Constructing completely independent spanning trees in crossed cubes

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Constructing completely independent spanning trees in crossed cubes Baolei Cheng a,b , Dajin Wang c , Jianxi Fan a,b,∗ a

School of Computer Science and Technology, Soochow University, Suzhou 215006, China

b

Provincial Key Laboratory for Computer Information Processing Technology, Soochow University, Suzhou 215006, China

c

Department of Computer Science, Montclair State University, Upper Montclair, NJ 07043, USA

article

info

Article history: Received 14 April 2016 Received in revised form 16 November 2016 Accepted 16 November 2016 Available online xxxx Keywords: Completely independent spanning tree Crossed cube Fault tolerance Reliable broadcasting

abstract The Completely Independent Spanning Trees (CISTs) are a very useful construct in a computer network. It can find applications in many important network functions, especially in reliable broadcasting, i.e., guaranteeing broadcasting operation in the presence of faulty nodes. The question for the existence of two CISTs in an arbitrary network is an NP-hard problem. Therefore most research on CISTs to date has been concerning networks of specific structures. In this paper, we propose an algorithm to construct two CISTs in the crossed cube, a prominent, widely studied variant of the well-known hypercube. The construction algorithm will be presented, and its correctness proved. Based on that, the existence of two CISTs in a special Bijective Connection network based on crossed cube is also discussed. © 2016 Elsevier B.V. All rights reserved.

1. Introduction The Completely Independent Spanning Trees (CISTs for short) of a computer network is a set of spanning trees

{T1 , T2 , . . . , Tk } of the network, such that for any two nodes u and v , the u-to-v paths belonging to Ti and Tj respectively have no common intermediate nodes, for all i ̸= j. CISTs can find applications in important network functions such as multinode broadcasting [3], one-to-all broadcasting [33], reliable broadcasting, secure message distribution [14,20], and increase of bandwidth [19]. For example, the network bandwidth can be increased by using the multiple paths between any two nodes in CISTs. Some examples of CISTs are illustrated in Fig. 1. The pairs in (a) and (b) are CISTs, while the pair in (c) is not because the paths from u to v in both trees contain node w . The decision problem as to whether there exist two CISTs in a general graph G has been shown NP-hard [15], and conditions have been proposed for CIST-containing graphs of certain structures. It was once conjectured that a k-connected graph (k ≥ 2) always contains two CISTs, only to be disproved recently by Péterfalvi and Pai [24,31]. Therefore most research on CISTs to date has been concerning networks of specific structures. Especially, the problem of constructing multiple CISTs in a certain regularly-structured network has received much attention [15,24,7,16,18,28]. As an important variant of the hypercube, the best-known structure for the interconnection network, the crossed cube (CQn for short) has attracted the interest of many researchers [6,5,4,8,12,25,26,29,30,34,35]. It was proved in [21] that an n-dimensional crossed cube CQn is not node-transitive, and [22] showed that the node set of CQn can be divided into 2⌈(n−4)/2⌉ equivalence classes, where the nodes in one equivalence class are all similar. Since there are two equivalence classes in the

∗

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

http://dx.doi.org/10.1016/j.dam.2016.11.019 0166-218X/© 2016 Elsevier B.V. All rights reserved.

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(a) T1 , T2 are CISTs.

(b) T1 , T2 are CISTs.

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(c) T1 , T2 are not CISTs.

Fig. 1. Example of CISTs in (a) and (b); non-CISTs in (c).

