra conditional diagnosability t /k-diagnosability

a b s t r a c t Connectivity and diagnosability are two important metrics in evaluating the fault tolerability of a network. The g-extra connectivity and the g-extra conditional diagnosability are both defined under the restraint that every component of the network removing a faulty vertex set has at least g + 1 fault-free vertices. The t /k-diagnosability is an outstanding diagnosis strategy, in which the identified faulty vertex set is allowed to contain at most k fault-free vertices. As a well-known model for a large-scale data center network (DCN) with a server-centric structure, the m-dimensional DCell with n-port switches and tm,n servers, D m,n , has many desirable properties. In this paper, we first investigate the g-extra connectivity of D m,n for 0 ≤ g ≤ n − 1. Based on this, we establish the g-extra conditional diagnosability of D m,n under the PMC model for 0 ≤ g ≤ n − 1. Finally, we evaluate the t /k-diagnosability of D m,n under the PMC model for 1 ≤ k ≤ n − 1. © 2018 Elsevier B.V. All rights reserved.

1. Introduction As the key infrastructure of cloud computing, data center networks (DCNs) have been built to provide increasingly popular online application services, such as GFS [30], Bigtable [9], and Dryad [22]. This leads to the proposal of data center networks (DCNs) supporting millions of servers with high network capacity by using only commodity switches. To accelerate the process of network innovation, many server-centric DCNs have been proposed and studied. A novel server-centric DCN, namely the DCell network, was proposed by Guo et al. [4], which can provide good network performance by using inexpensive commodity switches and a large number of servers. The DCell network has many desirable properties such as good expandability, high network capacity, small diameter, and large bisection width. Fault tolerance is the ability such that a network operates properly in the event of the failure of some of its components. The notion of connectivity is the key graph-theoretic concept for fault tolerance. The higher the connectivity, the more reliable the network. However, it is limited that all vertices incident to the same vertex can potentially fail at the same time. To compensate for this shortcoming, Harary [11] proposed the concept of conditional connectivity, which is used to better measure the reliability of multiprocessor systems. A famous conditional connectivity, g-extra connectivity, was proposed by Fàbrega and Fiol [14], which is the minimum number of vertices whose removal will disconnect a network, and every remaining component has at least g + 1 vertices. Zhu et al. [25] established the g-extra connectivity of the

*

Corresponding author at: School of Computer Science and Technology, Soochow University, Suzhou 215006, China. E-mail address: [email protected] (J. Fan).

https://doi.org/10.1016/j.tcs.2018.09.014 0304-3975/© 2018 Elsevier B.V. All rights reserved.

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n-dimensional BC network for 0 ≤ g ≤ n − 4. Hsieh and Chang [28] determined the 2-extra connectivity of the k-ary n-cube network. Chang and Hsieh [42] evaluated the {2, 3}-extra connectivities of hypercube-like networks. Lin et al. [20] proved the g-extra connectivity of the alternating group network for 1 ≤ g ≤ 3. Lin et al. [21] also showed the g-extra connectivity of the split-star network for 1 ≤ g ≤ 3. In fact, it is difficult to determine the g-extra connectivity of a given network. With the number of servers in a DCN increases, server failures will inevitably occur. Server fault identification (diagnosis) becomes an important issue for DCNs reliable communication and computing. The server diagnosis of DCNs, in fact, may be enlightened by the diagnosis of multiprocessor systems, where each server represents a processor of multiprocessor systems, and each link between servers represents a link between processors (switches can be regarded as transparent network devices). Therefore, the server diagnosis of a DCN can be equivalent to the processor diagnosis of a multiprocessor system. Identifying all faulty processors in a multiprocessor system is as known as system-level diagnosis. In 1967, Preparata, Metze, and Chien [10] first established a foundation of system diagnosis and an original diagnostic model, namely the PMC model, which is a test-based diagnosis model. Under the PMC model, a processor performs the diagnosis by testing the neighboring processors via the links between them, and a test result is reliable (respectively, unreliable) if the testing processor is fault-free (respectively, faulty). A system is t-diagnosable if all the faulty processors can be pointed out given that the number of faulty processors is at most t. The maximum number of faulty processors that the system can guarantee to identify is called the diagnosability of the system. Wang [6] studied the diagnosability of the enhanced hypercube under the PMC model. Fan et al. [13,15] investigated the diagnosabilities of the Möbius cube and the DCC linear congruential graph under the PMC model. The diagnosability of a system is quite small owing to the fact that it is limited as the minimum degree in interconnection network of multiprocessor system. To increase the degree of diagnosability, several different diagnostic strategies were proposed. The g-extra conditional diagnosability and the t /k-diagnosability are two important diagnostic strategies for system-level diagnosis, which can conspicuously improve the capability of a system’s self-diagnosing. The g-extra conditional diagnosability, was proposed by Zhang and Yang [31]. It is introduced under the assumption that every component of the system removing a set of faulty vertices has at least g + 1 vertices. Zhang and Yang [31] obtained the g-extra conditional diagnosability of the n-dimensional hypercube under the PMC model for 0 ≤ g ≤ n − 4. Wang et al. [26] studied the 2-extra conditional diagnosability of bubble-sort star graph networks under the PMC model. Han et al. [32] investigated the g-extra conditional diagnosability of the n-dimensional folded hypercube under the PMC model for 0 ≤ g ≤ n − 4. Wang et al. [27] showed the 2-extra conditional diagnosability of alternating group graph networks under the PMC model. The t /k-diagnosis strategy was proposed by Somani and Peleg [3], under this strategy, all the faulty processors can be isolated into the faulty set, but the faulty set may contain at most k fault-free processors. Definition 1. [3] A multiprocessor system G = ( V (G ), E (G )) is t /k-diagnosable if, given any test syndrome produced by the system under the presence of a faulty set F , all the faulty processors can be isolated to within a set of processors F  , out of which at most k processors can possibly be fault-free, and | F  | ≤ | F | + k. The t /k-diagnosability of hypercubes, star graphs, and mesh-based systems were analyzed by Somani and Peleg [3]. Fan and Lin [16] obtained the t /k-diagnosability of the n-dimensional BC network for 0 ≤ k ≤ n, and Yang et al. [33] extended the result on the t /k-diagnosability of the n-dimensional BC network for n + 1 ≤ k ≤ 2n − 1. Yang and Tang [39] suggested a (4n − 9)/3 diagnosis algorithm on the n-dimensional hypercube. Zhou et al. [29] investigated the t /k-diagnosability of the star graph network for 1 ≤ k ≤ 3. Recently, Lin et al. [18] studied the t /k-diagnosability of regular networks under some conditions. Some basic properties of the m-dimensional DCell built with n-port switches, D m,n , have been studied, such as connectivity [4,23], diameter [4,24], symmetry [24], broadcasting [4], Hamiltonian properties [34], the restricted h-connectivity [35], vertex-disjoint paths [37], disjoint path cover algorithm [36], diagnosability and g-good-neighbor diagnosability [38]. In this paper, we establish 1) the g-extra connectivity; 2) the g-extra conditional diagnosability; and 3) the t /k-diagnosability for the m-dimensional DCell with n-port switches, denoted by D m,n , under the PMC model. It is motivated by the following three aspects:

• The recent studies on the extra connectivity, extra conditional diagnosability, and t /k-diagnosability of some networks, including hypercube-like networks, folded hypercubes, k-ary n-cube networks, alternating group networks, and star graph networks [32], [28], [42], [27], [16], [29]. • The extra connectivity, extra conditional diagnosability and t /k-diagnosability of arrangement graphs [19] and the extra, restricted connectivity and conditional diagnosability of split-star networks [21]. • The unknown extra connectivity, extra conditional diagnosability and t /k-diagnosability of D m,n . We can find the common characterization in the process of proving the extra connectivity and the t /k-diagnosability. Furthermore, based on the extra connectivity, we can establish the extra conditional diagnosability. Therefore, this paper aims to evaluate the extra connectivity, extra conditional diagnosability, and t /k-diagnosability of D m,n . The major contributions are as follows:

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• For any integers n ≥ 2, m ≥ 2, and 0 ≤ g ≤ n − 1, the g-extra connectivity of D m,n is k g ( D m,n ) = ( g + 1)(m − 1) + n. • For any integers n ≥ 2, m ≥ 2, and 0 ≤ g ≤ n − 1, the g-extra conditional diagnosability of D m,n under the PMC model is  t g ( D m,n ) = ( g + 1)m + n − 1. • For any integers n ≥ 2, m ≥ 2, and 1 ≤ k ≤ n − 1, D m,n is [(k + 1)(m − 1) + n]/k-diagnosable under the PMC model. Furthermore, we give a t /k-diagnostic algorithm, named t /k-D m,n -DIAG. The remainder of this paper is organized as follows. Section 2 provides the terms and notations used throughout the paper; Section 3 evaluates the g-extra connectivity of D m,n ; Section 4 establishes the g-extra conditional diagnosability of D m,n under the PMC model; Section 5 determines the t /k-diagnosability of D m,n under the PMC model; Section 6 presents our conclusion. 2. Preliminaries In this section, we first present some terms and notations used throughout the paper. Then we review the PMC model. Finally, we introduce the DCell network and its basic properties. 2.1. Terminology and notations In this subsection, we give some terminology and notations of combinatorial network theory. For terminology and notations not defined here, we follow [17]. Let G = ( V (G ), E (G )) be a graph. The vertex set V (G ) is a nonempty and finite set, and the edge set E (G ) is a subset of {(u , v )|(u , v ) is an unordered pair of V (G )}. Two vertices u and v are adjacent if (u , v ) ∈ E (G ). The neighborhood set of a vertex u is the set of all vertices adjacent to u in G, denoted by N G (u ) = { v ∈ V (G )|(u , v ) ∈ E (G )}. The degree of u in G is denoted by deg G (u ) = | N G (u )| and δ(G ) = min{deg G (u )|u ∈ V (G )}. If S ⊆ V (G ), let G [ S ] be the subgraph of G induced by the vertex subset S in G. We use G − S to denote G [ V (G )\ S ]. The  neighborhood set of S is defined as N G ( S ) = ( N G (u )) − S. Let N G [ S ] = N G ( S ) ∪ S. For any subset S  ⊂ V (G ), the private u∈ S

neighbors of u is denoted by P N S  (u ) = N G (u ) − ( N G ( S  − {u }) ∪ S  ). Let K n denote the complete graph with n vertices. A path P is a sequence of distinct vertices, and any two consecutive vertices in P are adjacent. The distance between two distinct vertices u and v of G, denoted by d G (u , v ), is the length of the shortest path between u and v in G. If the graphs G 1 and G 2 are two isomorphic graphs, we will write G 1 ∼ = G2. For any subset F ⊆ V (G ), G − F is defined as a graph by deleting all vertices in F from G and removing those edges with at least one end vertex in F , i.e., V (G − F ) = V (G ) − F , E (G − F ) = {(u , v ) ∈ E (G )|u , v ∈ ( V (G ) − F )}. The maximal connected subgraphs of G − F are called components. Let mc (G ) be the number of vertices in the largest component of G. F is called a vertex-cut if G − F is disconnected. The connectivity κ (G ) of G is defined as the minimum cardinality over all vertex-cuts of G. Let g be a non-negative positive integer, for any subset F such that G − F is disconnected and each remaining component has at least g + 1 vertices, then F is called a g-extra cut (in brief, R g -cut). The g-extra connectivity of G, denoted by κ g (G ), is the minimum cardinality over all R g -cuts of G. 2.2. The PMC model Preparata, Metze, and Chien [10] proposed the PMC model, which performs diagnosis by testing the neighboring processor via the links between them. 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 vertices 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 vertices 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 ). If the vertices u and v are adjacent in G, then the test u , v ∈ L. The collection of all test results from the test assignment T is termed as a syndrome 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 can 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, if σ ( F 1 ) ∩ σ ( F 2 ) = ∅, ( F 1 , F 2 ) is a distinguishable pair; otherwise, ( F 1 , F 2 ) is an indistinguishable pair. Supposed 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. Let F 1  F 2 denote the symmetric difference ( F 1 − F 2 ) ∪ ( F 2 − F 1 ) between F 1 and F 2 . In [1], Dahbura and Masson proposed a sufficient and necessary condition of t-diagnosable systems. Lemma 1. [1] Let G = ( V (G ), E (G )) be a multiprocessor system. For any two distinct sets F 1 , F 2 ⊆ V (G ), F 1 and F 2 are distinguishable under the PMC model 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).

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Fig. 1. An illustration for Lemma 1.

Fig. 2. The 1-dimensional DCell with 4-port switches.

