Cycles embedding in hypercubes with node failures

doi:10.1016/j.ipl.2006.12.016

Information Processing Letters

Volume 102, Issue 6, 15 June 2007, Pages 242-246

https://doi.org/10.1016/j.ipl.2006.12.016 Get rights and content

Abstract

The hypercube has been widely used as the interconnection network in parallel computers. The n-dimensional hypercube $Q_{n}$ is a graph having $2^{n}$ vertices each labeled with a distinct n-bit binary strings. Two vertices are linked by an edge if and only if their addresses differ exactly in the one bit position. Let $f_{v}$ denote the number of faulty vertices in $Q_{n}$ . For $n ⩾ 3$ , in this paper, we prove that every fault-free edge and fault-free vertex of $Q_{n}$ lies on a fault-free cycle of every even length from 4 to $2^{n} - 2 f_{v}$ inclusive even if $f_{v} ⩽ n - 2$ . Our results are optimal.

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Cited by (27)

Vertex-disjoint paths joining adjacent vertices in faulty hypercubes
2019, Theoretical Computer Science
Let $Q_{n}$ denote the n-dimensional hypercube and the set of faulty edges and faulty vertices in $Q_{n}$ be denoted by $F_{e}$ and $F_{v}$ , respectively. In this paper, we investigate $Q_{n}$ ( $n \geq 3$ ) with $| F_{e} | + | F_{v} | \leq n - 3$ faulty elements, and demonstrate that there are two fault-free vertex-disjoint paths $P [a, b]$ and $P [c, d]$ satisfying that $2 \leq ℓ (P [a, b]) + ℓ (P [c, d]) \leq 2^{n} - 2 | F_{v} | - 2$ , where $2 | (ℓ (P [a, b]) + ℓ (P [c, d]))$ , $(a, b), (c, d) \in E (Q_{n})$ . The contribution of this paper is: (1) we can quickly obtain the interesting result that $Q_{n} - F_{e}$ is bipancyclic, where $| F_{e} | \leq n - 2$ and $n \geq 3$ ; (2) this result is a complement to Chen's part result (Chen (2009) [2]) in that our result shows that there are all kinds of two disjoint-free $(S, T)$ -paths which contain $4, 6, 8, \dots, 2^{n} - 2 | F_{v} |$ vertices respectively in $Q_{n}$ when $S = {a, c}$ , $T = {b, d}$ , and $(a, b), (c, d) \in E (Q_{n})$ . Our result is optimal with respect to the number of fault-tolerant elements.
Conditional fault-tolerant edge-bipancyclicity of hypercubes with faulty vertices and edges
2016, Theoretical Computer Science
Let F be a faulty set in an n-dimensional hypercube $Q_{n}$ such that in $Q_{n} - F$ each vertex is incident to at least two edges, and let $f_{v}$ , $f_{e}$ be the numbers of faulty vertices and faulty edges in F, respectively. In this paper, we consider the fault-tolerant edge-bipancyclicity of hypercubes. It is shown that each edge in $Q_{n} - F$ for $n \geq 3$ lies on a fault-free cycle of any even length from 6 to $2^{n} - 2 f_{v}$ if $f_{v} + f_{e} \leq 2 n - 5$ . This gives an answer for a problem proposed by Yang et al. (2016) [33].
Hamiltonicity of hypercubes with faulty vertices
2016, Information Processing Letters
Citation Excerpt :
There is a large amount of literature on cycle and/or path embedding into interconnection networks with or without faulty elements, see a survey [22] and references therein. For (fault-tolerant) path or cycle embedding of hypercubes, also see references [2–14,17–21]. In 2001, Locke [15] considered Hamiltonicity of hypercubes with faulty vertices and proposed the following conjecture that remains open.
Let $Q_{n}$ denote the n-dimensional hypercube with the bipartition W and B. Assume $W_{0} = {a_{1}, a_{2}, \dots, a_{k}} \subset W$ and $B_{0} = {b_{1}, b_{2}, \dots, b_{k}} \subset B$ , where $k \geq 1$ and $n \geq k + 2$ . The following is a long-standing conjecture (Locke, 2001 [15]): The graph $G = Q_{n} - (W_{0} \cup B_{0})$ contains a Hamiltonian cycle. Very recently, Locke' conjecture was proven when $k \leq 4$ [3] or $n \geq 2 k + 2$ [14]. In this paper, we obtain the following two results: (1) If $a_{i}$ is adjacent to $b_{i}$ for each i with $1 \leq i \leq k$ , then Locke' conjecture holds (2) Moreover, if $n \geq k + 3$ , then the graph G is Hamilton laceable, i.e., for any $a \in W \ W_{0}$ and $b \in B \ B_{0}$ , the graph G contains a Hamiltonian path connecting a and b, and the lower bound $k + 3$ is optimal. Our results may find some applications.
Fault-tolerant edge-bipancyclicity of faulty hypercubes under the conditional-fault model
2016, Information Sciences
Citation Excerpt :