n-dimensional locally twisted cube LTQn [27], and there are only one equivalence class [32] in the n-dimensional hypercube Qn , CQn is considered more complex than LTQn and Qn . Some characterizations have been proposed for the existence of CISTs in arbitrary graphs. Recently, Araki [1] proved that there exist two CISTs in a graph with n ≥ 3 nodes if the degree of each node is ≥ n/2, and Fan et al. [11] proved the existence of two CISTs in a graph with n ≥ 3 nodes if the sum of degrees of any two nodes is ≥ n. H.-Y. Chang et al. proved that for graphs of order n, with n ≥ 6, if the minimum degree is at least n − 2, then there are at least ⌊n/3⌋ CISTs [2]. T. Hasunuma showed that for any graph G with n ≥ 7 nodes, if the minimum degree of a vertex is n − k, where 3 ≤ k ≤ n/2, then there are ⌊n/k⌋ CISTs in G [17]. These are rather strong sufficient conditions because the needed degree of every node/pair of nodes is very large comparing the number of nodes. Furthermore, they are all existential proofs rather than constructional. In this paper, we study the problem of constructing CISTs in a CQn , the n-dimensional crossed cube. More specifically, we will (a) show that there exist two CISTs in a CQn if n ≥ 4; (b) present a recursive algorithm to construct two CISTs in a CQn , n ≥ 4, with the diameter of either tree being no more than n + 4. Furthermore, we will discuss the existence of CISTs in a more generalized network – the CQ-based BC network – and show that it contains two CISTs. The rest of this paper is organized as follows. In Section 2, we introduce terminology and notation that will be used throughout the paper, define the crossed cube CQn , and give some relevant properties of the CQn . Section 3 presents the main work of the paper: Proposing the algorithm that constructs two CISTs in a CQn , and proving the correctness of the algorithm. Section 4 discusses the existence of CISTs in CQ-based BC networks. Section 5 gives summarizing remarks and concludes the paper. 2. Preliminaries 2.1. Graph terminology and notation A computer interconnection network can be represented by a graph, where nodes represent processors and edges represent links between processors. Given a graph G, we use V (G) and E (G) to denote the node set and the edge set, respectively, in G. Let S be a nonempty subset of V (G). The subgraph of G induced by S, denoted by G[S ], is the subgraph with the node set S and those edges of G with both ends in S. A graph G1 is said to be isomorphic to graph G2 , if there is a bijection ψ : V (G1 ) → V (G2 ) such that (ψ(x), ψ(y)) ∈ E (G2 ) if and only if (x, y) ∈ E (G1 ). Let u, v ∈ V (G). A path from u to v is called ⟨u, v⟩-path, denoted by R: u = x(0) , x(1) , . . . , x(k) = v , where (x(i) , x(i+1) ) ∈ E (G)) for all 0 ≤ i ≤ k − 1. R can also be denoted by R: u = x(0) , x(1) , . . . , x(i−1) , R′ , x(j+1) , x(j+2) , . . . , x(k) = v , where R′ is a ⟨x(i) , x(j) ⟩-path, a subpath of R. The subpath R′ , starting from x(i) and ending with x(j) , can be denoted by R′ = (R; x(i) , x(j) ). We use V (R) and E (R) to denote the node set and the edge set in R, respectively. Two ⟨x, y⟩-paths P and Q are edge-disjoint if E (P ) ∩ E (Q ) = ∅. Two ⟨x, y⟩-paths P and Q are internally node-disjoint if they are edge-disjoint and V (P ) ∩ V (Q ) = {x, y}. Two spanning trees T1 and T2 rooted at a node u in graph G are independent if the ⟨u, v⟩-path in T1 and the ⟨u, v⟩-path in T2 are internally node-disjoint for each v ∈ V (G)\{u}. If for any two nodes x and y in V (G), the ⟨x, y⟩-path in T1 and the ⟨x, y⟩-path in T2 are internally node-disjoint, then T1 and T2 are said to be completely independent. A set of spanning trees in G are independent (completely independent) if they are pairwisely independent (completely independent). In this paper, we will use CISTs (ISTs) to refer to completely independent spanning trees (independent spanning trees). 2.2. The crossed cube Let bn−1 bn−2 . . . bi+1 bi bi−1 . . . b0 be a binary string of length n, where bi ∈ {0, 1}. bi is called the ith bit of the string, with bn−1 being the most significant bit and b0 the least significant bit. An n-dimensional crossed cube denoted CQn , has 2n nodes. Each node in CQn is represented by a unique binary string of length n, called the address of the node. The node addresses . . . 1. range from  00  . . . 0 to 11   n

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Fig. 2. (a) CQ3 ; (b) CQ4 .