2.3. The DCell network The data center network DCell, which interconnects servers by using recursively defined style. That is, a high-dimensional DCell can be recursively constructed from many low-dimensional ones. Due to this structure, DCell uses only mini-switches instead of using high-end switches. For any integers m ≥ 0 and n ≥ 2, we use DCellm,n to denote a m-dimensional DCell with n-port switches. Let tm,n denote the number of servers in DCellm,n , where t 0,n = n and tm,n = tm−1,n × (tm−1,n + 1) with m ≥ 1. The DCellm,n is constructed from tm,n servers and (t 0,n + 1) n-port switches. A server x of DCellm,n can be labeled by xm xm−1 · · · x2 x1 x0 . According to the definition of DCellm,n [4], we give a recursive definition of it. Definition 2. [4] The DCellm,n is defined recursively as follows. (1) A DCell0,n consists of n servers connected to a common switch. (2) For any integer m ≥ 1, DCellm,n is built from tm−1,n + 1 disjoint copies DCellm−1,n , according to the following steps. i (2.1) We use DCellm −1,n to denote one copy of DCellm−1,n , by prefixing the label of each server with i for i ∈ {0, 1, · · · , tm−1,n }. u (2.2) For any um , v m ∈ {0, 1, · · · , tm−1,n } with um < v m , server u = um um−1 · · · u 2 u 1 u 0 in DCellmm−1,n is adjacent to server v

v = v m v m−1 · · · v 2 v 1 v 0 in DCellmm−1,n if and only if um = v 0 +

m −1 j =1

( v j × t j −1,n ) and v m = u 0 +

m −1 j =1

(u j × t j −1,n ) + 1.

The 1-dimensional DCell with 4-port switches is shown in Fig. 2. Suppose that m is a non-negative integer, we use m to denote the set {0, 1, · · · , m} and use [m] to denote the set {1, 2, · · · , m}. Let I 0,n = n − 1 and I i ,n = t i −1,n for i ∈ [m]. The definition of m-dimensional DCell with n-port switches whose switches are regarded as transparent network devices (in brief, transparent DCell network) can be defined as follows. Definition 3. [4] For any integers m ≥ 0 and n ≥ 2, the m-dimensional transparent DCell network with n-port switches is denoted by a simple graph D m,n = ( V m,n , E m,n ) with tm,n vertices, where V m,n = {um um−1 · · · u 0 |u i ∈ I i ,n and i ∈ m }. For

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Fig. 3. Several transparent DCells with small parameters n and m.

any two vertices u = um um−1 · · · u 2 u 1 u 0 and v = v m v m−1 · · · v 2 v 1 v 0 , (u , v ) ∈ E m,n if and only if there exists an integer l with 1 ≤ l ≤ m + 1 such that

• um um−1 · · · ul = v m v m−1 · · · v l , • ul−1 = v l−1 , l −2 l −2 • ul−1 = v 0 + ( v j × t j −1,n ) and v l−1 = u 0 + (u j × t j −1,n ) + 1 with l > 1 and ul−1 < v l−1 . j =1

j =1

The transparent DCell networks with parameters n and m are shown in Fig. 3. For any integers n ≥ 2 and m ≥ 1, we call edges incident to vertices in the same copy of D m−1,n to be internal edges and edges incident to vertices in disjoint copies of D m−1,n to be external edges; we also call vertices adjacent to vertices in the same copy of D m−1,n to be internal l neighbor vertices and vertices adjacent to vertices in disjoint copies of D m−1,n to be external neighbor vertices. Let V m ,n = l {um um−1 · · · ul |u i ∈ I i ,n and i ∈ {l, l + 1, · · · , m}} with l ∈ [m]. For any α ∈ V m ,n with l ∈ [m], the graph obtained by prefixing α each vertex of D l−1,n with α is denoted by D lα−1,n . Clearly, D l−1,n ∼ D . = l−1,n If all conditions of Definition 3 hold, we say that two adjacent vertices u and v have a leftmost differing element at position l − 1. For any two adjacent vertices u and v, if u and v have a leftmost differing element at position d (d ≥ 0), we write as d = ldiff(u , v ). Obviously, ldiff(u , v ) = 0 represents that u and v are in the same complete subgraph D α 0,n , and ldiff(u , v ) ≥ 1 represents that u and v are not in the i 1 same complete subgraph D α , where α ∈ V . For any subset F ⊆ V m,n , let F i = F ∩ V ( D m m,n −1,n ) and J = {i : | F i | ≥ n + m − 2} 0,n    i J with i ∈ I m,n . Clearly, F J = F i , J = I m,n − J , F J = F i , and D m−1,n = D m−1,n . i∈ J

i∈ J

i∈ J

According to the structure characterizations of the DCell network, some properties of D m,n have been derived in the literatures. Lemma 2. [4] For any integers n ≥ 2 and m ≥ 0, D m,n has the following combinatorial properties. (n+m−1)t

m,n 1. D m,n is (n + m − 1)-regular with tm,n vertices and edges. 2 2. κ ( D m,n ) = λ( D m,n ) = n + m − 1. 3. For any integer m ≥ 0, there is no cycle of length 3 in D m,2 , and for any integers n ≥ 3 and m ≥ 0, there exist cycles of length 3 in D m,n . m 4. The number of vertices in D m,n satisfies tm,n ≥ (n + 12 )2 − 12 .

Lemma 3. [35] (1) For any integers n ≥ 2 and m ≥ 1, let u and v be any two distinct vertices in D m,n . Then

⎧ ⎨ = n − 2 if ldiff(u , v ) = 0; if ldiff(u , v ) ≥ 1; | N D m,n (u ) ∩ N D m,n ( v )| = 0 ⎩ if (u , v ) ∈ / E ( D m,n ). ≤1

β

(2) For any integers n ≥ 2 and m ≥ 1, H 0 ⊆ V ( D lα−1,n ) and H 1 ⊆ V ( D l−1,n ) with distinct H 1 | ≤ 1.

α , β ∈ V ml ,n , we have | N D m,n ( H 0 ) ∩

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j

i Lemma 4. [35] There exist tm−1,n disjoint paths (in which any two paths have no common vertices) joining D m −1,n and D m−1,n with j i i , j ∈ I m,n and i = j, denoted by P ( D m , D ) . m−1,n −1,n

Lemma 5. [35] Let F ⊆ V ( D m,n ) be a faulty vertex set. If | F | ≤ ( g + 1)(m − 1) + n with n ≥ 2, m ≥ 2, and 0 ≤ g ≤ n − 1, then J

| J | ≤ g + 1 and D m−1,n − F J is connected. Lemma 6. [35] For any integers n ≥ 2, m ≥ 2, and 0 ≤ g ≤ n − 1, there exists a complete graph H of order g + 1 in D m,n such that | N D m,n ( V ( H ))| = ( g + 1)(m − 1) + n and D m,n − N D m,n ( V ( H )) has exactly two connected components: one is D m,n [ V ( H )] and the other is D m,n − N D m,n [ V ( H )], where δ( D m,n − N D m,n [ V ( H )]) ≥ g. 3. The g-extra connectivity of the DCell network In this section, we first discuss a boundary problem of the DCell network, then we obtain a fault tolerant property by this boundary result. After that, the g-extra connectivity of the DCell network will be established for 0 ≤ g ≤ n − 1. Lemma 7. For any integers n ≥ 2, m ≥ 0, and 0 ≤ g ≤ n − 1, if H ⊆ V ( D m,n ) with | H | = g + 1, then | N D m,n ( H )| ≥ ( g + 1)(m − 1) + n. Proof. We conduct induction on m. If m = 0 and n ≥ 2, D m,n is a complete graph of order n. Clearly, D m,n [ H ] is isomorphic to K g +1 . By the definition of D m,n , | N D m,n ( H )| ≥ n − ( g + 1) for 0 ≤ g ≤ n − 1. Hence, the lemma holds. Suppose that the lemma holds for m = τ − 1 (τ ≥ 1). We will prove that the lemma holds for m = τ . Obviously, H can be αi divided into λ disjoint vertex subsets, say H 1 , H 2 , · · · , H λ , where H i = H ∩ V ( D τ − 1,n ) for 1 ≤ i ≤ λ, αi ∈ I τ ,n , and λ ∈ [ g + 1]. Suppose that | H i | = g i , it is easy to verify that

