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Fault-tolerant vertex-pancyclicity of locally twisted cubes LTQn
J. Parallel Distrib. Comput. 88 (2016) 57–62
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Fault-tolerant vertex-pancyclicity of locally twisted cubes LTQn ✩ Xirong Xu a,∗ , Yazhen Huang a , Peng Zhang b , Sijia Zhang a a
School of Computer Science and Technology, Dalian University of Technology, Dalian, 116024, PR China
b
School of Computer and Information Engineering, Zhongshan College of Dalian Medical University, Dalian, 116085, PR China
highlights • • • •
The structure of LTQ n and some definitions and notations. We introduce some properties of LTQ n . Investigates the fault-tolerant vertex-pancyclicity of LTQ n . Prove that LTQ n is n-3 fault-tolerant 4-vertex-pancyclicity.
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Article history: Received 16 July 2014 Received in revised form 27 March 2015 Accepted 12 November 2015 Available online 30 November 2015
abstract The n-dimensional locally twisted cube LTQn is a variant of the hypercube, which possesses some properties superior to the hypercube. This paper investigates the fault-tolerant vertex-pancyclicity of LTQn , and shows that if LTQn (n > 3) contains at most n − 3 faulty vertices and/or edges then, for any fault-free vertex u and any integer ℓ with 4 6 ℓ 6 2n − fv except for 5, there is a fault-free cycle of length ℓ containing the vertex u, where fv is the number of faulty vertices. The result is optimal in some senses. © 2015 Elsevier Inc. All rights reserved.
Keywords: Locally twisted cubes Vertex-pancyclicity Fault-tolerant
1. Introduction Interconnection networks play an important role in parallel processing systems. An interconnection network can be represented by a graph G = (V , E ), where V represents the vertex set and E represents the edge set. The capacity of embedding other existing network into an interconnection network is a critical issue in evaluating an interconnection network. Suppose that some process can be naturally decomposed into a collection of subprocesses that can be executed concurrently with certain communication among subprocesses. One obtains a graph by denoting each subprocess by a vertex and each communication between subprocesses by an edge between the corresponding vertices. The problem of allocating the subprocesses to processors in the given network can be modeled by the following graph embedding problem: given a host graph G2 = (V2 , E2 ), which represents the network into which other networks are to be embedded, and a guest
✩ The work was supported by NNSF of China (No.61472465, 61170303, 61562066) and Scientific Research Fund of Liaoning Provincial Education Department (No.L2013337). ∗ Corresponding author. E-mail address: [email protected] (X. Xu).
http://dx.doi.org/10.1016/j.jpdc.2015.11.002 0743-7315/© 2015 Elsevier Inc. All rights reserved.
graph G1 = (V1 , E1 ), which represents the network to be embedded, the problem is to find a mapping from each node of G1 to a node of G2 , and a mapping from each edge of G1 to a path in G2 [27]. One common measure of effectiveness of an embedding is the dilation. The dilation of embedding ψ is defined as dil(G1 , G2 , ψ) = max{dist(G2 , ψ(u), ψ(v))|(u, v) ∈ E1 }, where dist(G2 , ψ(u), ψ(v)) denotes the distance between the two nodes ψ(u) and ψ(v) in G2 . The smaller the dilation of an embedding is, the shorter the communication delay that the graph G2 simulates the graph G1 [1]. As two common guest graphs, linear arrays (i.e. paths) [9,8,7] and rings (i.e. cycles) [2,6,15] are two fundamental networks for parallel and distributed computing. In large interconnection networks, nodes or edges tend to become faulty. It is important to find an embedding of a guest graph into a host graph where all faulty nodes and edges have been removed. This is called fault-tolerant embedding. Much work has been done on the fault-tolerant embedding [3,10,12,19,21,20,17, 4,13]. The locally twisted cube has many properties superior to hypercube. Though both the locally twisted cube and the ordinary hypercube have the same number of vertices and the same vertex degree, the diameter of the locally twisted cube is approximately half that of the ordinary hypercube.
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In this paper, we are interested in the path and/or cycle embedding properties of the n-dimensional locally twisted cube LTQ n . Yang et al. proposed this new network [26] and proved that LTQ n contains cycles of all lengths from 4 to 2n [27]. Ma, Xu [23] and Hu et al. [16], independently, improved this result by proving that for any edge in LTQ n there are cycles of all lengths containing it. Ma and Xu [22] further improved this result by showing that for any two different vertices x and y with distance d in LTQ n , there exist xy-paths of all lengths from d to 2n − 1 except for d + 1. Even when faulty elements occur, Chang et al. [5] and Park et al. [24], independently, showed that LTQ n still contains fault-free cycles of all lengths provided that faulty elements do not exceed n − 2. Very recently, Han et al. [11] have showed that LTQ n with at most n − 3 faulty elements contains paths of all lengths from 2n−1 − 1 to 2n −fv −1 between any two distinct fault-free vertices, where fv is the number of faulty vertices. Hsieh and Wu [14] have considered more faulty edges and showed that LTQ n contains a fault-free Hamiltonian cycle provided that faulty edges do not exceed 2n − 5 and each vertex is incident with at least two fault-free edges. This condition is natural since, in practical applications, the probability is small for a vertex x being isolated (all links incident with x are faulty) or pendant (only one link incident with x is fault-free and the others are all faulty). In this paper, we show that for any vertex u ∈ V (LTQ n ) and any integer 4 6 ℓ 6 2n − fv except for 5, there exists cycle C of length ℓ in LTQ n such that u is in C if fe + fv 6 n − 3. The approach we use is based on the recursive construction of LTQ n . The remaining part of this paper is organized as follows. In Section 2, we recall the structure of LTQ n , and some definitions and notations. In Section 3, we introduce some properties of LTQ n to be used in our proofs. In Section 4, we give the proof of our result. Finally, we give some concluding remarks in Section 5. 2. Preliminaries In this section, we will give some definitions and properties about LTQ n . A graph G = (V , E ) consists of a vertex-set V and an edge-set E, where V = V (G) is a finite set and E = E (G) is a subset of xy–xy is an unordered pair of V . Two vertices x and y are adjacent if xy is an edge of G, and which are also the endvertices of xy. For a vertex x, the vertex adjacent to x is called as the neighbor of x. The degree of a vertex x is the number of edges incident with it. A graph is called k-regular if each vertex has degree k. For two distinct vertices x and y, an xy-path between x and y is a sequence of distinct vertices in which any two consecutive vertices are adjacent. The length of a path is the number of edges on the path. An xy-path of length at least three is called a cycle if x = y. A connected subgraph of G is called a spanning tree if it contains all vertices of G and no cycles, in which a distinguished vertex is called the root of the spanning tree. The distance between two distinct vertices x and y in G is the length of a shortest xy-path in G, and the diameter of G is the maximum distance between any two vertices. A non-empty subset M of E (G) is called a matching of G if no two of its elements have a common end vertices in G. A matching M is perfect if every vertex of G is an end-vertex of some edge in M. We now recall the definition of the n-dimensional locally twisted cube, proposed by Yang, Evans and Megson [26], which has 2n vertices, and each vertex is an n-string on {0, 1}. Definition 1 ([26]). The n-dimensional locally twisted cube, denoted by LTQ n (n > 2), is recursively defined as follows.