Definition 2.1 ([9,10]). Two binary strings x = x1 x0 and y = y1 y0 of length two are said to be pair-related (denoted by x ∼ y) if and only if (x, y) ∈ {(00, 00), (10, 10), (01, 11), (11, 01)}. Definition 2.2 ([10]). The n-dimensional crossed cube CQn is recursively defined as follows. CQ1 is the complete (undirected) graph on two nodes whose addresses are 0 and 1. CQn consists of two subcubes CQn0−1 and CQn1−1 . The most significant bit of the addresses of the nodes in CQn0−1 and CQn1−1 are 0 and 1, respectively. The nodes u = un−1 un−2 . . . u0 ∈ CQn0−1 and v = vn−1 vn−2 . . . v0 ∈ CQn1−1 , where un−1 = 0 and vn−1 = 1, are joined by an edge in CQn if and only if (1) un−2 = vn−2 if n is even, and (2) u2i+1 u2i ∼ v2i+1 v2i , for ⌊(n − 1)/2⌋ > i ≥ 0. In Fig. 2, (a) and (b) illustrate CQ3 and CQ4 , respectively. Definition 2.3 ([23]). For integer m with 1 ≤ m ≤ n − 1 and a string s ∈ {0, 1}m , CQns−|s| is defined as the subgraph of CQn induced by the set of nodes whose addresses have the common prefix s, where |s| denotes the length of string s. Lemma 2.1 ([23]). CQns−|s| is isomorphic to CQn−|s| . As suggested by Lemma 2.1, for an integer n with n ≥ 2, if u = un−1 un−2 . . . u0 is an arbitrary node in CQn , u belongs to u u ...u the subgraph of CQn : CQn−n−i 1 n−2 n−i for integer i with 1 ≤ i ≤ n − 1. Suppose that u = un−1 un−2 . . . u0 and v = un−1 un−2 . . . ul ul−1 vl−2 vl−3 . . . v0 are two nodes in CQn . We say that u and v have a leftmost differing bit at position l − 1. When two adjacent nodes u and v have a leftmost differing bit at position d, we say that v is the d-neighbor of u or that the edge (u, v) is an edge of dimension d. Let N (u, d) denote the d-neighbor of u. We adopt the following lemma, which is used to show the relation between u and its neighbors. u

Lemma 2.2 ([5]). Given u = un−1 un−2 . . . u0 ∈ V (CQn ), we have N (u, k) ∈ V (CQk n−1

un−2 ...uk+1 uk

) for 0 ≤ k ≤ n − 1.

3. Construction of completely independent spanning trees in CQn In this section, we first construct two CISTs in a CQ4 . We will then present an algorithm, called 2CIST, to construct two CISTs in CQn , for any n ≥ 4. We will go over the algorithm with some examples, and prove the correctness of Algorithm 2CIST. 3.1. CISTs in CQ4 We can partition CQ4 into four subgraphs: CQ200 , CQ201 , CQ210 , CQ211 . Fig. 3 shows the two completely independent spanning trees in CQ4 , constructed as follows. We first make four Hamiltonian cycles C1 , C2 , C3 , and C4 in CQ200 , CQ201 , CQ210 , and CQ211 so that (0000, 0010), (0100, 0110), (1000, 1010), and (1100, 1110) are edges in these cycles, respectively, shown in Fig. 3(a). C1 C2 C3 C4

: 0000 → 0001 → 0011 → 0010 → 0000 : 0100 → 0101 → 0111 → 0110 → 0100 : 1000 → 1001 → 1011 → 1010 → 1000 : 1100 → 1101 → 1111 → 1110 → 1100.

T1 is obtained by the following steps: First, connect C1 and C2 with edges {(v, N (v, 2))|v ∈ V (C1 ) \ {0010}}; C2 and C4 with {(0100, 1100)}; C3 and C4 with {(v, N (v, 2))|v ∈ V (C3 ) \ {1010}}. Next, remove edges in sets E (C1 ) \ {(0000, 0010)}, {(0100, 0110), (1000, 1010)}, and E (C4 ) \ {(1100, 1110)}. The result is T1 in Fig. 3(b). With similar steps, we can construct another tree T2 , also shown in Fig. 3(b). Note in Fig. 3(b), that if we replace edge (0010, 1010) with (0010, 0110) in T2 , T1 and T2 are also CISTs. In [1], Araki proposed the condition for a network G to

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Fig. 3. Construction of two CISTs in CQ4 .