λ 

i =1

g i = | H | = g + 1 and | N D αi

τ −1,n

( H i )| ≥ g i (τ − 2) + n. We separately deal

with the cases below. Case 1. λ = 1. α1 Without loss of generality, let H = H 1 ⊆ V ( D τ − 1,n ). By the induction hypothesis, the lemma holds for | N D α1 ( H )| ≥ ( g + 1)(τ − 2) + n. By the definition of D τ ,n , we have

τ − 1, and thus,

τ −1,n

| N D τ ,n ( H )| = | N D α1

τ −1,n

( H )| + | N D τ ,n − D α1

τ −1,n

( H )|

≥ ( g + 1)(τ − 2) + n + ( g + 1) = ( g + 1)(τ − 1) + n. Therefore, the lemma holds. Case 2. 2 ≤ λ ≤ g + 1. α2 α1 αλ By the definition of D τ ,n , D τ − 1,n , D τ −1,n , . . . , D τ −1,n are λ disjoint subgraphs of D τ ,n , we have

| N D τ ,n ( H )| ≥ | N D α1

τ −1,n

( H 1 )| + | N D α1

τ −1,n

( H 2 )| + · · · + | N D α1

τ −1,n

( H λ )|

≥ [ g 1 (τ − 2) + n] + [ g 2 (τ − 2) + n] + · · · + [ g λ (τ − 2) + n] [ g i (τ − 2) + n] = 1≤i ≤λ

= ( g + 1)(τ − 2) + nλ. ∂ f (λ)

Let f (λ) = ( g + 1)(τ − 2) + nλ be a function on λ with 2 ≤ λ ≤ g + 1. We have ∂λ = n. Since n ≥ 2, we have Then f (λ) is an increasing function. Thus, for 2 ≤ λ ≤ g + 1 and n ≥ g + 1, we have

| N D τ ,n ( H )| ≥ ( g + 1)(τ ≥ ( g + 1)(τ ≥ ( g + 1)(τ = ( g + 1)(τ

∂ f (λ) ∂λ

> 0.

− 2) + n λ − 2) + 2n − 2) + n + ( g + 1) − 1) + n .

In summary, the lemma holds for m = τ . Thus, the lemma holds.

2

Lemma 8. For any integers n ≥ 2, m ≥ 2, and 1 ≤ h ≤ n, let f (h) = h(m − 1) + n be a function of h, then f (h) is strictly monotonically increasing on h.

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∂ f (h)

Proof. Since ∂ h = m − 1 > 0 for m ≥ 2, f (h) is a strictly monotonically increasing function on h with 1 ≤ h ≤ n. Then f (h ) < f (h) for any two positive integers h and h with 1 ≤ h < h ≤ n. 2 α Lemma 9. For any integers n ≥ 2, m ≥ 2, and 1 ≤ h ≤ n, if H α ⊆ V ( D m −1,n ) with | H α | ≤ n and

there are at least two of H 0 , H 1 , · · · , H tm−1,n are non-empty sets, then

tm −1,n

α =0

Proof. For any non-empty set H α , let | H α | = hα ≥ hα ≥ 1 such that

tm −1,n

α =0

| H α | ≥ h, where α ∈ I m,n , and

| N D mα −1,n ( H α )| ≥ h(m − 1) + n.

tm −1,n

α =0

hα = h, where

α ∈ I m,n . Clearly, 1 ≤ hα ≤ h − 1 ≤

n − 1. Let f (h) = h(m − 1) + n be a function of h, by Lemma 8, we have f (hα ) ≤ f (hα ). Then hα (m − 1) + n ≤ hα (m − 1) + n. Since there are at least two of H 0 , H 1 , · · · , H tm−1,n are non-empty sets, without loss of generality, suppose that H 0 and H 1 are non-empty sets. By Lemma 7, for any integers 1 ≤ x ≤ tm−1,n , n ≥ 2, and 1 ≤ h ≤ n, we have tm −1,n

α =0

= |N D 0

| N D mα −1,n ( H α )| − [h(m − 1) + n]

m−1,n

( H 0 )| + | N D 1

m−1,n

( H 1 )| + · · · + | N

tm−1,n

D m−1,n

( H tm−1,n )| − [h(m − 1) + n]

( H 2 )| + · · · + ≥ h0 (m − 2) + n + h1 (m − 2) + n + | N D 2 m−1,n | N tm−1,n ( H tm−1,n )| − [h(m − 1) + n] ≥ = = = ≥ Then

D m−1,n

h0 (m − 2) + n + (h − h0 )(m − 2) + xn − [h(m − 1) + n] h(m − 2) + (x + 1)n − h(m − 1) − n −h + xn −h + n + (x − 1)n 0.

tm −1,n

α =0

| N D mα −1,n ( H α )| ≥ h(m − 1) + n, the lemma holds. 2

Lemma 10. For any integers n ≥ 2, m ≥ 2, and 0 ≤ g ≤ n − 1, if F ⊆ V ( D m,n ) with | F | ≤ ( g + 1)(m − 1) + n − 1, then D m,n − F has a largest component containing at least tm,n − | F | − g vertices. Proof. We conduct induction on g. If g = 0, D m,n − F is connected since | F | ≤ n + m − 2 < n + m − 1 = κ ( D m,n ), the lemma holds. Suppose that the lemma holds for g = i − 1, where n ≥ 2, m ≥ 2, and 1 ≤ i ≤ n − 1, we will prove that the following claim holds: Claim 1. The lemma holds for n ≥ 2, m ≥ 2, and g = i with 1 ≤ i ≤ n − 1. Let C 1 , C 2 , · · · , C η , C η+1 be the components of D m,n − F , where C η+1 is the largest component. By Lemma 5, | J | ≤ i + 1 J