(1) LTQ 2 is a graph isomorphic to Q2 . (2) For n > 3, LTQ n is built from disjoint copies of LTQ n−1 according to the following steps. Let LTQ 0n−1 and LTQ 1n−1 denote graphs obtained by prefixing labels of each vertex of one copy of LTQ n−1 with 0 and with 1, respectively, and connect a vertex x = 0x2 x3 . . . xn of LTQ 0n−1 with another vertex y = 1(x2 + xn )x3 . . . xn of LTQ 1n−1 by an edge xy, where ‘+’ represents the modulo 2 addition. The graphs shown in Fig. 1 are LTQ 3 and LTQ 4 . The locally twisted cube LTQ n can be equivalently defined with the following non-recursive fashion. Definition 2 ([26]). For n > 2, the n-dimensional locally twisted cube LTQ n is a graph with n-strings on {0, 1} as the vertex set. Two vertex x = x1 x2 . . . xn−1 xn and y = y1 y2 . . . yn−1 yn of LTQ n are adjacent if and only if either (a) xi = y¯ i and xi+1 = yi+1 + xn for some 1 6 i 6 n − 2, and xj = yj for all the remaining bits, where ‘+’ represents the modulo 2 addition, or (b) xi = y¯ i for some i ∈ {n − 1, n}, and xi = yi for all the remaining bits. According to the above definition, it is not difficult to see that LTQ n is an n-regular graph with 2n vertices and n2n−1 edges. From the definition, LTQ n can be expressed as the union of two disjoint copies of LTQ n−1 by adding a perfect matching between them according to the specified rule. For short, we often write LTQ n = L ⊕ R, where L ∼ = LTQ 0n−1 and R ∼ = LTQ 1n−1 . We now make some remarks on the n-dimensional locally twisted cube. Firstly, like to many variants of the hypercube such as the twisted cube, the crossed cube, the augmented cube and otherwise, the locally twisted cube not only keeps many nice properties of the hypercube such as regularity, high connectivity and high recursive constructability, but also has diameter of about half of that of the hypercube of the same size. Secondly, the locally twisted cube also keeps a nice property of the hypercube, that is, the labels of any two adjacent vertices differ in at most two successive bits. However, a common feature of the above-mentioned variants is that the labels of some neighbor vertices may differ in a large number of bits. As a result, a portion of good properties of hypercube are lost in these variants. For example, the design of efficient parallel algorithms on these variants turns out to be a difficult task [26]. Thirdly, the locally twisted cube LTQ n contains cycles of all lengths from 4 to 2n [27], but the hypercube Qn contains only even cycles since it is a bipartite graph. Thus, LTQ n is superior to Qn in cycle embedding property. Fourthly, the construction of the locally twisted cube LTQ n is quite different from that of the twisted cube TQn . The former is defined for any positive integer n, while the latter only for odd integer. Lastly, it should be noted that, like to many variants of the hypercube, the locally twisted cube LTQ n is not vertex-transitive for n > 4 proved by Liu et al. [18]. 3. Properties In this section, we introduce some properties of LTQ n to be used in our proofs in Section 4. Yang, Evans and Megson [26] found an isomorphic expression of LTQ n . For example, two graphs shown in Fig. 2 are other expressions of LTQ 3 and LTQ 4 , respectively. In general, they proved the following result.
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Fig. 1. The locally twisted cubes LTQ 3 and LTQ 4 .
Fig. 2. Another painting of LTQ 3 and LTQ 4 .
Lemma 1 ([26]). Let L′ be the graph obtained from Qn−1 by suffixing the labels of all vertices with 0, R′ be the graph obtained from a graph isomorphic to Qn−1 by suffixing the labels of all vertices with 1. Then LTQ n is isomorphic to the graph obtained from L′ and R′ by adding a perfect matching M between them, denoted by LTQ n = L′ ⊕ R′ , where L′ ∼ = Qn0−1 and R′ ∼ = Qn1−1 and M is the set of edges by linking two vertices with difference only in suffixes. Let LTQ n = L ⊕ R defined in Lemma 1. For convenience, for a vertex u in LTQ n , if u is in L, we write uL for u, and use uR to denote its neighbor in R. Let uL and vL be two adjacent vertices in L. We say vL is a strong neighbor of uL in L if their neighbors uR and vR are adjacent in R (see Fig. 3(a)), and a weak neighbor of uL in L if their neighbors uR and vR are not adjacent in R (see Fig. 3(b)). Similarly, we can define a strong neighbor or a weak neighbor of two adjacent vertices uR and vR in R. For example, in LTQ 4 (see Fig. 1), consider uL = 0000. Since vL = 0100 is the neighbor of uL in L, then vL is a strong neighbor of uL in L because their neighbors uR = 1000 and vR = 1100 are adjacent in R. Since vL′ = 0001 is the other neighbor of uL in L, then vL′ is a weak neighbor of uL in L because their neighbors uR = 1000 and vR′ = 1101 are not adjacent in R. Lemma 2 ([25]). Let LTQ n = L ⊕ R. If n > 4 then, for any vertex uL in L, there are n − 2 strong neighbors and one weak neighbor in L. Moreover, if wL is the weak neighbor of uL , then the distance between uR and wR is two in R. The same conclusion holds for any vertex uR in R. Lemma 3 ([22]). Let x and y be any two different vertices in LTQ n and the distance between them be d. Then for any integer ℓ with d 6 ℓ 6 2n − 1 except for d + 1, there exists an xy-path of length ℓ in LTQ n for n > 3.
Lemma 4 ([11]). If fv + fe 6 n − 3 and n > 3, then for any integer ℓ with 2n−1 − 1 6 ℓ 6 2n − fv − 1, there is a fault-free path of length ℓ between any two distinct vertices in LTQ n . Lemma 5 ([16]). Let u be any vertex in LTQ n and ℓ be any integer with 4 6 ℓ 6 2n . Then there is a cycle of length ℓ containing the vertex u in LTQ n for n > 2. 4. Proof of theorem In this section, we investigate the fault-tolerant vertexpancyclicity of LTQ n and show that LTQ n is (n − 3)-fault-tolerant vertex-pancyclic. We state this result as the following theorem. Theorem 1. If fv + fe 6 n − 3 and n > 3 then, for any fault-free vertex u in LTQ n and any integer ℓ with 4 6 ℓ 6 2n − fv except for 5, there is a fault-free cycle of length ℓ containing the vertex u in LTQ n . Proof of Theorem. We use the expression LTQ n = L ⊕ R, where L ∼ = LTQ 0n−1 and R ∼ = LTQ 1n−1 . Let F be a set of faulty elements in LTQ n with |F | = fv + fe , FL = F ∩ L, FR = F ∩ R, FC = F ∩ EC , EC represents the crossed edges connecting L and R. F v = F ∩ V (LTQ n ), FLv = FL ∩ V (L) and FRv = FR ∩ V (R). Without loss of generality, we may assume |FR | 6 |FL |. Let u be an arbitrary fault-free vertex in LTQ n and ℓ be any integer with 4 6 ℓ 6 2n − fv except for 5. We need to prove that there exists a fault-free cycle of length ℓ containing the vertex u in LTQ n if |F | 6 n − 3 and n > 3. We proceed by induction on n > 3. For n = 3, there are no faulty vertices or edges in LTQ 3 , and so the theorem holds by Lemma 5.
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Fig. 3. (a) vL is strong neighbor of uL in L. (b) vL is weak neighbor of uL in L.