have CIST-partition G[V1 ] and G[V2 ]: (1) both induced subgraphs G[V1 ] and G[V2 ] are connected, and (2) the bipartite graph B(V1 , V2 , G) has no tree components, that is, every connected component H of B(V1 , V2 , G) satisfies |E (H )| ≥ |V (H )|. We can verify that there exists a CIST-partition in CQ4 . See Fig. 3(c) for example: Let V1 = {0000, 0100, 0101, 0111, 1000, 1001, 1011, 1100} V2 = {0001, 0010, 0011, 0110, 1010, 1101, 1110, 1111}. It is clear that CQ4 satisfies above stated conditions: (1) CQ4 [V1 ] and CQ4 [V2 ] are both connected, and (2) the bipartite graph B(V1 , V2 , CQ4 ) has one component, which is not a tree. Due to the rather small size of T1 and T2 , it can be directly verified that they are indeed completely independent spanning trees. 3.2. The algorithm to construct two CISTs in CQn We now present the algorithm, called 2CIST, that constructs two completely independent spanning trees in CQn . 2CIST is a recursive procedure, with the recursion terminating at base case CQ4 . The two independent spanning trees constructed by 2CIST are of low diameters. During the recursion, a set of connection rules is used to construct two bigger CISTs from two smaller CISTs. The 2CIST algorithm is formally described below. The input n to 2CIST, n ≥ 4, is the dimension of the crossed cube CQn . For the sake of convenience, in the algorithm we will use decimal addresses instead of binary. 3.3. The correctness of Algorithm 2CIST A leaf-node of a tree is a node of degree 1. In the discussion of this section we will need the following theorem from [16] describing an important property of CISTs. Theorem 3.1 ([16]). T1 , T2 , . . . , Tk are CISTs in a graph G if and only if they are edge-disjoint spanning trees and for any node v ∈ V (G), there is at most one Ti such that v is not a leaf in Ti . By Definition 2.2, Lemma 2.2, and Algorithm 2CIST, we have the following lemma. Lemma 3.1. For any node v ∈ Tn0−1,i in CQn0−1 with i = 1, 2, N (v, n − 1) ∈ Tn1−1,i and Tn1−1,i is a tree in CQn1−1 . Let ψ be the operation of adding 2n to a node address, i.e., prefixing ‘‘1’’ to its binary address. E.g. for n = 4, ψ(3) = 3 + 24 = 19, or ψ(0011) = 10011.

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Algorithm 2CIST(n) Input: n, where n ≥ 4; Output: Two completely independent spanning trees in CQn ; Begin 1: if (n == 4) then // Base case Return two trees T4,1 and T4,2 with V (T4,1 ) = V (T4,2 ) = V (CQ4 ) = {0000, 0001, 0010, ..., 1110, 1111};

2:

E (T4,1 ) = {(0000,0010), (0000,0100), (0001,0111), (0011,0101), (0100,0101), (0100,1100), (0101,0111), (0110,0111), (1000,1001), (1000,1100), (1001,1011), (1001,1111), (1010,1011), (1011,1101), (1100,1110)};

3:

E (T4,2 ) = {(0000,0001), (0001,1011), (0001,0011), (0010,0011), (0010,1010), (0011,1001), (0100,0110), (0101,1111), (0110,1110), (0111,1101), (1000,1010), (1010,1110), (1100,1101), (1101,1111), (1110,1111)}; else // Recursion

5:

call 2CIST(n − 1); // Step 1: Recursive call; two CISTs in CQn−1 , Tn−1,1 and Tn−1,2 , returned Build two identical copies of Tn−1,1 /Tn−1,2 , name them as Tn0−1,1 /Tn0−1,2 and Tn1−1,1 /Tn1−1,2 , respectively

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Prefix a ‘‘0’’ to each node’s binary address in Tn0−1,1 , Tn0−1,2 ;

4:

// Re-label all node-addresses in Tn0−1,1 /Tn0−1,2 , Tn1−1,1 /Tn1−1,2

Prefix a ‘‘1’’ to each node’s binary address in Tn1−1,1 , Tn1−1,2 ;

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// Step 2: Combining step: Construct two CISTs Tn,1 and Tn,2 in CQn // out of Tn0−1,1 /Tn0−1,2 and Tn1−1,1 /Tn1−1,2 if (n == 5) then Let Wn,1 = {00000, 00100, 01000, 01100, 01001, 01011} Wn,2 = {00001, 00010, 00011, 00110, 01010, 01110}; else // n ≥ 6

8: 9:

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for i = 1 to 2 do Prefix a ‘‘0’’ to each node’s binary address in Wn−1,i ; Wn,i = Wn−1,i ∪ {N (v, n − 2)|v ∈ Wn−1,i }; // N (v, n − 2) is the (n − 2)-neighbor of v end for end if

11: 12: 13: 14: 15:

for i = 1 to 2 do Ei = E (Tn0−1,i ) ∪ E (Tn1−1,i ); for every w ∈ Wn,i , Ei = Ei ∪ {(w, N (w, n − 1))}; Ei = Ei \E (Tn1−1,i [{N (v, n − 1)|v ∈ Wn,i }]);

16: 17: 18: 19:

V (Ti ) = V ( ) ∪ V (Tn1−1,i ); E (Ti ) = Ei ; end for // End of Step 2 Tn0−1,i

20: 21: 22: 23:

end if

// [{N (v, n − 1)|v ∈ // by {N (v, n − 1)|v ∈ Wn,i }

Wn,i }] is the subgraph induced

// End of recursion

End

Let T ′ be a tree, and T ′′ be a tree obtained from T ′ by: 1. v ∈ V (T ′ ) ⇐⇒ ψ(v) ∈ V (T ′′ ); 2. (v, w) ∈ E (T ′ ) ⇐⇒ (ψ(v), ψ(w)) ∈ E (T ′′ ). That is, T ′′ is just a copy of T ′ with all its nodes’ binary address prefixed with a ‘‘1’’. To prove the correctness of Algorithm 2CIST, we introduce the notion of ψ -CQ -node-set. A ψ -CQ -node-set is a set of nodes in T ′ whose corresponding node-set in T ′′ by ψ -operation is the same as the corresponding node-set by the link rules of the crossed cube. Definition 3.1. Let T ′ be a tree in CQn0−1 , and W a subset in T ′ , for n ≥ 4. W is called a ψ -CQ subtree if T ′ [W ] is a subtree of T ′ and {ψ(v)|v ∈ W } = {N (v, n − 1)|v ∈ W } (i.e. W is a ψ -CQ -node-set). For example, in Fig. 5(a), let W = {0, 4, 8, 12}. {ψ(v)|v ∈ W } = {16, 20, 24, 28} = {N (v, n − 1)|v ∈ W } and T ′ [W ] is a subtree of T ′ , thus W is a ψ -CQ subtree. However in (b), W = {0, 4, 8}. {ψ(v)|v ∈ W } = {16, 20, 24} ̸= {N (v, n − 1)|v ∈ W } = {16, 28, 24}. Lemma 3.2. For any node xn−1 xn−2 . . . x0 ∈ W , if x2j = 1, xn−1 xn−2 ...x2j+3 x2j+2 x2j+1 x2j x2j−1 x2j−2 . . . x1 x0 ∈ W , then W is a

ψ -CQ -node-set in T ′ .

Proof. By Definition 3.1, we only need to prove that {ψ(v)|v ∈ W } = {N (v, n − 1)|v ∈ W }. By Definition 2.2, each node in T ′ u (a tree rooted at u in CQn−n−11 ) has one and exactly one (n − 1)-neighbor. Recall that ψ is an isomorphic mapping from T ′ to T ′′ .

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Fig. 4. Construction of isomorphic tree.

Then we have |{ψ(v)|v ∈ W }| = |{N (v, n − 1)|v ∈ W }| = |W |, which implies that {ψ(v)|v ∈ W } = {N (v, n − 1)|v ∈ W } if {ψ(v)|v ∈ W } ⊆ {N (v, n − 1)|v ∈ W }. The proof of {ψ(v)|v ∈ W } ⊆ {N (v, n − 1)|v ∈ W } is as follows. We only need to prove that for any node p ∈ {ψ(v)|v ∈ W }, there exists a node y ∈ W such that p = N (y, n − 1). Since ψ is an isomorphic mapping from T ′ to T ′′ , there exists a node v = vn−1 vn−2 . . . v0 ∈ W such that ψ(v) = p. We construct y = un−1 yn−2 . . . y0 as follows: For integer j with 0 ≤ j < ⌊(n)/2⌋, let y2j+1 y2j = v2j+1 v2j , if v2j = 1. y2j+1 y2j = v2j+1 v2j , otherwise. Then we can verify that y ∈ W . By Definition 2.2, p = ψ(v) = N (y, n − 1). Hence, the lemma holds.  For example, in Fig. 4, both sets {0, 4, 5, 7, 8, 12, 9, 11} in (a) and {1, 2, 3, 6, 10, 13, 14, 15} in (c) are not ψ -CQ -nodeset in the left tree. By Algorithm 2CIST, the selected ψ -CQ -node-sets shown in the left tree of Fig. 4(b) and the left tree of Fig. 4(d) are {0, 4, 8, 9, 11, 12} and {1, 2, 3, 6, 10, 14}, respectively. There may be many ψ -CQ -node-sets in a given tree. In Fig. 4(a), {0}, {0, 4, 12}, {0, 8}, {0, 9, 11}, {0, 4, 8, 12}, {0, 4, 9, 11, 12}, {0, 8, 9, 11}, {0, 4, 8, 9, 11, 12}, {4, 8, 12}, {4, 9, 11, 12}, {8}, {8, 9, 11}, {11, 12} are all ψ -CQ -node-sets. We present the following lemma regarding Algorithm 2CIST. Lemma 3.3. For any node v in CQn , if v is a leaf node in T1 , then v is an internal node in T2 . Proof. We prove the lemma by induction on n. Base: It is easy to verify that the lemma holds for n = 4, see Fig. 3(b) for example. Now we prove that the lemma holds for n = 5, and Wi constructed by Algorithm 2CIST is a ψ -CQ -node-set for i = 1, 2. Since W1 = {0, 4, 8, 9, 12, 11} and

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(a) W ={0, 4, 12, 8}.