J

and D m−1,n − F J is connected. Thus, V ( D m−1,n − F J ) ⊂ V (C η+1 ). Let r = | J | ≤ i + 1 and J = {α1 , α2 , · · · , αr } ∈ I m,n . For J

r = 0, by Lemma 5, D m−1,n − F J = D m,n − F is connected. Hence, the claim holds. In the following, we will consider three cases to prove the claim for 1 ≤ r ≤ i + 1. αj Case 1. For any integer 1 ≤ j ≤ r, D m−1,n − F α j is connected. αj

J

β

Since D m−1,n − F α j is connected for any 1 ≤ j ≤ r and D m−1,n − F J is connected, D m−1,n − F β is connected for any β ∈ I m,n . Next, we will show that D m,n − F is connected. s t To show that D m,n − F is connected, it suffices to show that D m −1,n − F s and D m−1,n − F t are connected for any s, t ∈ I m,n with s = t. For n ≥ 2, m ≥ 2, and 1 ≤ i ≤ n − 1, by Lemma 2 (4) and Lemma 4, we have m −1

1 2 s t | P (Dm −1,n , D m−1,n )| = tm−1,n ≥ (n + 2 ) > n(m − 1) + n ≥ (i + 1)(m − 1) + n ≥ | F |.

−

1 2

s t It easy to verify that there exists a fault-free path from D m −1,n to D m−1,n for any s, t ∈ I m,n . By the arbitrariness of s and t, D m,n − F is connected (see Fig. 4). Hence the claim holds. α α α Case 2. Exactly one of D m1−1,n − F α1 , D m2−1,n − F α2 , · · · , D mr−1,n − F αr is disconnected.

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Fig. 4. An illustration for Case 1 of Claim 1.

α

Without loss of generality, suppose that D m1−1,n − F α1 is disconnected. By the similar method used in the Case 1, D m,n − η α α D m−1,n − F is connected. Then V ( D m,n − D m1−1,n − F ) ⊂ V (C η+1 ). Hence, p =1 V (C p ) ⊆ V ( D m1−1,n − F α1 ). In the following, α1

we will prove that

η 

p =1

| V (C p )| ≤ i. We consider two cases as follows.

Case 2.1. r = 1. η Let H = p =1 V (C p ) and h = | H |. For the sake of contradiction, suppose that h ≥ i + 1 and let T ⊆ H with | T | = i + 1. By Lemma 7, we have | N D α1 ( T )| ≥ (i + 1)(m − 2) + n. Moreover, by the hypothesis, N D α1 ( T ) − ( H − T ) ⊆ F α1 . Then, we m−1,n

have

| F α1 | ≥ | N D α1 ≥ |N D

m−1,n

α1

m−1,n

m−1,n

( T ) − ( H − T )| ( T )| − [h − (i + 1)]

≥ (i + 1)(m − 2) + n − h + (i + 1) ≥ (i + 1)(m − 1) + n − h. α

Since each vertex of H has exactly one neighbor in D m,n − D m1−1,n and N D

| F α1 | ≤ | F | − | N D m,n − D α1

m−1,n

α1 m,n − D m−1,n

( H ) ⊆ F − F α1 , we have

( H )|

= |F | − |H | = |F | − h ≤ (i + 1)(m − 1) + n − 1 − h,

which contradicts with the above inequality | F α1 | ≥ (i + 1)(m − 1) + n − h. Hence, h =

η 

p =1

| V (C p )| ≤ i.

Case 2.2. r ≥ 2. Since | F αl | ≥ n + m − 2 for 2 ≤ l ≤ r, we have

| F α1 | ≤ | F | −

 2≤l≤r

| F αl |

≤ (i + 1)(m − 1) + n − 1 − (r − 1)(n + m − 2) ≤ (i + 1)(m − 1) + n − 1 − (n + m − 2) ≤ (i + 1)(m − 1) + n − 1 − (m − 1 + i ) = (m − 2)i + n − 1 for 1 ≤ i ≤ n − 1. α By the induction hypothesis, D m1−1,n − F α1 has a large connected component C α 1 containing at least tm−1,n −| F α1 | −(i − 1) vertices. Moreover, by Lemma 2 (4),

tm−1,n ≥ (n +

1 2

m −1

)2

for n ≥ 2 and m ≥ 2. Then

−

1 2

≥ m(2n − 1) − 1

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|C α 1 | ≥ tm−1,n − | F α1 | − (i − 1) m −1 − 12 − [(m − 2)i + n − 1] − (i − 1) ≥ (n + 12 )2 ≥ m(2n − 1) − 1 − [(m − 1)i + n − 2] = (m − 1)(2n − 1) + 2n − 2 − [(m − 1)i + n − 2] ≥ (m − 1)(2i + 1) + 2n − 2 − [(m − 1)i + n − 2] = (m − 1)(2i + 1) + 2n − 2 − (m − 1)i − n + 2 = (i + 1)(m − 1) + n > |F | for n ≥ 2, m ≥ 2, and 1 ≤ i ≤ n − 1. η α α Therefore, C α 1 is connected to D m,n − D m1−1,n − F . Then p =1 V (C p ) ⊆ V ( D m1−1,n − F α1 − C α 1 ). η Thus, | p =1 V (C p )| ≤ i − 1 < i, the Claim 1 holds.

Case 3. Exactly r  of D m1−1,n − F α1 , D m2−1,n − F α2 , · · · , D mr−1,n − F αr are disconnected for 2 ≤ r  ≤ r. α

α

α

αq j

Suppose that D m−1,n − F αq is disconnected, where q j ⊆ {1, 2, · · · , r } and 1 ≤ j ≤ r  . By the similar method used in the j

r 

αq j

Case 1, D m,n − j =1 D m−1,n − F is connected. Hence, V ( D m,n − 1 ≤ l ≤ r  , we have

| F αq j | ≤ | F | −



1≤l≤r  ,l= j

r 

αq j

D m−1,n − F ) ⊂ V (C η+1 ). Since | F αq | ≥ n + m − 2 for l

j =1

| F αql |

≤ (i + 1)(m − 1) + n − 1 − (r  − 1)(n + m − 2) ≤ (i + 1)(m − 1) + n − 1 − (n + m − 2) ≤ (i + 1)(m − 1) + n − 1 − (m − 1 + i ) = (m − 2)i + n − 1 for r  ≥ 2 and 1 ≤ i ≤ n − 1.

