Faulty Hypercubes Research Articles

Vertex-disjoint paths joining adjacent vertices in faulty hypercubes

Let Qn denote the n-dimensional hypercube and the set of faulty edges and faulty vertices in Qn be denoted by Fe and Fv, respectively. In this paper, we investigate Qn (n≥3) with |Fe|+|Fv|≤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≤ℓ(P[a,b])+ℓ(P[c,d])≤2n−2|Fv|−2, where 2|(ℓ(P[a,b])+ℓ(P[c,d])), (a,b),(c,d)∈E(Qn). The contribution of this paper is: (1) we can quickly obtain the interesting result that Qn−Fe is bipancyclic, where |Fe|≤n−2 and n≥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,…,2n−2|Fv| vertices respectively in Qn when S={a,c}, T={b,d}, and (a,b),(c,d)∈E(Qn). Our result is optimal with respect to the number of fault-tolerant elements.

Hamiltonian laceability in hypercubes with faulty edges

It is useful to consider faulty networks because node faults or link faults may occur in networks. In this paper, we investigate hamiltonian properties of conditional faulty hypercubes. Let F be a set of faulty edges in hypercube Qn with n≥4 and |F|≤3n−11. We prove that there still exists a hamiltonian path in Qn−F joining any two vertices of different partite sets if the following two constraints are satisfied: (1) the degree of every vertex in Qn−F is at least 2, and (2) there is at most one vertex with degree 2 in Qn−F.

Fault-tolerant edge-bipancyclicity of faulty hypercubes under the conditional-fault model

It is well-known that the n-dimensional hypercube Qn 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 Qn and let fv, fe 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 Qn for n ≥ 3 lies on a fault-free cycle of any even length from 6 to 2n−2fv if fv+fe≤2n−5,fe≤n−2 and both x and y are incident to at least two free edges. This result confirms a conjecture reported in the literature.

Fault-tolerant cycles embedding in hypercubes with faulty edges

Let Qn be an n-dimensional hypercube with fe⩽3n-8 faulty edges and n⩾5. In this paper, we consider the faulty hypercube under the following two additional conditions: (1) each vertex is incident to at least two fault-free edges, and (2) every 4-cycle does not have any pair of non-adjacent vertices whose degrees are both two after removing the faulty edges. We prove that there exists a fault-free cycle of every even length from 4 to 2n in Qn. Our result improves the result by Liu and Wang (2014) in terms of the lengths of embedding cycles, where under the same conditions, a fault-free Hamiltonian cycle was constructed.

Path Coverings with Prescribed Ends in Faulty Hypercubes

We discuss the existence of vertex disjoint path coverings with prescribed ends for the \(n\)-dimensional hypercube with or without deleted vertices. Depending on the type of the set of deleted vertices and desired properties of the path coverings we establish the minimal integer \(m\) such that for every \(n \ge m\) such path coverings exist. Using some of these results, for \(k \le 4\), we prove Locke’s conjecture that a hypercube with \(k\) deleted vertices of each parity is Hamiltonian if \(n \ge k +2.\) Some of our lemmas substantially generalize known results of I. Havel and T. Dvořák. At the end of the paper we formulate some conjectures supported by our results.

Paired many-to-many disjoint path covers in faulty hypercubes

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, Qm, 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+2k⩽m. This implies that Qm with m−2 or less faulty elements is strongly Hamiltonian-laceable.

Fault-tolerant cycle embedding in the faulty hypercubes

The hypercube is one of the best known interconnection networks. Embedding cycles of all possible lengths in faulty hypercubes has received much attention. Let Fv (respectively, Fe) denote the set of faulty vertices (respectively, faulty edges) and fv (respectively, fe) denote the number of faulty vertices (respectively, faulty edge) in an n-dimensional hypercube Qn. Let f(e) denote the number of faulty nodes and/or faulty edges incident with the end-vertices of an edge e∈E(Qn). 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∣Fv∣ if ∣Fv∣+∣Fe∣⩽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 fv+fe⩽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.

On the mutually independent Hamiltonian cycles in faulty hypercubes

Two ordered Hamiltonian paths in the n-dimensional hypercube Qn are said to be independent if ith vertices of the paths are distinct for every 1⩽i⩽2n. Similarly, two s-starting Hamiltonian cycles are independent if the ith vertices of the cycle are distinct for every 2⩽i⩽2n. A set S of Hamiltonian paths (s-starting Hamiltonian cycles) are mutually independent if every two paths (cycles, respectively) from S are independent. We show that for n pairs of adjacent vertices wi and bi, there are n mutually independent Hamiltonian paths with endvertices wi, bi in Qn. We also show that Qn contains n−f fault-free mutually independent s-starting Hamiltonian cycles, for every set of f⩽n−2 faulty edges in Qn and every vertex s. This improves previously known results on the numbers of mutually independent Hamiltonian paths and cycles in the hypercube with faulty edges.

On the maximum number of fault-free mutually independent Hamiltonian cycles in the faulty hypercube

Hsieh and Yu (2007) first claimed that an injured n-dimensional hypercube Q n contains (n?1?f)-mutually independent fault-free Hamiltonian cycles, where f≤n?2 denotes the total number of permanent edge-faults in Q n for n?4, and edge-faults can occur everywhere at random. Later, Kueng et al. (2009a) presented a formal proof to validate Hsieh and Yu's argument. This paper aims to improve this mentioned result by showing that up to (n?f)-mutually independent fault-free Hamiltonian cycles can be embedded under the same condition. Let F denote the set of f faulty edges. If all faulty edges happen to be incident with an identical vertex s, i.e., the minimum degree of the survival graph Q n ?F is equal to n?f, then Q n ?F contains at most (n?f)-mutually independent Hamiltonian cycles starting from s. From such a point of view, the presented result is optimal. Thus, not only does our improvement increase the number of mutually independent fault-free Hamiltonian cycles by one, but also the optimality can be achieved.