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(b) W ={0, 4, 8}.

Fig. 5. (a) {0, 4, 8, 12} is a ψ -CQ -node-set; (b) {0, 4, 8} is not a ψ -CQ -node-set.

W2 = {1, 2, 3, 6, 10, 14}, let T40,1 and T40,2 be two CISTs in CQ40 and T41,1 and T41,2 be two CISTs in CQ41 . By Lemma 3.2, W1 and W2 are two ψ -CQ -node-sets in T40,1 and T40,2 , respectively. Thus, the internal node set of T1 (resp. T2 ) is the union set of internal node sets of T40,1 and T40,2 (resp. T41,1 and T41,2 ). Consequently, we can verify that the lemma holds.

Induction: Suppose that the lemma holds for n = τ (τ ≥ 5) and W1′ and W2′ are two ψ -CQ -node-sets, that is: (i) For any node v in CQτ , if v is a leaf node in T1 ′ , then v is an internal node in T2 ′ . (ii) W1′ and W2′ are two ψ -CQ -node-sets. We will prove the lemma holds for n = τ + 1. By Definition 2.1, CQτ +1 is composed of CQτ0 and CQτ1 . By Algorithm 2CIST, let TA,i = 0Ti for i = 1, 2. By Algorithm 2CIST and Definition 2.1, we can verify that TB,1 and TB,2 are two CISTs in CQτ1 . As to the construction rule of the set Wi in Ti for i = 1, 2, by Algorithm 2CIST, Wi = Wi′ ∪ {N (v, τ )|v ∈ Wi′ }. By Lemma 3.1, we can verify that Wi is a ψ -CQ -node-set. Thus, each node in Wi is either an internal node in TA,i or an internal node in TB,i for i = 1, 2. As a consequence, we have proved the lemma for n = τ + 1.  Note that T1 and T2 in Fig. 3 are used as T4,1 and T4,2 in Algorithm 2CIST. Since the two trees Tn,1 and Tn,2 in Algorithm 2CIST are constructed recursively using ψ -CQ -node-sets, we can verify that when the input n of Algorithm 2CIST grows by 1, the diameter of the corresponding CISTs both increases by 1. Noting that the initial diameter of T4,1 /T4,2 in CQ4 is 8 (the length of path(T4,1 , 0110, 1010) or path(T4,2 , 0000, 1100) is 8), we state the following obvious lemma. Lemma 3.4. Both T1 and T2 of CQn obtained from Algorithm 2CIST are of diameter n + 4. According to Algorithm 2CIST, we can verify that there are no common edges in T1 and T2 , which implies the following lemma. Lemma 3.5. T1 and T2 are edge-disjoint spanning trees. Note that the two trees in Algorithm 2CIST are constructed separately. Thus they can be constructed in parallel. It is then straightforward to see that the time complexity of Algorithm 2CIST is O(2n ), which is the number of nodes in CQn . Based on Theorem 3.1, Lemmas 3.3–3.5, and Algorithm 2CIST, we have the conclusion in the following theorem. Theorem 3.2. For a CQn , n ≥ 4, Algorithm 2CIST generates two CISTs in O(2n ) steps, with diameter n + 4 for both trees. 4. CISTs in BC networks We now consider the existence of CISTs in networks of more general structures. To be more specific, we examine a particular type of BC networks. The bijective connection networks (BC networks for short) are a general form of networks that includes all networks recursively constructed by (1) combining two components of the same size G0 and G1 ; and (2) adding a unique and only edge between every pair of nodes u ∈ G0 and v ∈ G1 , respectively. The hypercube and all hypercube variants, including CQn , belong to BC networks, with different rules of adding edges. In this section, we first give the original definition of BC networks. Based on that, we define a less general form of BC networks, called CQ-based BC networks, and prove the existence of CISTs in the CQ-based BC networks.