αq j

By the induction hypothesis, D m−1,n − F αq has a large connected component C α q containing at least tm−1,n − | F αq | − j

tm−1,n ≥ (n +

1 2

m −1

)2

−

1 2

j

j

(i − 1) vertices for 1 ≤ j ≤ r  . Moreover, by Lemma 2 (4),

≥ m(2n − 1) − 1

for n ≥ 2 and m ≥ 2. Then

|C α q | ≥ tm−1,n − | F αq j | − (i − 1) j

m −1

− 12 − [(m − 2)i + n − 1] − (i − 1) ≥ (n + 12 )2 ≥ m(2n − 1) − 1 − [(m − 1)i + n − 2] = (m − 1)(2n − 1) + 2n − 2 − [(m − 1)i + n − 2] ≥ (m − 1)(2i + 1) + 2n − 2 − [(m − 1)i + n − 2] = (m − 1)(2i + 1) + 2n − 2 − (m − 1)i − n + 2 = (i + 1)(m − 1) + n > |F | for n ≥ 2, m ≥ 2, 1 ≤ i ≤ n − 1, and 1 ≤ j ≤ r  .

r 

η

αq j

αq j

Hence, C α q is connected to D m,n − j =1 D m−1,n − F . Thus, V (C α q ) ⊂ V (C η+1 ). Let U αq = ( p =1 V (C p )) ∩ V ( D m−1,n ) j j j for 1 ≤ j ≤ r  . By the similar method used in the Case 2, we have |U αq | ≤ i − 1 ≤ n − 2 with i ≤ n − 1. In the following, we will prove that

r  j =1

j

|U αq j | ≤ i. On the contrary, we assume that

r  j =1

|U αq j | ≥ i + 1. By Lemma 9, we

have r  j =1

|N

αq j (U α )| qj D m−1,n

≥ (i + 1)(m − 1) + n > | F |,

which results in a contraction. Thus, |

r 

j =1

U αq | ≤ i. Then D m,n − F has a component containing at least tm,n − | F | − i j

vertices for 1 ≤ i ≤ n − 1. In summary, the Claim 1 holds for n ≥ 2, m ≥ 2, and g = i with 1 ≤ i ≤ n − 1. By Claim 1, the lemma holds. 2 Theorem 1. For any integers n ≥ 2, m ≥ 2, and 0 ≤ g ≤ n − 1, the g-extra connectivity of D m,n is k g ( D m,n ) = ( g + 1)(m − 1) + n.

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Proof. We first establish the upper bound of k g ( D m,n ). By Lemma 6, for any integers n ≥ 2, m ≥ 2, and 0 ≤ g ≤ n − 1, there exists a complete graph H of order g + 1 in D m,n such that | N D m,n ( V ( H ))| = ( g + 1)(m − 1) + n and D m,n − N D m,n ( V ( H )) has exactly two connected components: one is D m,n [ V ( H )] and the other is D m,n − N D m,n [ V ( H )], where δ( D m,n − N D m,n [ V ( H )]) ≥ g. It is easy to verify that | V ( H )| = g + 1 and | V ( D m,n − N D m,n [ V ( H )])| ≥ g + 1. Thus, N D m,n ( H ) is an R g -cut of D m,n . Then k g ( D m,n ) ≤ | N D m,n ( V ( H ))| = ( g + 1)(m − 1) + n. Next, we explore the lower bound of k g ( D m,n ). For the sake of contradiction, suppose that F is an R g -cut of D m,n with | F | ≤ ( g + 1)(m − 1) + n − 1, then each component of D m,n − F has at least g + 1 vertices. However, by Lemma 10, D m,n − F has a largest component containing at least tm,n − | F | − g, where n ≥ 2, m ≥ 2, and 0 ≤ g ≤ n − 1. Thus, D m,n − F has a union of small components containing at most g vertices. As a result, F is not an R g -cut of D m,n with | F | ≤ ( g + 1)(m − 1) + n − 1, a contradiction. Then k g ( D m,n ) ≥ ( g + 1)(m − 1) + n. Hence, the g-extra connectivity of D m,n is k g ( D m,n ) = ( g + 1)(m − 1) + n for n ≥ 2, m ≥ 2, and 0 ≤ g ≤ n − 1. 2 4. The g-extra conditional diagnosability of the DCell network under the PMC model In this section, we will determine the g-extra conditional diagnosability of D m,n under the PMC model for 0 ≤ g ≤ n − 1. We first review the definition of the g-extra conditional diagnosability of a multiprocessor system G, which was proposed by Zhang and Yang [31]. Definition 4. [31] Let G = ( V (G ), E (G )) be a multiprocessor system. A faulty set F is called a g-extra conditional faulty set if G − F is disconnected and every component of G − F has at least g + 1 vertices. Definition 5. [31] A system G = ( V (G ), E (G )) is g-extra conditional t-diagnosable if each distinct pair of g-extra conditional faulty sets F 1 and F 2 of V (G ) with | F 1 | ≤ t, | F 2 | ≤ t are distinguishable. t g (G ) of G is the Definition 6. [31] Let G = ( V (G ), E (G )) be a multiprocessor system. The g-extra conditional diagnosability  maximum value of t such that G is g-extra conditionally t-diagnosable. The following lemma provides an upper bound of g-extra conditional diagnosability of D m,n under the PMC model for 0 ≤ g ≤ n − 1. Lemma 11. For any integer n ≥ 2, m ≥ 1, and 0 ≤ g ≤ n − 1, the g-extra conditional diagnosability of D m,n under the PMC model is  t g ( D m,n ) ≤ ( g + 1)m + n − 1. Proof. Let H be a complete graph of order g + 1 in D m,n . Suppose F 1 = N D m,n ( V ( H )) and F 2 = N D m,n [ V ( H )]. By Lemma 6, we have | F 1 | = | N D m,n ( V ( H ))| = ( g + 1)(m − 1) + n and | F 2 | = | N D m,n [ V ( H )]| = ( g + 1)m + n, where D m,n − F 2 is a connected component and δ( D m,n − F 2 ) ≥ g. It is easy to verify that | V ( D m,n − F 2 )| ≥ g + 1. Therefore, F 1 and F 2 are two g-extra conditional faulty sets of D m,n with | F 1 | ≤ ( g + 1)m + n and | F 2 | ≤ ( g + 1)m + n. On the other hand, since V ( H ) = F 1  F 2 and N D m,n ( V ( H )) = F 1 , there is no edge between F 1  F 2 and V ( D m,n ) − ( F 1 ∪ F 2 ). By Lemma 1, F 1 and F 2 are indistinguishable under the PMC model. By Definition 5, D m,n is not g-extra conditional [( g + 1)m + n]-diagnosable under the PMC model. Thus,  t g ( D m,n ) ≤ ( g + 1)m + n − 1 under the PMC model for 0 ≤ g ≤ n − 1. 2 Next, we explore the lower bound of g-extra conditional diagnosability of D m,n under the PMC model for 0 ≤ g ≤ n − 1. Lemma 12. For any integers n ≥ 2, m ≥ 2, and 0 ≤ g ≤ n − 1, the g-extra conditional diagnosability of D m,n under the PMC model is

 t g ( D m,n ) ≥ ( g + 1)m + n − 1.