Here, Qn can have only faulty vertices, only faulty edges or both faulty vertices and faulty edges. There is also an interesting fault-tolerant cycle embedding problem which is the fault-tolerant edge-bipancyclicity of hypercubes [13,18,22–26,30,31]. As a comparative viewpoint, for the fault-tolerant vertex-bipancyclicity of hypercubes, one may refer to [23,32].
It is well-known that the n-dimensional hypercube Q_n is one of the most versatile and efficient interconnection network architecture yet discovered for building massively parallel or distributed systems. Let F be the faulty set of Q_n and let f_v, f_e be the numbers of faulty vertices and faulty edges in F, respectively. An edge $e = (x, y)$ is said to be free if e, x, y are not in F, and a cycle is said to be fault-free if there is no faulty vertex or faulty edge on the cycle. In this paper, we prove that each free edge (x, y) in Q_n for n ≥ 3 lies on a fault-free cycle of any even length from 6 to $2^{n} - 2 f_{v}$ if $f_{v} + f_{e} \leq 2 n - 5,$ $f_{e} \leq n - 2$ and both x and y are incident to at least two free edges. This result confirms a conjecture reported in the literature.
Odd cycles embedding on folded hypercubes with conditional faulty edges
2014, Information Sciences
Let ${FF}_{e}$ be the set of $| {FF}_{e} | ⩽ 2 n - 5$ faulty edges in an n-dimensional folded hypercube ${FQ}_{n}$ such that each vertex of ${FQ}_{n}$ is incident to at least two fault-free edges, where $n ⩾ 4$ and n is even. Under this assumption, we show that every fault-free edge of ${FQ}_{n}$ lies on a fault-free cycle of every odd length from $n + 1$ to $2^{n} - 1$ . In terms of the number of tolerant faulty edges and embedding odd cycles in ${FQ}_{n}$ , our result improves not only the result in Xu and Ma (2006) where $| {FF}_{e} | = 0$ , but also the previous best result gotten by Xu et al. (2006) where $| {FF}_{e} | ⩽ n - 1$ .
Fault-tolerant cycle embedding in the faulty hypercubes
2013, Information Sciences
Citation Excerpt :
Most of the works on graph embedding take paths, cycles, trees, meshes as guest graphs [12,13,21,22], because these interconnection networks are widely used in parallel computing systems. It is important to have efficient ring embedding in a network, since some parallel applications are originally designated on a ring or Torus architecture such as those in image and signal processing [16]. The cycle-embedding problem deals with all possible lengths of the cycles in a given graph.
The hypercube is one of the best known interconnection networks. Embedding cycles of all possible lengths in faulty hypercubes has received much attention. Let F_v (respectively, F_e) denote the set of faulty vertices (respectively, faulty edges) and f_v (respectively, f_e) denote the number of faulty vertices (respectively, faulty edge) in an n-dimensional hypercube Q_n. Let f(e) denote the number of faulty nodes and/or faulty edges incident with the end-vertices of an edge e ∈ E(Q_n). In this paper, we assume that each node is incident with at least three fault-free neighbors and at least three fault-free edges. Under this assumption, we show that every fault-free edge lies on a fault-free cycle of every even length from 4 to 2ⁿ − 2∣F_v∣ if ∣F_v∣ + ∣F_e∣ ⩽ 2n − 7 and f(e) ⩽ n − 2, where n ⩾ 5. Under our condition, our result not only improves the previously best known result of Hsieh et al. [S.-Y. Hsieh, T.-H. Shen, Edge-bipancyclicity of a hypercube with faulty vertices and edges, Discrete Applied Mathematics 156 (10) (2008) 1802–1808] where f_v + f_e ⩽ n − 2 was assumed, but also extends the result of Tsai [C.-H. Tsai, Linear array and ring embedding in conditional faulty hypercubes, Theoretical Computer Science 314 (3) (2004) 431–443] where only the faulty edges were considered and Tsai [C.-H. Tsai, Cycle embedding in hypercubes with node failures, Information Processing Letters 102 (6) (2007) 242–246] where only the faulty vertices were considered.

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^☆: This work was supported in part by the National Science Council of the Republic of China under Contract NSC 95-2221-E-026-002.

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Cycles embedding in hypercubes with node failures☆

Abstract

Parallel Comput.

Parallel Comput.

Inform. Process. Lett.

Theoret. Comput. Sci.

Inform. Process. Lett.