Assume that the theorem holds for LTQ n−1 . We consider LTQ n for n > 4. Case 1. |FL | = n − 3. In this case |FC | = |FR | = 0. Let u be a fault-free vertex in LTQ n . Since LTQ n is n-regular and |F | = n − 3, there exists a fault-free edge joined with u, denoted by (u, u1 ). Let ℓ = ℓ′ + 1. If 2n−1 − 1 6 ℓ′ 6 2n − fv − 1 then, by Lemma 4, there exists a fault-free uu1 -path P of length ℓ′ in LTQ n . Thus, P + uu1 is a fault-free cycle of length ℓ containing the vertex u in LTQ n . Thus, we only need to consider ℓ with 4 6 ℓ 6 2n−1 except for 5. Subcase 1.1. The fault-free vertex u is in R. Since |FR | = 0 and by Lemma 5, there is a fault-free cycle of length ℓ containing the vertex u for any ℓ with 4 6 ℓ 6 2n−1 . Thus, we only need to consider one case according as the fault-free vertex u is in L. Subcase 1.2. The fault-free vertex u is in L. Since LTQ 0n−1 is (n − 1)-regular, then the fault-free vertex u has n − 1 neighbor vertices in LTQ 0n−1 , in which there are n − 2 strong neighbors and one weak neighbor by Lemma 2. Moreover, since the faulty elements have n − 3 in LTQ 0n−1 , the worst case is that n − 3 faulty elements are all faulty vertices which are linked with the fault-free vertex u. Then there exist at least two fault-free edges joined with the fault-free vertex u, denoted by (u, u1 ) and (u, u2 ). It implies that u1 and u2 are both strong neighbors of u, or, one of u1 and u2 is strong neighbor and another is weak neighbor of u, without loss of generality, assume u1 is strong neighbor and u2 is weak neighbor of u. Suppose that u1 and u2 are both strong neighbors of u in L. Consider the strong neighbor u1 of u, let v and v1 be neighbors of u and u1 in R, respectively. By Lemma 2, vv1 ∈ E (R). Since |FC | = |FR | = 0, the cycle (u, u1 , v1 , v) of length 4 contains the vertex u and is fault-free. For any ℓ with 4 6 ℓ 6 2n−1 except for 5, let ℓ = ℓ′ + 3. Then 1 6 ℓ′ 6 2n−1 − 3 except 2. By Lemma 3, there exists a path P (v, v1 ) of length ℓ′ in R. So P (v, v1 )+ u1 v1 + uu1 + uv is a fault-free cycle of length ℓ containing the vertex u (see Fig. 4). Suppose that u1 is the strong neighbor and u2 is the weak neighbor of u in L. Similarly, it suffices to consider the strong neighbor u1 of u that we can find a fault-free cycle of length ℓ with 4 6 ℓ 6 2n−1 except for 5 containing the fault-free vertex u. Case 2. |FL | 6 n − 4. Since the fault-free vertex u ∈ V (LTQ n ), it may be in L or R. Without loss of generality, we may assume that the vertex u is in L. Since |FR | 6 n − 4 and according to Case 1, let uu1 is a faultfree edge joined with u. If 4 6 ℓ 6 2n−1 − |FLv | except 5 then, by the induction hypothesis, there is a fault-free cycle of length ℓ containing the vertex u in L, so in LTQ n . Thus, we only need to consider such an ℓ that satisfies 2n−1 − |FLv | + 1 6 ℓ 6 2n − fv . If n = 4, since 0 6 |FLv | 6 |FL | 6 n − 4 = 0, then 24−1 + 1 = 9 6 ℓ 6 24 − fv . Let ℓ = ℓ′ + 1. Then 8 6 ℓ′ 6 24 − fv − 1. By Lemma 4, there is a fault-free uu1 -path P of length ℓ′ in LTQ 4 , and so P + uu1 is a fault-free cycle of length ℓ containing the vertex u (see Fig. 5(a)).
Fig. 4. The illustration of Subcase 1.2.
Fig. 5. The illustration of Case 2. Table 1 Cycles of length 5 containing the vertex uL = 00001. (00001 00000 00010 00110 00111 00001) (00001 00000 00100 00101 00111 00001) (00001 00000 00100 00101 00011 00001) (00001 00000 00100 00110 00111 00001) (00001 00000 00100 01100 01101 00001) (00001 00000 01000 01100 01101 00001) (00001 00000 01000 11000 11001 00001) (00001 00000 10000 11000 11001 00001) (00001 00011 00010 00110 00111 00001)
Now, assume n > 5 and write ℓ = ℓ1 + 1 + ℓ2 , where 2n−2 − |FLv | 6 ℓ1 6 2n−1 − |FLv | and 2n−2 6 ℓ2 6 2n−1 − |FRv | − 1. Since 2n−2 − |FLv | > 2n−2 − |FL | > 2n−2 − n + 4 > 4 for n > 5, by the induction hypothesis, there is a fault-free cycle C of length ℓ1 containing the vertex u in L. Consider the following inequality.
ℓ1 − |FC | − |FR | − |u| > 2n−2 − |FLv | − |FC | − |FR | − 1 > 2n−2 − |F | − 1 > 2n−2 − n + 2.
Let f (x) = 2x−2 − x + 2. Since f ′ (x) = 2x−2 ln 2 − 1 > 0 for x ≥ 5, f (x) is an increasing function, which implies that ℓ1 − |FC | − |FR | > f (5) = 25−2 − 5 + 2 = 5. In other words, there exists at least two vertices u1 and u2 that distinct u on C and two edges (u1 , v1 ), (u2 , v2 ) are fault-free. Since |FR | 6 n − 4, by Lemma 4, there is a fault-free v1 v2 -path P of length ℓ2 in R. So C − u1 u2 + u1 v1 + u2 v2 + P is a fault-free cycle of length ℓ(=ℓ1 + 1 + ℓ2 ) containing u (see Fig. 5(b)). The proof of the theorem is complete.
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Table 2 Cycles of length 6 containing the vertex u = 0010. (0010 0000 0001 0111 0101 0011 0010) (0010 0000 1000 1001 1111 0011 0010) (0010 0110 0111 0101 0100 0000 0010) (0010 0110 0100 1100 1000 0000 0010) (0010 0110 1110 1100 0100 0000 0010) (0010 1010 1011 1001 1000 0000 0010) (0010 1010 1011 1001 0101 0011 0010) (0010 1010 1011 1101 0001 0000 0010) (0010 1010 1011 0111 0101 0011 0010) (0010 1010 1011 0111 0001 0011 0010) (0010 1010 1000 1001 0101 0011 0010) (0010 1010 1000 1100 0100 0110 0010) (0010 1010 1000 0000 0001 0011 0010) (0010 1010 1110 1100 1000 0000 0010) (0010 1010 1110 1100 0100 0000 0010)
5. Conclusion and remarks As one of the most fundamental networks for parallel and distributed computation, cycles are suitable for developing simple algorithms with low communication cost. Edge and/or vertex failures are inevitable when a large parallel computer system is put in use. Therefore, the fault-tolerant capacity of a network is a critical issue in parallel computing. The fault-tolerant pancyclicity of an interconnection network is a measure of its capability of implementing ring-structured parallel algorithms in a communication-efficient fashion in the presence of faults. In view of the fact that the hypercube network Qn contains only even cycles, LTQ n is superior to Qn in fault-tolerant pancyclicity. This shows that, when the locally twisted cube is used to model the topological structure of a large-scale parallel processing system, our result implies that the system has larger capability of implementing ring-structured parallel algorithms in a communication-efficient fashion in the hybrid presence of edge and vertex failures than one of the hypercube network. We make some remarks on the optimality of our result in the following sense. (1) Consider the vertex uL ∈ L if we write LTQ n = L ⊕ R. By Lemma 3, there are only two kind distinct cycles of length ℓ = 5 containing the vertex uL , which are obtained by the strong neighbor vL or the weak neighbor vL′ . If vL or/and vL′ is faulty, then there are no fault-free cycles of length ℓ = 5 containing the vertex uL in LTQ n . For example, in LTQ 5 , taking uL = 00001, all the cycles of length ℓ = 5 containing the vertex uL = 00001 calculated by the computer are given in Table 1. According to above results, we find that if the vertex x = 00000 and the vertex y = 00011 are faulty, then there are no fault-free cycles of length ℓ = 5 containing the vertex uL = 00001 in LTQ 5 . (2) If fv + fe = n − 2, then the theorem does not hold for n ≥ 4. For any n ≥ 3, we claim that there exists a vertex of LTQ n such that, with n − 2 faulty vertices and/or edges, there is no fault-free cycle of length ℓ = 4 containing the vertex. Let u = u1 u2 . . . un = 0n (n consecutive 0’s) be a vertex of LTQ n . One can check that any cycle of length ℓ = 4 containing u must contain v = u1 u2 . . . ui−1 u¯ i . . . un for some 2 6 i 6 n. Therefore, if F = {u1 u2 . . . ui−1 u¯ i . . . un |i is an integer for 2 6 i 6 n}, there is no cycle of length ℓ = 4 containing u in LTQ n − F . For example, let n = 4 and u = 0010, the four cycles of length ℓ = 4 containing u = 0010 are {0010 0011 0001 0000 0010}, {0010 0110 0100 0000 0010}, {0010 1010 1000 0000 0010}, {0010 1010 1110 0110 0010}. With F = {0000, 0110} or F = {0000, (1110, 0110)}, there is no faultfree cycle of length ℓ = 4 containing u = 0010 in LTQ 4 − F . Consider the vertex u = 0010 in LTQ 4 , all the cycles of length ℓ = 6 containing the vertex u = 0010 calculated by the computer are listed in Table 2. According to the listed results, we find that if the vertex x = 0000 and the vertex y = 1010 are both faulty