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Definition 4.1 ([13]). Let G be a network, V (G) = V1 ∪ V2 , V1 ̸= ∅, V2 ̸= ∅, and V1 ∩ V2 = ∅. If the following two conditions hold: (1) For any node u ∈ V1 , there exists a unique node v ∈ V2 such that (u, v) ∈ E (G), and (2) For any node v ′ ∈ V2 , there exists a unique node u′ ∈ V1 such that (u′ , v ′ ) ∈ E (G), G

then there exists a bijective connection (BC for short) between the subsets V1 and V2 , denoted by V1 ←→ V2 . Definition 4.2 ([13]). The 1-dimensional BC network X ′ 1 is a complete graph on two nodes 0 and 1. The family of the 1dimensional BC network is defined as L′ 1 = {X ′ 1 }. Let G be a graph. G is an n-dimensional BC network, denoted by X ′ n , if there exist V0 , V1 ⊂ V (G) such that the following three conditions hold: ′ ′ ′ n −1 (1) V0 = 0V0′ and  V1 = 1V1 , where V0 = V1 = {0, 1} ; (2) V (G) = V0 V1 , V0 ̸= ∅, V1 ̸= ∅, and V0 V1 = ∅; and G

(3) V0 ←→ V1 , G[V0 ] ∈ L′ n−1 , and G[V1 ] ∈ L′ n−1 . The family of the n-dimensional BC networks is defined as L′ n = {G|G is an n-dimensional BC network}. Note that BC networks are defined recursively, with X ′ 1 being the base. We now give the definition of the CQ-based BC Networks. Different from general BC networks, the base of CQ-based BC networks is CQ4 , the 4-dimensional crossed cube. Definition 4.3. The 4-dimensional CQ-based BC network X4 is CQ4 . The family of the 4-dimensional CQ-based BC network is defined as L4 = {X4 }. Let G be a graph. G is an n-dimensional CQ-based BC network, denoted by Xn (n ≥ 4), if there exist V0 , V1 ⊂ V (G) such that the following three conditions hold: ′ ′ ′ n−1 (1) V0 = 0V0′ and  V1 = 1V1 , where V0 = V1 = {0, 1} ; (2) V (G) = V0 V1 , V0 ̸= ∅, V1 ̸= ∅, and V0 V1 = ∅; and G

(3) V0 ←→ V1 , G[V0 ] ∈ Ln−1 , and G[V1 ] ∈ Ln−1 . The family of the n-dimensional CQ-based BC networks is defined as Ln = {G|G is an n-dimensional CQ-based BC network}. Intuitively, a CQ-based BC network is composed as follows: 1. The base of the CQ-based BC network is a CQ4 ; 2. When n > 4, an n-dimensional CQ-based BC network is composed by adding a set of edges between two (n − 1)dimensional CQ-based BC networks; 3. The added edges must be bijective. Other than that, there is no any particular rules for adding the edges. We will show that two CISTs can be constructed in such a network. Theorem 4.1. There exist two CISTs in an n-dimensional CQ-based BC network Xn , n ≥ 4. And the number of leaf nodes in each CIST is half the number of nodes in Xn . Proof. We prove the lemma by induction on n. Base: Fig. 3 shows the two CISTs in X4 (X4 is CQ4 ). There are 8 leaf nodes in both T1 and T2 , which is half of the number of nodes in X4 . Thus, the theorem holds for n = 4. Hypothesis: The theorem holds for n = τ (τ ≥ 4), that is, there exist two CISTs in any τ -dimensional CQ-based BC network Xτ , where τ ≥ 4. The number of leaf nodes in each CIST is half of the number of nodes in Xτ . Induction: We will show that the theorem holds for n = τ + 1. By Definition 4.3, suppose Xτ +1 is composed of 0Xτ and 1Xτ′ . By Hypothesis, let T1 and T2 be the two CISTs in 0Xτ , T1 ′ and T2 ′ the two CISTs in 1Xτ′ . Pick an arbitrary internal node u from T1 . Since u is a node in 0Xτ , by Definition 4.3, u has a unique adjacent node w in 1Xτ′ . Since both T1 ′ and T2 ′ in 1Xτ′ have half internal nodes and half leaf nodes, and by Theorem 3.1, all internal nodes in T1 ′ must be leaf nodes in T2 ′ , and vice versa. That means any node w in 1Xτ′ is either an internal node in T1 ′ or an internal node in T2 ′ . We then have the following two cases. Case 1. w is an internal node in T1 ′ . Then, add the bijective edge between T1 and T1 ′ joining nodes u and w , and call the resulting spanning tree T1 ′′ . See Fig. 6. By Theorem 3.1, u must be a leaf node in T2 , and w must be a leaf node in T2 ′ , as illustrated in Fig. 6. Claim. There must exist a bijective edge (v, x), such that v ∈ T2 , x ∈ T2 ′ are both internal nodes. To see this claim, observe Fig. 6 again. There are equal number of internal (black) and leaf (white) nodes in T2 and T2 ′ , respectively. That is,

