Embedding a ring in a hypercube with both faulty links and faulty nodes

doi:10.1016/0020-0190(96)00114-7

Information Processing Letters

Volume 59, Issue 4, 26 August 1996, Pages 217-222

https://doi.org/10.1016/0020-0190(96)00114-7 Get rights and content

Abstract

In this paper, we show that given a binary n-cube with f_e ⩽ n − 4 faulty edges and f_v ⩽ n − 1 faulty vertices such that f_e + f_v ⩽ n − 1, a ring of length at least 2ⁿ − 2f_v can be obtained. The best known results can tolerate only faulty edges or only faulty vertices.

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Paired many-to-many disjoint path covers in faulty hypercubes
2013, Theoretical Computer Science
A paired many-to-many k-disjoint path cover (k-DPC for short) of a graph is a set of k disjoint paths joining k distinct source–sink pairs that cover all the vertices of the graph. Extending the notion of DPC, we define a paired many-to-many bipartite k-DPC of a bipartite graph G to be a set of k disjoint paths joining k distinct source–sink pairs that altogether cover the same number of vertices as the maximum number of vertices covered when the source–sink pairs are given in the complete bipartite, spanning supergraph of G. We show that every m-dimensional hypercube, $Q_{m}$ , under the condition that f or less faulty elements (vertices and/or edges) are removed, has a paired many-to-many bipartite k-DPC joining any k distinct source–sink pairs for any f and $k ⩾ 1$ subject to $f + 2 k ⩽ m$ . This implies that $Q_{m}$ with $m - 2$ or less faulty elements is strongly Hamiltonian-laceable.
Paths and cycles embedding on faulty enhanced hypercube networks
2013, Acta Mathematica Scientia
Let Q_n,k (n ≥ 3, 1 ≤ k ≤ n−1) be an n-dimensional enhanced hypercube which is an attractive variant of the hypercube and can be obtained by adding some complementary edges, f_v and f_e be the numbers of faulty vertices and faulty edges, respectively. In this paper, we give three main results. First, a fault-free path P[u,v] of length at least 2ⁿ−2f_v−1 (respectively, 2ⁿ−2f_v−2) can be embedded on Q_n,k with f_v+f_e ≤ n−1 when d_{Q_n,k}(u,v) is odd (respectively, d_{Q_n,k}(u,v) is even). Secondly, an Q_n,k is (n−2) edge-fault-free hyper Hamiltonian-laceable when n (≥ 3) and k have the same parity. Lastly, a fault-free cycle of length at least 2ⁿ−2f_v can be embedded on Q_n,k with f_e ≤ n−1 and f_v+f_e ≤ 2n−4.
A note on cycle embedding in hypercubes with faulty vertices
2011, Information Processing Letters
Let $f_{v}$ denote the number of faulty vertices in an n-dimensional hypercube. This note shows that a fault-free cycle of length of at least $2^{n} - 2 f_{v}$ can be embedded in an n-dimensional hypercube with $f_{v} = 2 n - 3$ and $n ⩾ 5$ . This result not only enhances the previously best known result, and also answers a question in [J.-S. Fu, Fault-tolerant cycle embedding in the hypercube, Parallel Computing 29 (2003) 821–832].
Long paths and cycles in hypercubes with faulty vertices
2009, Information Sciences
A fault-free path in the $n$ -dimensional hypercube $Q_{n}$ with $f$ faulty vertices is said to be long if it has length at least $2^{n} - 2 f - 2$ . Similarly, a fault-free cycle in $Q_{n}$ is long if it has length at least $2^{n} - 2 f$ . If all faulty vertices are from the same bipartite class of $Q_{n}$ , such length is the best possible. We show that for every set of at most $2 n - 4$ faulty vertices in $Q_{n}$ and every two fault-free vertices $u$ and $v$ satisfying a simple necessary condition on neighbors of $u$ and $v$ , there exists a long fault-free path between $u$ and $v$ . This number of faulty vertices is tight and improves the previously known results. Furthermore, we show for every set of at most $n^{2} / 10 + n / 2 + 1$ faulty vertices in $Q_{n}$ where $n ⩾ 15$ that $Q_{n}$ has a long fault-free cycle. This is a first quadratic bound, which is known to be asymptotically optimal.
Long paths in hypercubes with conditional node-faults
2009, Information Sciences
Let F be a set of $f ⩽ 2 n - 5$ faulty nodes in an n-cube $Q_{n}$ such that every node of $Q_{n}$ still has at least two fault-free neighbors. Then we show that $Q_{n} - F$ contains a path of length at least $2^{n} - 2 f - 1$ (respectively, $2^{n} - 2 f - 2$ ) between any two nodes of odd (respectively, even) distance. Since the n-cube is bipartite, the path of length $2^{n} - 2 f - 1$ (or $2^{n} - 2 f - 2$ ) turns out to be the longest if all faulty nodes belong to the same partite set. As a contribution, our study improves upon the previous result presented by [J.-S. Fu, Longest fault-free paths in hypercubes with vertex faults, Information Sciences 176 (2006) 759–771] where only $n - 2$ faulty nodes are considered.
Longest fault-free paths in hypercubes with vertex faults
2006, Information Sciences
The hypercube is one of the most versatile and efficient interconnection networks (networks for short) so far discovered for parallel computation. Let f denote the number of faulty vertices in an n-cube. This study demonstrates that when f ⩽ n − 2, the n-cube contains a fault-free path with length at least 2ⁿ − 2f − 1 (or 2ⁿ − 2f − 2) between two arbitrary vertices of odd (or even) distance. Since an n-cube is a bipartite graph with two partite sets of equal size, the path is longest in the worst-case. Furthermore, since the connectivity of an n-cube is n, the n-cube cannot tolerate n − 1 faulty vertices. Hence, our result is optimal.

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^☆: This research was supported by the National Science Council of the Republic of China under Grant # NSC85-2213-E-216-021.

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Embedding a ring in a hypercube with both faulty links and faulty nodes☆

Abstract

J. Parallel Distributed Comput.

J. Parallel Distributed Comput.

J. Parallel Distributed Comput.

J. Parallel Distributed Comput.

J. Parallel Distributed Comput.

On the mapping problem

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Fault-tolerant embeddings of complete binary trees in hypercubes

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