(0010 0000 0001 1101 1111 0011 0010) (0010 0000 1000 1001 0101 0011 0010) (0010 0110 0100 0000 0001 0011 0010) (0010 0110 1110 1100 1000 0000 0010) (0010 0110 1110 1010 1000 0000 0010) (0010 1010 1011 1001 1111 0011 0010) (0010 1010 1011 1101 1111 0011 0010) (0010 1010 1011 1101 0001 0011 0010) (0010 1010 1011 0111 0001 0000 0010) (0010 1010 1000 1001 1111 0011 0010) (0010 1010 1000 1100 1110 0110 0010) (0010 1010 1000 1100 0100 0000 0010) (0010 1010 1000 0000 0100 0110 0010) (0010 1010 1110 1100 0100 0110 0010) (0010 1010 1110 0110 0100 0000 0010)
or the vertex x = 0000 and the edge e = (0000, 1010) are both faulty, namely, F = {0000, 1010} or F = {0000, (0000, 1010)}, then there is no fault-free cycles of length ℓ = 6 containing the vertex u = 0010 in LTQ 4 − F . Acknowledgments The authors would like to express their gratitude to the anonymous referees for their kind suggestions and comments on the original manuscript, which resulted in this version. References [1] L. Auletta, A. Rescigno, V. Scarano, Embedding graphs onto the supercube, IEEE Trans. Comput. 44 (1995) 593–597. http://dx.doi.org/10.1109/12.376173. [2] M. Bae, B. Bose, Edge disjoint Hamiltonian cycles in k-ary n-cubes and hypercubes, IEEE Trans. Comput. 52 (10) (2003) 1271–1284. http://dx.doi.org/10.1109/TC.2003.1234525. [3] M. Chan, S. Lee, Fault-tolerant embedding of complete binary trees in hypercubes, IEEE Trans. Parallel Distrib. Syst. 4 (3) (1993) 277–288. http://dx.doi.org/10.1109/71.210811. [4] N.-W. Chang, S.-Y. Hsieh, Fault-tolerant bipancyclicity of faulty hypercubes under the generalized conditional-fault model, IEEE Trans. Commun. 59 (2011) 3400–3409. http://dx.doi.org/10.1109/TCOMM.2011.093011.100321. [5] Q. Chang, M. Ma, J. Xu, Fault-tolerant pancyclicity of locally twisted cubes, J. Univ. Sci. Technol. China 36 (2006) 607–610 (in Chinese). [6] J.-M. Chang, J.-S. Yang, Fault-tolerant cycle-embedding in alternating group graphs, Appl. Math. Comput. 197 (2) (2008) 760–767. http://dx.doi.org/10.1016/j.amc.2007.08.010. [7] J. Fan, X. Jia, Edge-pancyclicity and path-embeddability of bijective connection graphs, Inform. Sci. 178 (2) (2008) 340–351. http://dx.doi.org/10.1016/j.ins.2007.08.012. [8] J. Fan, X. Jia, X. Lin, Optimal embeddings of paths with various lengths in twisted cubes, IEEE Trans. Parallel Distrib. Syst. 18 (4) (2007) 511–521. http://dx.doi.org/10.1109/TPDS.2007.1003. [9] J. Fan, X. Lin, X. Jia, Optimal path embedding in crossed cubes, IEEE Trans. Parallel Distrib. Syst. 16 (12) (2005) 1190–1200. http://dx.doi.org/10.1109/TPDS.2005.151. [10] J. Fan, X. Lin, Y. Pan, X. Jia, Optimal fault-tolerant embedding of paths in twisted cubes, J. Parallel Distrib. Comput. 67 (2) (2007) 205–214. http://dx.doi.org/10.1016/j.jpdc.2006.04.004. [11] Y. Han, J. Fan, J. Yang, P. Qian, Path embedding in faulty locally twisted cubes, in: 2009 2nd IEEE International Conference on Computer Science and Information Technology, Vol. 3, 2009, pp. 214–218. [12] S.-Y. Hsieh, N.-W. Chang, Hamiltonian path embedding and pancyclicity on the Mobius cube with faulty nodes and faulty edges, IEEE Trans. Comput. 55 (7) (2006) 854–863. http://dx.doi.org/10.1109/TC.2006.104. [13] S.-Y. Hsieh, C.-W. Lee, Pancyclicity of restricted hypercube-like networks under the conditional fault model, SIAM J. Discrete Math. 23 (4) (2010) 2100–2119. http://dx.doi.org/10.1137/090753747. [14] S.-Y. Hsieh, C.-Y. Wu, Edge-fault-tolerant hamiltonicity of locally twisted cubes under conditional edge faults, J. Comb. Optim. 19 (1) (2010) 16–30. http://dx.doi.org/10.1007/s10878-008-9157-x. [15] S.-Y. Hsieh, P.-Y. Yu, Cycle embedding on twisted cubes, Extended abstract, in: Seventh International Conference on Parallel and Distributed Computing, Applications and Technologies, Proceedings, 2006, pp. 102–104. [16] K.S. Hu, S.-S. Yeoh, C. Chen, L.-H. Hsu, Node-pancyclicity and edge-pancyclicity of hypercube variants, Inform. Process. Lett. 102 (2007) 1–7. [17] C.-W. Lee, S.-Y. Hsieh, Pancyclicity of matching composition networks under the conditional fault model, IEEE Trans. Comput. 61 (2012) 278–283. http://dx.doi.org/10.1109/TC.2010.229.
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[18] Y.-J. Liu, W.Y. Chou, J.K. Lan, C. Chen, Constructing independent spanning trees for hypercubes and locally twisted cubes, in: Proceedings of the 2009 10th International Symposium on Pervasive Systems, Algorithms, and Networks, ISPAN 2009, 2009, pp. 17–22. http://dx.doi.org/10.1109/I-SPAN.2009.97. [19] R. Lo, G. Chen, Embedding Hamiltonian paths in faulty arrangement graphs with the backtracking method, IEEE Trans. Parallel Distrib. Syst. 12 (2) (2001) 209–222. http://dx.doi.org/10.1109/71.910874. [20] M. Ma, G. Liu, X. Pan, Path embedding in faulty hypercubes, Appl. Math. Comput. 192 (1) (2007) 233–238. http://dx.doi.org/10.1016/j.amc.2007.03.003. [21] M. Ma, G. Liu, J.-M. Xu, Fault-tolerant embedding of paths in crossed cubes, Theoret. Comput. Sci. 407 (1–3) (2008) 110–116. http://dx.doi.org/10.1016/j.tcs.2008.05.002. [22] M. Ma, J.-M. Xu, Panconnectivity of locally twisted cubes, Appl. Math. Lett. 19 (7) (2006) 673–677. http://dx.doi.org/10.1016/j.aml.2005.08.021. [23] M. Ma, J.-M. Xu, Weak edge-pancyclicity of locally twisted cubes, Ars Combin. 89 (2008) 89–94. [24] J.-H. Park, H.-S. Lim, H.-C. Kim, Panconnectivity and pancyclicity of hypercubelike interconnection networks with faulty elements, Theoret. Comput. Sci. 377 (1–3) (2007) 170–180. http://dx.doi.org/10.1016/j.tcs.2007.02.029. [25] X. Xu, W. Zhai, J.-M. Xu, A. Deng, Y. Yang, Fault-tolerant edge-pancyclicity of locally twisted cubes, Inform. Sci. 181 (11) (2011) 2268–2277. http://dx.doi.org/10.1016/j.ins.2011.01.031. [26] X. Yang, D. Evans, G. Megson, The locally twisted cubes, Int. J. Comput. Math. 82 (4) (2005) 401–413. http://dx.doi.org/10.1080/0020716042000301752. [27] X. Yang, G. Megson, D. Evans, Locally twisted cubes are 4-pancyclic, Appl. Math. Lett. 17 (8) (2004) 919–925. http://dx.doi.org/10.1016/j.aml.2003.10.009.
interest includes Theory and Application of Graphs, Topological Structure and Analysis of Interconnection Networks, Fault Tolerance, Algorithms Design and Analysis.