Proof. We prove this lemma by contradiction. Suppose that there are two distinct g-extra conditional faulty sets F 1 and F 2 , which are indistinguishable with | F 1 | ≤ ( g + 1)m + n − 1 and | F 2 | ≤ ( g + 1)m + n − 1. In what follows, we consider two cases. Case 1. V ( D m,n ) = F 1 ∪ F 2 . By Lemma 2 (4), we have

1

1

m

| V ( D m,n )| = tm,n ≥ (n + )2 − . 2

2

Since | F 1 ∪ F 2 | ≤ | F 1 | + | F 2 | ≤ 2[( g + 1)m + n − 1],

1

m

| V ( D m,n )| − | F 1 ∪ F 2 | ≥ (n + )2 − 2

1 2

− 2[( g + 1)m + n − 1].

JID:TCS AID:11738 /FLA

Doctopic: Algorithms, automata, complexity and games

m

Let f ( g ) = (n + 12 )2 − decreasing function. Then

1 2

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∂ f (g) ∂g

= −2m < 0 for m ≥ 1. Thus, f ( g ) is a

− 2[( g + 1)m + n − 1] for 0 ≤ g ≤ n − 1. We obtain

f ( g ) ≥ f (n − 1) = (n +

1 2

m

)2 −

1 2

− 2(nm + n − 1) > 0

for n ≥ 2 and m ≥ 2. Then | V ( D m,n )| − | F 1 ∪ F 2 | > 0, which induces a contradiction since V ( D m,n ) = F 1 ∪ F 2 . Case 2. V ( D m,n ) = F 1 ∪ F 2 . Since F 1 and F 2 are indistinguishable, there is no edge between V ( D m,n ) − F 1 ∪ F 2 and F 1  F 2 . Moreover, since F 1 and F 2 are two g-extra conditional faulty sets, every component of D m,n − F 1 and D m,n − F 2 contains at least g + 1 vertices. Thus, F 1 ∩ F 2 is also a g-extra conditional faulty set, every component of D m,n − F 1 ∩ F 2 also contains at least g + 1 vertices. Then F 1 ∩ F 2 is an R g -cut of D m,n . By Theorem 1, we have | F 1 ∩ F 2 | ≥ ( g + 1)(m − 1) + n for 0 ≤ g ≤ n − 1. Since F 1 = F 2 , we may assume that F 2 − F 1 = ∅. There are at least g + 1 vertices in the components of D m,n − F 1 since F 1 is a g-extra conditional faulty set. Thus, | F 2 − F 1 | ≥ g + 1. Then

| F 2| = | F 2 − F 1| + | F 1 ∩ F 2| ≥ g + 1 + ( g + 1)(m − 1) + n = ( g + 1)m + n, which results in a contradiction since | F 2 | ≤ ( g + 1)m + n − 1. Hence, the lemma holds.

2

Combining Lemma 11 and Lemma 12, we have the following theorem. Theorem 2. For any integers n ≥ 2, m ≥ 2, and 0 ≤ g ≤ n − 1, the g-extra conditional diagnosability of D m,n under the PMC model is

 t g ( D m,n ) = ( g + 1)m + n − 1.

5. The t /k-diagnosability of the DCell network under the PMC model In this section, we will evaluate the t /k-diagnosability of D m,n under the PMC model for 1 ≤ k ≤ n − 1. Furthermore, we will present a relevant t /k-diagnosis algorithm. The following results are useful in this section and thus presented here. Definition 7. [39] Given a graph G = ( V (G ), E (G )) and a syndrome on G produced by a faulty set. The 0-test subgraph of G, denoted T 0 (G ), is a subgraph of G defined by V ( T 0 (G )) ⊆ V (G ) and E ( T 0 (G )) = {(u , v ) ∈ E (G )|σ (u , v ) = σ ( v , u ) = 0}. Lemma 13. [39] Given a graph G = ( V (G ), E (G )) and a syndrome σ on G under the PMC model. 1. Let u , v be two adjacent vertices of G. If σ (u , v ) = σ ( v , u ) = 0, then either both vertices u and v are fault-free, or both u and v are faulty. 2. Let C be a component of T 0 (G ). Then either all vertices of C are fault-free, or all vertices of C are faulty. In what follows, we establish the t /k-diagnosability of D m,n under the PMC model for 1 ≤ k ≤ n − 1. Moreover, we give a t /k-diagnosis algorithm of D m,n under the PMC model, named t /k-D m,n -DIAG. Lemma 14. For any integers n ≥ 2, m ≥ 2, and 1 ≤ k ≤ n − 1, let F be a faulty vertex set of D m,n and let C be a largest component of T 0 ( D m,n ). If | F | ≤ (k + 1)(m − 1) + n − 1, then all vertices in C are fault-free with | V (C )| ≥ tm,n − | F | − k. Proof. We need to prove | V (C )| = mc ( D m,n − F ). Let C  be one of the largest component of D m,n − F . Clearly, C  is a component of T 0 ( D m,n ). By Lemma 10, D m,n − F has a largest component containing at least tm,n − | F | − k vertices, we have | V (C  )| = mc ( D m,n − F ) ≥ tm,n − | F | − k. Let the edge-set E  = {(u , v ) ∈ E ( D m,n )|u ∈ F , v ∈ ( D m,n − F )}. Since the edge-set E  does not belong to E ( T 0 ( D m,n )), C  is the sole largest component of T 0 ( D m,n ) with | V (C  )| = mc ( T 0 ( D m,n )). That is, C is the unique largest component C  of D m,n − F . For n ≥ 2, m ≥ 2, and 1 ≤ k ≤ n − 1, by Lemma 2 (4), we have m tm,n ≥ (n + 12 )2 − 12 > 2[(k + 1)(m − 1) + n − 1] + k. Then

| V (C )| = mc ( D m,n − F ) ≥ tm,n − | F | − k ≥ tm,n − [(k + 1)(m − 1) + n − 1] − k > 2[(k + 1)(m − 1) + n − 1] + k − [(k + 1)(m − 1) + n − 1] − k = (k + 1)(m − 1) + n − 1 ≥ | F |. By Lemma 13 (2), all vertices in C are fault-free. Hence the lemma holds.

2

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Algorithm 1 t /k-D m,n -DIAG. Input: The syndrome σ on D m,n under the PMC model, and the upper bound on the number of faulty vertices | F | ≤ t = (k + 1)(m − 1) + n, where integers

n ≥ 2, k ≥ 2, and 1 ≤ k ≤ n − 1. Output: A faulty vertex set F  and a fault-free vertex set H with F  ∪ H = V ( D m,n ).