|V (T2 )| 2

black nodes in T2 , |V (T2 ′ )| 2

|V (T2 ′ )| 2

white nodes in T2 ′ , and

|V (T2 )| 2

=

|V (T2 ′ )| 2

. Since the leaf node w already

has an edge from a leaf u, at most − 1 white nodes can have edges from black nodes. That means at least one black node in T2 will be bijectioned to a black node in T2 ′ . Let (v, x) be this internal-to-internal edge. Using (v, x) to join T2 and T2 ′ , we get spanning tree T2 ′′ . We now have two bigger spanning trees T1 ′′ and T2 ′′ , using an internal-to-internal edge in each tree. The internal nodes remain internal, leaf nodes remain leaf, in both bigger spanning trees. The property of Theorem 3.1 is preserved. By Theorem 3.1, T1 ′′ and T2 ′′ are two CISTs in Xτ +1 . Obviously, the number of leaf nodes in T1 ′′ and T2 ′′ are both half the number of nodes in Xτ +1 . Case 2. w is an internal node in T2 ′ . The proof is similar to Case 1. 

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Fig. 6. Use internal-to-internal edges to build bigger spanning trees.

5. Conclusion We have studied the problem of constructing completely independent spanning trees (CISTs) in the crossed cube. In particular, we presented an algorithm to construct two CISTs in a crossed cube. Based on that, the existence of two CISTs in a crossed cube-based BC network is also discussed. A spanning tree is a very important structure for many network tasks. With two CISTs, one can always designate two entirely different paths between any pair of source/destination nodes. That adds to the fault-tolerability of the underlying network. With some modification, the proposed recursive algorithm could also be applied to other networks of regular structures. The crossed cube, along with other hypercube variant, achieves lower diameter than the hypercube. However, some regularity of the hypercube is lost in the crossed cube. It is of both theoretical and practical interest to understand how this loss of regularity affects the hypercube’s original capability. The result of this paper provides one more perspective on the crossed cube’s competency in terms of fault tolerance. Acknowledgments This work is supported by National Natural Science Foundation of China (No. 61572337), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 14KJB520034), China Postdoctoral Science Foundation Funded Project (No. 2015M581858), and the Jiangsu Planned Projects for Postdoctoral Research Funds (No. 1501089B). References [1] T. Araki, Dirac’s condition for completely independent spanning trees, J. Graph Theory 77 (3) (2014) 171–179. [2] H.-Y. Chang, W. Hung-Lung, Y. Jinn-Shyong, J.-M. Chang, A note on the degree condition of completely independent spanning trees, IEICE Trans. Fundam. Electron., Commun. Comput. Sci. 98 (10) (2015) 2191–2193. [3] Y.-S. Chen, C.-Y. Chiang, C.-Y. Chen, Multi-node broadcasting in all-ported 3-d wormhole-routed torus using an aggregation-then-distribution strategy, J. Syst. Archit. 50 (9) (2004) 575–589. [4] B. Cheng, J. Fan, X. Jia, Dimensional-permutation-based independent spanning trees in bijective connection networks, IEEE Trans. Parallel Distrib. Syst. 26 (1) (2015) 45–53. [5] B. Cheng, J. Fan, X. Jia, J. Wang, Dimension-adjacent trees and parallel construction of independent spanning trees on crossed cubes, J. Parallel Distrib. Comput. 73 (5) (2013) 641–652. [6] B. Cheng, J. Fan, X. Jia, S. Zhang, Independent spanning trees in crossed cubes, Inform. Sci. 233 (2013) 276–289. [7] B. Darties, N. Gastineau, O. Togni, Completely independent spanning trees in some regular networks, 2014, arXiv preprint arXiv:1409.6002. [8] Q. Dong, X. Yang, J. Zhao, Y.Y. Tang, Embedding a family of disjoint 3d meshes into a crossed cube, Inform. Sci. 178 (11) (2008) 2396–2405. [9] K. Efe, A variation on the hypercube with lower diameter, IEEE Trans. Comput. 40 (11) (1991) 1312–1316. [10] K. Efe, The crossed cube architecture for parallel computation, IEEE Trans. Parallel Distrib. Syst. 3 (5) (1992) 513–524. [11] G. Fan, Y. Hong, Q. Liu, Ore’s condition for completely independent spanning trees, Discrete Appl. Math. 177 (2014) 95–100. [12] J. Fan, X. Jia, Embedding meshes into crossed cubes, Inform. Sci. 177 (15) (2007) 3151–3160. [13] J. Fan, X. Jia, X. Liu, S. Zhang, J. Yu, Efficient unicast in bijective connection networks with the restricted faulty node set, Inform. Sci. 181 (11) (2011) 2303–2315.

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