Xirong Xu received her B.S. degree in Applied Mathematics from East China University of Science and Technology, China in June 1990 and her M.S. degree in Computer Application Technology from Dalian University of Technology, China in June 2002, and Ph.D. degree in Computer Software and Theory, Dalian University of Technology, China in June 2005. She worked in mathematics postdoctoral research station at the University of Science and Technology of China from 2005 to 2007. She is an associate professor at School of Computer Science and Technology, Dalian University of Technology, China. Her current research
Sijia Zhang received her B.S. degree in Information Management and Information System from Liaoning Technical University, China in June 2005 and her M.S. degree in Computer Application Technology from Dalian University of Technology, China in June 2010. She is currently working towards her Ph.D. degree in Computer Software and Theory at Dalian University of Technology, China. Her current research interest includes Topological Structure and Analysis of Interconnection Networks, Graph Theory, Algorithms Design and Analysis.
Yazhen Huang received her B.S. degree in Computer science and Technology from Henan Normal University, China in June 2012. She is currently a graduate student and working towards her M.S. degree in Computer Applied Technology at Dalian University of Technology, China. Her current research interest includes Topological Structure and Analysis of Interconnection Networks, Graph Theory, Algorithms Design and Analysis and Fault Tolerance.
Peng Zhang received his B.S. degree in Clinical Medicine from Dalian Medical University, China in June 2004 and his M.S. degree in Computer Applied Technology from Dalian University of Technology, China in June 2013. He is an associate professor at College of Computer and Information Engineering, Dalian Medical University, China. His current research interest includes Computer Networking Technology, Computer Applied Technology, Combination of information technology network and medicine.
Contents lists available at ScienceDirect
J. Parallel Distrib. Comput. journal homepage: www.elsevier.com/locate/jpdc
Fault-tolerant vertex-pancyclicity of locally twisted cubes LTQn ✩ Xirong Xu a,∗ , Yazhen Huang a , Peng Zhang b , Sijia Zhang a a
School of Computer Science and Technology, Dalian University of Technology, Dalian, 116024, PR China
b
School of Computer and Information Engineering, Zhongshan College of Dalian Medical University, Dalian, 116085, PR China
highlights • • • •
The structure of LTQ n and some definitions and notations. We introduce some properties of LTQ n . Investigates the fault-tolerant vertex-pancyclicity of LTQ n . Prove that LTQ n is n-3 fault-tolerant 4-vertex-pancyclicity.
article
info
Article history: Received 16 July 2014 Received in revised form 27 March 2015 Accepted 12 November 2015 Available online 30 November 2015
abstract The n-dimensional locally twisted cube LTQn is a variant of the hypercube, which possesses some properties superior to the hypercube. This paper investigates the fault-tolerant vertex-pancyclicity of LTQn , and shows that if LTQn (n > 3) contains at most n − 3 faulty vertices and/or edges then, for any fault-free vertex u and any integer ℓ with 4 6 ℓ 6 2n − fv except for 5, there is a fault-free cycle of length ℓ containing the vertex u, where fv is the number of faulty vertices. The result is optimal in some senses. © 2015 Elsevier Inc. All rights reserved.
Keywords: Locally twisted cubes Vertex-pancyclicity Fault-tolerant
1. Introduction Interconnection networks play an important role in parallel processing systems. An interconnection network can be represented by a graph G = (V , E ), where V represents the vertex set and E represents the edge set. The capacity of embedding other existing network into an interconnection network is a critical issue in evaluating an interconnection network. Suppose that some process can be naturally decomposed into a collection of subprocesses that can be executed concurrently with certain communication among subprocesses. One obtains a graph by denoting each subprocess by a vertex and each communication between subprocesses by an edge between the corresponding vertices. The problem of allocating the subprocesses to processors in the given network can be modeled by the following graph embedding problem: given a host graph G2 = (V2 , E2 ), which represents the network into which other networks are to be embedded, and a guest
✩ The work was supported by NNSF of China (No.61472465, 61170303, 61562066) and Scientific Research Fund of Liaoning Provincial Education Department (No.L2013337). ∗ Corresponding author. E-mail address: [email protected] (X. Xu).
http://dx.doi.org/10.1016/j.jpdc.2015.11.002 0743-7315/© 2015 Elsevier Inc. All rights reserved.
graph G1 = (V1 , E1 ), which represents the network to be embedded, the problem is to find a mapping from each node of G1 to a node of G2 , and a mapping from each edge of G1 to a path in G2 [27]. One common measure of effectiveness of an embedding is the dilation. The dilation of embedding ψ is defined as dil(G1 , G2 , ψ) = max{dist(G2 , ψ(u), ψ(v))|(u, v) ∈ E1 }, where dist(G2 , ψ(u), ψ(v)) denotes the distance between the two nodes ψ(u) and ψ(v) in G2 . The smaller the dilation of an embedding is, the shorter the communication delay that the graph G2 simulates the graph G1 [1]. As two common guest graphs, linear arrays (i.e. paths) [9,8,7] and rings (i.e. cycles) [2,6,15] are two fundamental networks for parallel and distributed computing. In large interconnection networks, nodes or edges tend to become faulty. It is important to find an embedding of a guest graph into a host graph where all faulty nodes and edges have been removed. This is called fault-tolerant embedding. Much work has been done on the fault-tolerant embedding [3,10,12,19,21,20,17, 4,13]. The locally twisted cube has many properties superior to hypercube. Though both the locally twisted cube and the ordinary hypercube have the same number of vertices and the same vertex degree, the diameter of the locally twisted cube is approximately half that of the ordinary hypercube.
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In this paper, we are interested in the path and/or cycle embedding properties of the n-dimensional locally twisted cube LTQ n . Yang et al. proposed this new network [26] and proved that LTQ n contains cycles of all lengths from 4 to 2n [27]. Ma, Xu [23] and Hu et al. [16], independently, improved this result by proving that for any edge in LTQ n there are cycles of all lengths containing it. Ma and Xu [22] further improved this result by showing that for any two different vertices x and y with distance d in LTQ n , there exist xy-paths of all lengths from d to 2n − 1 except for d + 1. Even when faulty elements occur, Chang et al. [5] and Park et al. [24], independently, showed that LTQ n still contains fault-free cycles of all lengths provided that faulty elements do not exceed n − 2. Very recently, Han et al. [11] have showed that LTQ n with at most n − 3 faulty elements contains paths of all lengths from 2n−1 − 1 to 2n −fv −1 between any two distinct fault-free vertices, where fv is the number of faulty vertices. Hsieh and Wu [14] have considered more faulty edges and showed that LTQ n contains a fault-free Hamiltonian cycle provided that faulty edges do not exceed 2n − 5 and each vertex is incident with at least two fault-free edges. This condition is natural since, in practical applications, the probability is small for a vertex x being isolated (all links incident with x are faulty) or pendant (only one link incident with x is fault-free and the others are all faulty). In this paper, we show that for any vertex u ∈ V (LTQ n ) and any integer 4 6 ℓ 6 2n − fv except for 5, there exists cycle C of length ℓ in LTQ n such that u is in C if fe + fv 6 n − 3. The approach we use is based on the recursive construction of LTQ n . The remaining part of this paper is organized as follows. In Section 2, we recall the structure of LTQ n , and some definitions and notations. In Section 3, we introduce some properties of LTQ n to be used in our proofs. In Section 4, we give the proof of our result. Finally, we give some concluding remarks in Section 5. 2. Preliminaries In this section, we will give some definitions and properties about LTQ n . A graph G = (V , E ) consists of a vertex-set V and an edge-set E, where V = V (G) is a finite set and E = E (G) is a subset of xy–xy is an unordered pair of V . Two vertices x and y are adjacent if xy is an edge of G, and which are also the endvertices of xy. For a vertex x, the vertex adjacent to x is called as the neighbor of x. The degree of a vertex x is the number of edges incident with it. A graph is called k-regular if each vertex has degree k. For two distinct vertices x and y, an xy-path between x and y is a sequence of distinct vertices in which any two consecutive vertices are adjacent. The length of a path is the number of edges on the path. An xy-path of length at least three is called a cycle if x = y. A connected subgraph of G is called a spanning tree if it contains all vertices of G and no cycles, in which a distinguished vertex is called the root of the spanning tree. The distance between two distinct vertices x and y in G is the length of a shortest xy-path in G, and the diameter of G is the maximum distance between any two vertices. A non-empty subset M of E (G) is called a matching of G if no two of its elements have a common end vertices in G. A matching M is perfect if every vertex of G is an end-vertex of some edge in M. We now recall the definition of the n-dimensional locally twisted cube, proposed by Yang, Evans and Megson [26], which has 2n vertices, and each vertex is an n-string on {0, 1}. Definition 1 ([26]). The n-dimensional locally twisted cube, denoted by LTQ n (n > 2), is recursively defined as follows.