1: Initialize “F  = ∅”, “H = ∅”, “U = V ( D m,n )”, where the symbol ∅ denotes the empty set. 2: Check all test outcomes on D m,n . Delete all edges with 1 ↔ 0, 0 ↔ 1, 1 ↔ 1, and add all edges with 0 ↔ 0 to E  . Let T 0 ( D m,n ) be the induced subgraph by E  . 3: (Obtaining the set C of fault-free vertices) Obtain the largest component C by Breadth-First Search in T 0 ( D m,n ), and add all vertices in C to H . Let U = U − H. 4: For each undiagnosed vertex v in the previous steps, if it has a neighbor u such that u ∈ C , then add v to F  according to σ (u , v ) = 1. Let U = U − F  . 5: For each undiagnosed vertex v in the previous steps, if | F  | = (k + 1)(m − 1) + n, then add all vertices in U to H ; otherwise, add all (suspicious) vertices in U to F  . 6: return The faulty vertex set F  and the fault-free vertex set H .

Theorem 3. For any integers n ≥ 2, m ≥ 2, and 1 ≤ k ≤ n − 1, the time complexity of algorithm t /k-D m,n -DIAG is O ( NlogN ), where N = tm,n is the number of vertices in D m,n . Proof. By algorithm t /k-D m,n -DIAG, the main time cost is on step 2. The BFS algorithm runs at most O ( N (n + m − 1)) time. By Lemma 2 (4), for n ≥ 2 and m ≥ 2, we have,

N = tm,n

m

≥ (n + 12 )2 − > 2n+m−1 .

1 2

Then log(2n+m−1 ) = n + m − 1 ≤ log( N ). Therefore, the step 2 costs O ( NlogN ) time. The step 3, step 4, and step 5 cost O ( N ) time. Hence, the total time of algorithm t /k-D m,n -DIAG is O ( NlogN ). 2 Theorem 4. For any integers n ≥ 2, m ≥ 2, and 1 ≤ k ≤ n − 1, let F be a faulty vertex set of D m,n with | F | ≤ t = (k + 1)(m − 1) + n, and σ be a syndrome on D m,n under the PMC model. Then the algorithm t /k-D m,n -DIAG isolates all faulty vertices to within a set, say F  , containing at most k fault-free vertices. Proof. Depending on the size of | F |, we have the following cases. Case 1. | F | = t. Suppose that D m,n − F has a largest component and an union of small components. Let M be the vertex set of the union of small components, i.e., M ⊆ F  and N D m,n ( M ) ⊆ F . In the following, we consider the size of M. If | M | ≤ k, then the theorem holds. If | M | ≥ k + 1, it suffices to show that | N D m,n ( M )| ≥ t. That is, N D m,n ( M ) = F and M ∩ F = ∅. We can deduce that every vertex of M is fault-free. Hence the set of vertex that could be incorrectly identified is empty. Then the theorem holds. In the following, we need to prove that for any vertex set M ⊆ V ( D m,n ) with | M | ≥ k + 1, | N D m,n ( M )| ≥ (k + 1)(m − 1) + n = t. By contradiction, if | N D m,n ( M )| ≤ (k + 1)(m − 1) + n − 1, by Lemma 10, D m,n − F has a largest component and an union H of small components with | H | ≤ k. However, M ⊆ H and | H | ≥ | M | ≥ k + 1, a contradiction. Case 2. | F | ≤ t − 1. By Lemma 14, | F  | ≤ | F | + k, which implies that F  contains at most k fault-free vertices. Hence, the theorem holds. 2 Theorem 5. For any integers n ≥ 2, m ≥ 2, and 1 ≤ k ≤ n − 1, D m,n is [(k + 1)(m − 1) + n]/k-diagnosable under the PMC model. Proof. If | F | ≤ (k + 1)(m − 1) + n, by Theorem 4, the algorithm t /k-D m,n -DIAG isolates all faulty vertices to within a set, say F  (| F  | ≤ | F | + k), containing at most k fault-free vertices, where n ≥ 2, m ≥ 2, and 1 ≤ k ≤ n − 1. If | F | = (k + 1)(m − 1) + n, these faulty vertices can surround k fault-free vertices. Therefore, (k + 1)(m − 1) + n is the upper bound of [(k + 1)(m − 1) + n]/k-diagnosis with 1 ≤ k ≤ n − 1. 2 6. Conclusion In this paper, our work demonstrates that connectivity and diagnosability problems of the DCell network can be studied by using the method of multi-processor interconnection networks, which provides an elicitation for the further research on these problems of other DCNs. We first show that the g-extra connectivity of D m,n is k g ( D m,n ) = ( g + 1)(m − 1) + n for n ≥ 2, m ≥ 2, and 0 ≤ g ≤ n − 1. Then we use this result to establish the g-extra conditional diagnosability of D m,n under the PMC model, and we obtain that  t g ( D m,n ) = ( g + 1)m + n − 1 = k g ( D m,n ) + g for n ≥ 2, m ≥ 2, and 0 ≤ g ≤ n − 1. After that, we evaluate the t /k-diagnosability of D m,n under the PMC model for 1 ≤ k ≤ n − 1. By analysis and comparison, we can deduce that the t /k-diagnosability of D m,n under the PMC model is equal to the k-extra connectivity of D m,n for n ≥ 2, m ≥ 2, and 1 ≤ k ≤ n − 1. Furthermore, we give a fast t /k-diagnosis algorithm under the PMC model. These results can provide a more accurate measure for evaluating a large-scale DCN’s reliability and availability.

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On the other hand, there are a number of open questions enlightened from this paper that we will investigate in the future. A non-comprehensive list is as follows.

• Under what conditions, the relationship between the g-extra connectivity, the g-extra conditional diagnosability, and the t /k-diagnosability of general graphs also have the same relationship between the three in the DCell network? • Further investigate the g-extra connectivity and g-extra conditional diagnosability of D m,n for general integers g ≥ n, and further study the t /k-diagnosability of D m,n for k ≥ n. • Explore the g-extra conditional diagnosability and the t /k-diagnosability of D m,n under the comparison model [12,2]. • Some novel data center networks related to the DCell network have been proposed recently [24,41,5], whether the method developed in this paper can be expanded to determine extra connectivity and diagnosabilities on them?

• As a reliability problem, how to design efficient fault-tolerant routing algorithms [7,40,8] for the DCell network? Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 61572337, No. 61702351, and No. 61872257), Opening Foundation of Jiangsu High Technology Research Key Laboratory for Wireless Sensor Networks (No. WSNLBKF201701), Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX17_2005), Science and Technology Project of Fujian Provincial Education Department (No. JAT160544), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 18KJA520009). References [1] A.T. Dahbura, G.M. Masson, An O (n2.5 ) faulty identification algorithm for diagnosable systems, IEEE Trans. Comput. 33 (6) (1984) 486–492. [2] A. Sengupta, A. Dahbura, On self-diagnosable multiprocessor system: diagnosis by the comparison approach, IEEE Trans. Comput. 41 (11) (1992) 1386–1396. [3] A.K. Somani, O. 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