(1) LTQ 2 is a graph isomorphic to Q2 . (2) For n > 3, LTQ n is built from disjoint copies of LTQ n−1 according to the following steps. Let LTQ 0n−1 and LTQ 1n−1 denote graphs obtained by prefixing labels of each vertex of one copy of LTQ n−1 with 0 and with 1, respectively, and connect a vertex x = 0x2 x3 . . . xn of LTQ 0n−1 with another vertex y = 1(x2 + xn )x3 . . . xn of LTQ 1n−1 by an edge xy, where ‘+’ represents the modulo 2 addition. The graphs shown in Fig. 1 are LTQ 3 and LTQ 4 . The locally twisted cube LTQ n can be equivalently defined with the following non-recursive fashion. Definition 2 ([26]). For n > 2, the n-dimensional locally twisted cube LTQ n is a graph with n-strings on {0, 1} as the vertex set. Two vertex x = x1 x2 . . . xn−1 xn and y = y1 y2 . . . yn−1 yn of LTQ n are adjacent if and only if either (a) xi = y¯ i and xi+1 = yi+1 + xn for some 1 6 i 6 n − 2, and xj = yj for all the remaining bits, where ‘+’ represents the modulo 2 addition, or (b) xi = y¯ i for some i ∈ {n − 1, n}, and xi = yi for all the remaining bits. According to the above definition, it is not difficult to see that LTQ n is an n-regular graph with 2n vertices and n2n−1 edges. From the definition, LTQ n can be expressed as the union of two disjoint copies of LTQ n−1 by adding a perfect matching between them according to the specified rule. For short, we often write LTQ n = L ⊕ R, where L ∼ = LTQ 0n−1 and R ∼ = LTQ 1n−1 . We now make some remarks on the n-dimensional locally twisted cube. Firstly, like to many variants of the hypercube such as the twisted cube, the crossed cube, the augmented cube and otherwise, the locally twisted cube not only keeps many nice properties of the hypercube such as regularity, high connectivity and high recursive constructability, but also has diameter of about half of that of the hypercube of the same size. Secondly, the locally twisted cube also keeps a nice property of the hypercube, that is, the labels of any two adjacent vertices differ in at most two successive bits. However, a common feature of the above-mentioned variants is that the labels of some neighbor vertices may differ in a large number of bits. As a result, a portion of good properties of hypercube are lost in these variants. For example, the design of efficient parallel algorithms on these variants turns out to be a difficult task [26]. Thirdly, the locally twisted cube LTQ n contains cycles of all lengths from 4 to 2n [27], but the hypercube Qn contains only even cycles since it is a bipartite graph. Thus, LTQ n is superior to Qn in cycle embedding property. Fourthly, the construction of the locally twisted cube LTQ n is quite different from that of the twisted cube TQn . The former is defined for any positive integer n, while the latter only for odd integer. Lastly, it should be noted that, like to many variants of the hypercube, the locally twisted cube LTQ n is not vertex-transitive for n > 4 proved by Liu et al. [18]. 3. Properties In this section, we introduce some properties of LTQ n to be used in our proofs in Section 4. Yang, Evans and Megson [26] found an isomorphic expression of LTQ n . For example, two graphs shown in Fig. 2 are other expressions of LTQ 3 and LTQ 4 , respectively. In general, they proved the following result.
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Fig. 1. The locally twisted cubes LTQ 3 and LTQ 4 .
Fig. 2. Another painting of LTQ 3 and LTQ 4 .
Lemma 1 ([26]). Let L′ be the graph obtained from Qn−1 by suffixing the labels of all vertices with 0, R′ be the graph obtained from a graph isomorphic to Qn−1 by suffixing the labels of all vertices with 1. Then LTQ n is isomorphic to the graph obtained from L′ and R′ by adding a perfect matching M between them, denoted by LTQ n = L′ ⊕ R′ , where L′ ∼ = Qn0−1 and R′ ∼ = Qn1−1 and M is the set of edges by linking two vertices with difference only in suffixes. Let LTQ n = L ⊕ R defined in Lemma 1. For convenience, for a vertex u in LTQ n , if u is in L, we write uL for u, and use uR to denote its neighbor in R. Let uL and vL be two adjacent vertices in L. We say vL is a strong neighbor of uL in L if their neighbors uR and vR are adjacent in R (see Fig. 3(a)), and a weak neighbor of uL in L if their neighbors uR and vR are not adjacent in R (see Fig. 3(b)). Similarly, we can define a strong neighbor or a weak neighbor of two adjacent vertices uR and vR in R. For example, in LTQ 4 (see Fig. 1), consider uL = 0000. Since vL = 0100 is the neighbor of uL in L, then vL is a strong neighbor of uL in L because their neighbors uR = 1000 and vR = 1100 are adjacent in R. Since vL′ = 0001 is the other neighbor of uL in L, then vL′ is a weak neighbor of uL in L because their neighbors uR = 1000 and vR′ = 1101 are not adjacent in R. Lemma 2 ([25]). Let LTQ n = L ⊕ R. If n > 4 then, for any vertex uL in L, there are n − 2 strong neighbors and one weak neighbor in L. Moreover, if wL is the weak neighbor of uL , then the distance between uR and wR is two in R. The same conclusion holds for any vertex uR in R. Lemma 3 ([22]). Let x and y be any two different vertices in LTQ n and the distance between them be d. Then for any integer ℓ with d 6 ℓ 6 2n − 1 except for d + 1, there exists an xy-path of length ℓ in LTQ n for n > 3.
Lemma 4 ([11]). If fv + fe 6 n − 3 and n > 3, then for any integer ℓ with 2n−1 − 1 6 ℓ 6 2n − fv − 1, there is a fault-free path of length ℓ between any two distinct vertices in LTQ n . Lemma 5 ([16]). Let u be any vertex in LTQ n and ℓ be any integer with 4 6 ℓ 6 2n . Then there is a cycle of length ℓ containing the vertex u in LTQ n for n > 2. 4. Proof of theorem In this section, we investigate the fault-tolerant vertexpancyclicity of LTQ n and show that LTQ n is (n − 3)-fault-tolerant vertex-pancyclic. We state this result as the following theorem. Theorem 1. If fv + fe 6 n − 3 and n > 3 then, for any fault-free vertex u in LTQ n and any integer ℓ with 4 6 ℓ 6 2n − fv except for 5, there is a fault-free cycle of length ℓ containing the vertex u in LTQ n . Proof of Theorem. We use the expression LTQ n = L ⊕ R, where L ∼ = LTQ 0n−1 and R ∼ = LTQ 1n−1 . Let F be a set of faulty elements in LTQ n with |F | = fv + fe , FL = F ∩ L, FR = F ∩ R, FC = F ∩ EC , EC represents the crossed edges connecting L and R. F v = F ∩ V (LTQ n ), FLv = FL ∩ V (L) and FRv = FR ∩ V (R). Without loss of generality, we may assume |FR | 6 |FL |. Let u be an arbitrary fault-free vertex in LTQ n and ℓ be any integer with 4 6 ℓ 6 2n − fv except for 5. We need to prove that there exists a fault-free cycle of length ℓ containing the vertex u in LTQ n if |F | 6 n − 3 and n > 3. We proceed by induction on n > 3. For n = 3, there are no faulty vertices or edges in LTQ 3 , and so the theorem holds by Lemma 5.
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Fig. 3. (a) vL is strong neighbor of uL in L. (b) vL is weak neighbor of uL in L.
Assume that the theorem holds for LTQ n−1 . We consider LTQ n for n > 4. Case 1. |FL | = n − 3. In this case |FC | = |FR | = 0. Let u be a fault-free vertex in LTQ n . Since LTQ n is n-regular and |F | = n − 3, there exists a fault-free edge joined with u, denoted by (u, u1 ). Let ℓ = ℓ′ + 1. If 2n−1 − 1 6 ℓ′ 6 2n − fv − 1 then, by Lemma 4, there exists a fault-free uu1 -path P of length ℓ′ in LTQ n . Thus, P + uu1 is a fault-free cycle of length ℓ containing the vertex u in LTQ n . Thus, we only need to consider ℓ with 4 6 ℓ 6 2n−1 except for 5. Subcase 1.1. The fault-free vertex u is in R. Since |FR | = 0 and by Lemma 5, there is a fault-free cycle of length ℓ containing the vertex u for any ℓ with 4 6 ℓ 6 2n−1 . Thus, we only need to consider one case according as the fault-free vertex u is in L. Subcase 1.2. The fault-free vertex u is in L. Since LTQ 0n−1 is (n − 1)-regular, then the fault-free vertex u has n − 1 neighbor vertices in LTQ 0n−1 , in which there are n − 2 strong neighbors and one weak neighbor by Lemma 2. Moreover, since the faulty elements have n − 3 in LTQ 0n−1 , the worst case is that n − 3 faulty elements are all faulty vertices which are linked with the fault-free vertex u. Then there exist at least two fault-free edges joined with the fault-free vertex u, denoted by (u, u1 ) and (u, u2 ). It implies that u1 and u2 are both strong neighbors of u, or, one of u1 and u2 is strong neighbor and another is weak neighbor of u, without loss of generality, assume u1 is strong neighbor and u2 is weak neighbor of u. Suppose that u1 and u2 are both strong neighbors of u in L. Consider the strong neighbor u1 of u, let v and v1 be neighbors of u and u1 in R, respectively. By Lemma 2, vv1 ∈ E (R). Since |FC | = |FR | = 0, the cycle (u, u1 , v1 , v) of length 4 contains the vertex u and is fault-free. For any ℓ with 4 6 ℓ 6 2n−1 except for 5, let ℓ = ℓ′ + 3. Then 1 6 ℓ′ 6 2n−1 − 3 except 2. By Lemma 3, there exists a path P (v, v1 ) of length ℓ′ in R. So P (v, v1 )+ u1 v1 + uu1 + uv is a fault-free cycle of length ℓ containing the vertex u (see Fig. 4). Suppose that u1 is the strong neighbor and u2 is the weak neighbor of u in L. Similarly, it suffices to consider the strong neighbor u1 of u that we can find a fault-free cycle of length ℓ with 4 6 ℓ 6 2n−1 except for 5 containing the fault-free vertex u. Case 2. |FL | 6 n − 4. Since the fault-free vertex u ∈ V (LTQ n ), it may be in L or R. Without loss of generality, we may assume that the vertex u is in L. Since |FR | 6 n − 4 and according to Case 1, let uu1 is a faultfree edge joined with u. If 4 6 ℓ 6 2n−1 − |FLv | except 5 then, by the induction hypothesis, there is a fault-free cycle of length ℓ containing the vertex u in L, so in LTQ n . Thus, we only need to consider such an ℓ that satisfies 2n−1 − |FLv | + 1 6 ℓ 6 2n − fv . If n = 4, since 0 6 |FLv | 6 |FL | 6 n − 4 = 0, then 24−1 + 1 = 9 6 ℓ 6 24 − fv . Let ℓ = ℓ′ + 1. Then 8 6 ℓ′ 6 24 − fv − 1. By Lemma 4, there is a fault-free uu1 -path P of length ℓ′ in LTQ 4 , and so P + uu1 is a fault-free cycle of length ℓ containing the vertex u (see Fig. 5(a)).
Fig. 4. The illustration of Subcase 1.2.
Fig. 5. The illustration of Case 2. Table 1 Cycles of length 5 containing the vertex uL = 00001. (00001 00000 00010 00110 00111 00001) (00001 00000 00100 00101 00111 00001) (00001 00000 00100 00101 00011 00001) (00001 00000 00100 00110 00111 00001) (00001 00000 00100 01100 01101 00001) (00001 00000 01000 01100 01101 00001) (00001 00000 01000 11000 11001 00001) (00001 00000 10000 11000 11001 00001) (00001 00011 00010 00110 00111 00001)
Now, assume n > 5 and write ℓ = ℓ1 + 1 + ℓ2 , where 2n−2 − |FLv | 6 ℓ1 6 2n−1 − |FLv | and 2n−2 6 ℓ2 6 2n−1 − |FRv | − 1. Since 2n−2 − |FLv | > 2n−2 − |FL | > 2n−2 − n + 4 > 4 for n > 5, by the induction hypothesis, there is a fault-free cycle C of length ℓ1 containing the vertex u in L. Consider the following inequality.
ℓ1 − |FC | − |FR | − |u| > 2n−2 − |FLv | − |FC | − |FR | − 1 > 2n−2 − |F | − 1 > 2n−2 − n + 2.
Let f (x) = 2x−2 − x + 2. Since f ′ (x) = 2x−2 ln 2 − 1 > 0 for x ≥ 5, f (x) is an increasing function, which implies that ℓ1 − |FC | − |FR | > f (5) = 25−2 − 5 + 2 = 5. In other words, there exists at least two vertices u1 and u2 that distinct u on C and two edges (u1 , v1 ), (u2 , v2 ) are fault-free. Since |FR | 6 n − 4, by Lemma 4, there is a fault-free v1 v2 -path P of length ℓ2 in R. So C − u1 u2 + u1 v1 + u2 v2 + P is a fault-free cycle of length ℓ(=ℓ1 + 1 + ℓ2 ) containing u (see Fig. 5(b)). The proof of the theorem is complete.
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Table 2 Cycles of length 6 containing the vertex u = 0010. (0010 0000 0001 0111 0101 0011 0010) (0010 0000 1000 1001 1111 0011 0010) (0010 0110 0111 0101 0100 0000 0010) (0010 0110 0100 1100 1000 0000 0010) (0010 0110 1110 1100 0100 0000 0010) (0010 1010 1011 1001 1000 0000 0010) (0010 1010 1011 1001 0101 0011 0010) (0010 1010 1011 1101 0001 0000 0010) (0010 1010 1011 0111 0101 0011 0010) (0010 1010 1011 0111 0001 0011 0010) (0010 1010 1000 1001 0101 0011 0010) (0010 1010 1000 1100 0100 0110 0010) (0010 1010 1000 0000 0001 0011 0010) (0010 1010 1110 1100 1000 0000 0010) (0010 1010 1110 1100 0100 0000 0010)
5. Conclusion and remarks As one of the most fundamental networks for parallel and distributed computation, cycles are suitable for developing simple algorithms with low communication cost. Edge and/or vertex failures are inevitable when a large parallel computer system is put in use. Therefore, the fault-tolerant capacity of a network is a critical issue in parallel computing. The fault-tolerant pancyclicity of an interconnection network is a measure of its capability of implementing ring-structured parallel algorithms in a communication-efficient fashion in the presence of faults. In view of the fact that the hypercube network Qn contains only even cycles, LTQ n is superior to Qn in fault-tolerant pancyclicity. This shows that, when the locally twisted cube is used to model the topological structure of a large-scale parallel processing system, our result implies that the system has larger capability of implementing ring-structured parallel algorithms in a communication-efficient fashion in the hybrid presence of edge and vertex failures than one of the hypercube network. We make some remarks on the optimality of our result in the following sense. (1) Consider the vertex uL ∈ L if we write LTQ n = L ⊕ R. By Lemma 3, there are only two kind distinct cycles of length ℓ = 5 containing the vertex uL , which are obtained by the strong neighbor vL or the weak neighbor vL′ . If vL or/and vL′ is faulty, then there are no fault-free cycles of length ℓ = 5 containing the vertex uL in LTQ n . For example, in LTQ 5 , taking uL = 00001, all the cycles of length ℓ = 5 containing the vertex uL = 00001 calculated by the computer are given in Table 1. According to above results, we find that if the vertex x = 00000 and the vertex y = 00011 are faulty, then there are no fault-free cycles of length ℓ = 5 containing the vertex uL = 00001 in LTQ 5 . (2) If fv + fe = n − 2, then the theorem does not hold for n ≥ 4. For any n ≥ 3, we claim that there exists a vertex of LTQ n such that, with n − 2 faulty vertices and/or edges, there is no fault-free cycle of length ℓ = 4 containing the vertex. Let u = u1 u2 . . . un = 0n (n consecutive 0’s) be a vertex of LTQ n . One can check that any cycle of length ℓ = 4 containing u must contain v = u1 u2 . . . ui−1 u¯ i . . . un for some 2 6 i 6 n. Therefore, if F = {u1 u2 . . . ui−1 u¯ i . . . un |i is an integer for 2 6 i 6 n}, there is no cycle of length ℓ = 4 containing u in LTQ n − F . For example, let n = 4 and u = 0010, the four cycles of length ℓ = 4 containing u = 0010 are {0010 0011 0001 0000 0010}, {0010 0110 0100 0000 0010}, {0010 1010 1000 0000 0010}, {0010 1010 1110 0110 0010}. With F = {0000, 0110} or F = {0000, (1110, 0110)}, there is no faultfree cycle of length ℓ = 4 containing u = 0010 in LTQ 4 − F . Consider the vertex u = 0010 in LTQ 4 , all the cycles of length ℓ = 6 containing the vertex u = 0010 calculated by the computer are listed in Table 2. According to the listed results, we find that if the vertex x = 0000 and the vertex y = 1010 are both faulty
(0010 0000 0001 1101 1111 0011 0010) (0010 0000 1000 1001 0101 0011 0010) (0010 0110 0100 0000 0001 0011 0010) (0010 0110 1110 1100 1000 0000 0010) (0010 0110 1110 1010 1000 0000 0010) (0010 1010 1011 1001 1111 0011 0010) (0010 1010 1011 1101 1111 0011 0010) (0010 1010 1011 1101 0001 0011 0010) (0010 1010 1011 0111 0001 0000 0010) (0010 1010 1000 1001 1111 0011 0010) (0010 1010 1000 1100 1110 0110 0010) (0010 1010 1000 1100 0100 0000 0010) (0010 1010 1000 0000 0100 0110 0010) (0010 1010 1110 1100 0100 0110 0010) (0010 1010 1110 0110 0100 0000 0010)
or the vertex x = 0000 and the edge e = (0000, 1010) are both faulty, namely, F = {0000, 1010} or F = {0000, (0000, 1010)}, then there is no fault-free cycles of length ℓ = 6 containing the vertex u = 0010 in LTQ 4 − F . Acknowledgments The authors would like to express their gratitude to the anonymous referees for their kind suggestions and comments on the original manuscript, which resulted in this version. References [1] L. Auletta, A. Rescigno, V. Scarano, Embedding graphs onto the supercube, IEEE Trans. Comput. 44 (1995) 593–597. http://dx.doi.org/10.1109/12.376173. [2] M. Bae, B. Bose, Edge disjoint Hamiltonian cycles in k-ary n-cubes and hypercubes, IEEE Trans. Comput. 52 (10) (2003) 1271–1284. http://dx.doi.org/10.1109/TC.2003.1234525. [3] M. Chan, S. Lee, Fault-tolerant embedding of complete binary trees in hypercubes, IEEE Trans. Parallel Distrib. Syst. 4 (3) (1993) 277–288. http://dx.doi.org/10.1109/71.210811. [4] N.-W. Chang, S.-Y. Hsieh, Fault-tolerant bipancyclicity of faulty hypercubes under the generalized conditional-fault model, IEEE Trans. Commun. 59 (2011) 3400–3409. http://dx.doi.org/10.1109/TCOMM.2011.093011.100321. [5] Q. Chang, M. Ma, J. Xu, Fault-tolerant pancyclicity of locally twisted cubes, J. Univ. Sci. Technol. China 36 (2006) 607–610 (in Chinese). [6] J.-M. Chang, J.-S. Yang, Fault-tolerant cycle-embedding in alternating group graphs, Appl. Math. Comput. 197 (2) (2008) 760–767. http://dx.doi.org/10.1016/j.amc.2007.08.010. [7] J. Fan, X. Jia, Edge-pancyclicity and path-embeddability of bijective connection graphs, Inform. Sci. 178 (2) (2008) 340–351. http://dx.doi.org/10.1016/j.ins.2007.08.012. [8] J. Fan, X. Jia, X. Lin, Optimal embeddings of paths with various lengths in twisted cubes, IEEE Trans. Parallel Distrib. Syst. 18 (4) (2007) 511–521. http://dx.doi.org/10.1109/TPDS.2007.1003. [9] J. Fan, X. Lin, X. Jia, Optimal path embedding in crossed cubes, IEEE Trans. Parallel Distrib. Syst. 16 (12) (2005) 1190–1200. http://dx.doi.org/10.1109/TPDS.2005.151. [10] J. Fan, X. Lin, Y. Pan, X. Jia, Optimal fault-tolerant embedding of paths in twisted cubes, J. Parallel Distrib. Comput. 67 (2) (2007) 205–214. http://dx.doi.org/10.1016/j.jpdc.2006.04.004. [11] Y. Han, J. Fan, J. Yang, P. Qian, Path embedding in faulty locally twisted cubes, in: 2009 2nd IEEE International Conference on Computer Science and Information Technology, Vol. 3, 2009, pp. 214–218. [12] S.-Y. Hsieh, N.-W. Chang, Hamiltonian path embedding and pancyclicity on the Mobius cube with faulty nodes and faulty edges, IEEE Trans. Comput. 55 (7) (2006) 854–863. http://dx.doi.org/10.1109/TC.2006.104. [13] S.-Y. Hsieh, C.-W. Lee, Pancyclicity of restricted hypercube-like networks under the conditional fault model, SIAM J. Discrete Math. 23 (4) (2010) 2100–2119. http://dx.doi.org/10.1137/090753747. [14] S.-Y. Hsieh, C.-Y. Wu, Edge-fault-tolerant hamiltonicity of locally twisted cubes under conditional edge faults, J. Comb. Optim. 19 (1) (2010) 16–30. http://dx.doi.org/10.1007/s10878-008-9157-x. [15] S.-Y. Hsieh, P.-Y. Yu, Cycle embedding on twisted cubes, Extended abstract, in: Seventh International Conference on Parallel and Distributed Computing, Applications and Technologies, Proceedings, 2006, pp. 102–104. [16] K.S. Hu, S.-S. Yeoh, C. Chen, L.-H. Hsu, Node-pancyclicity and edge-pancyclicity of hypercube variants, Inform. Process. Lett. 102 (2007) 1–7. [17] C.-W. Lee, S.-Y. Hsieh, Pancyclicity of matching composition networks under the conditional fault model, IEEE Trans. Comput. 61 (2012) 278–283. http://dx.doi.org/10.1109/TC.2010.229.
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interest includes Theory and Application of Graphs, Topological Structure and Analysis of Interconnection Networks, Fault Tolerance, Algorithms Design and Analysis.
Xirong Xu received her B.S. degree in Applied Mathematics from East China University of Science and Technology, China in June 1990 and her M.S. degree in Computer Application Technology from Dalian University of Technology, China in June 2002, and Ph.D. degree in Computer Software and Theory, Dalian University of Technology, China in June 2005. She worked in mathematics postdoctoral research station at the University of Science and Technology of China from 2005 to 2007. She is an associate professor at School of Computer Science and Technology, Dalian University of Technology, China. Her current research
Sijia Zhang received her B.S. degree in Information Management and Information System from Liaoning Technical University, China in June 2005 and her M.S. degree in Computer Application Technology from Dalian University of Technology, China in June 2010. She is currently working towards her Ph.D. degree in Computer Software and Theory at Dalian University of Technology, China. Her current research interest includes Topological Structure and Analysis of Interconnection Networks, Graph Theory, Algorithms Design and Analysis.
Yazhen Huang received her B.S. degree in Computer science and Technology from Henan Normal University, China in June 2012. She is currently a graduate student and working towards her M.S. degree in Computer Applied Technology at Dalian University of Technology, China. Her current research interest includes Topological Structure and Analysis of Interconnection Networks, Graph Theory, Algorithms Design and Analysis and Fault Tolerance.
Peng Zhang received his B.S. degree in Clinical Medicine from Dalian Medical University, China in June 2004 and his M.S. degree in Computer Applied Technology from Dalian University of Technology, China in June 2013. He is an associate professor at College of Computer and Information Engineering, Dalian Medical University, China. His current research interest includes Computer Networking Technology, Computer Applied Technology, Combination of information technology network and medicine.