Bipanconnectivity of faulty hypercubes with minimum degree

In this paper, we consider the conditionally faulty hypercube Q n with n ⩾ 2 where each vertex of Q n is incident with at least m fault-free edges, 2 ⩽ m ⩽ n − 1. We shall generalize the limitation m ⩾ 2 in all previous results of edge-bipancyclicity. We also propose a new edge-fault-tolerant bipanconnectivity called k-edge-fault-tolerant bipanconnectivity. A bipartite graph is k-edge-fault-tolerant bipanconnected if G − F remains bipanconnected for any F ⊂ E( G) with ∣ F∣ ⩽ k. For every integer m, under the same hypothesis, we show that Q n is ( n − 2)-edge-fault-tolerant edge-bipancyclic and bipanconnected, and the results are optimal with respect to the number of edge faults tolerated. This not only improves some known results on edge-bipancyclicity and bipanconnectivity of hypercubes, but also simplifies the proof.

Computational complexity of long paths and cycles in faulty hypercubes

The problem of existence of an optimal-length (long) fault-free cycle in the n -dimensional hypercube with f faulty vertices is NP-hard. This holds even in case that f is bounded by a polynomial of degree three (six) with respect to n . On the other hand, there is a linear (quadratic) bound on f which guarantees that the problem is decidable in polynomial time. Similar results are obtained for paths as well as for paths between prescribed endvertices.

Extended Fault-Tolerant Cycle Embedding in Faulty Hypercubes

We consider fault-tolerant embedding, where an n-dimensional faulty hypercube, denoted by <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , acts as the host graph, and the longest fault-free cycle represents the guest graph. Let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> be a set of faulty nodes in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> . Also, let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> be a set of faulty edges in which at least one end-node of each edge is faulty, and let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Fe</i> be a set of faulty edges in which the end-nodes of each edge are both fault-free. An edge in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> is said to be critical if it is either fault-free or in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> . In this paper, we prove that there exists a fault-free cycle of length at least 2 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -2| <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> | in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ges 3) with | <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> | les 2n-5, and | <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> |+| <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> | les 2n-4 , in which each node is incident to at least two critical edges. Our result improves on the previously best known results reported in the literature, where only faulty nodes or faulty edges are considered.

Long paths and cycles in faulty hypercubes: existence, optimality, complexity

Abstract A fault-free cycle in the n-dimensional hypercube Q n with f faulty vertices 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 prove a conjecture of Castaneda and Gotchev [N. Castaneda and I. S. Gotchev. Embedded paths and cycles in faulty hypercubes. J. Comb. Optim., 2009. doi:10.1007/s10878-008-9205-6 .] asserting that f n = ( n 2 ) − 2 where f n for every set of at most f n faulty vertices, there exists a long fault-free cycle in Q n . Furthermore, we present several results on similar problems of long paths and long routings in faulty hypercubes and their complexity.

Many-to-many disjoint paths in faulty hypercubes

This paper considers the problem of many-to-many disjoint paths in the hypercube Qn with fv faulty vertices and fe faulty edges, and obtains the following result. For any integer k with 1⩽k⩽n-1, any two sets S and T of k fault-free vertices in different parts, if fv+fe⩽n-k-1, then there exist k disjoint fault-free (S,T)-paths in Qn which contains at least 2n-2fv vertices. This result is optimal in the worst case.

Embedded paths and cycles in faulty hypercubes

An important task in the theory of hypercubes is to establish the maximum integer f n such that for every set ? of f vertices in the hypercube ${\mathcal {Q}}_{n},$ with 0?f?f n , there exists a cycle of length at least 2 n ?2f in the complement of ?. Until recently, exact values of f n were known only for n?4, and the best lower bound available for f n with n?5 was 2n?4. We prove that f 5=8 and obtain the lower bound f n ?3n?7 for all n?5. Our results and an example provided in the paper support the conjecture that $f_{n}={n\choose 2}-2$ for each n?4. New results regarding the existence of longest fault-free paths with prescribed ends are also proved.

Partitions of Faulty Hypercubes into Paths with Prescribed Endvertices

Given a set $\pc=\{a_i,b_i\}_{i=1}^m$ of pairs of vertices in a graph $G$, is there a collection of paths $\{P_i\}_{i=1}^m$ such that $P_i$ connects $a_i$ with $b_i$ and $\{V(P_i)\}_{i=1}^m$ partitions $V(G)$? We study this problem for the graph $Q_n-\ff$ obtained from the $n$-dimensional hypercube $Q_n$ by removing a set $\ff$ of faulty vertices. We show that an obvious necessary condition for the existence of such a partition is also sufficient provided $2|\pc|+ 3|\ff|\le n-3$. As a corollary, we obtain a similar characterization for the existence of a hamiltonian cycle and a hamiltonian path of $Q_n-\ff$ provided $|\ff|\le(n-5)/3$. On the other hand, if the size of $\ff$ is not limited, the problems are NP-complete.

Edge-bipancyclicity of conditional faulty hypercubes

Xu et al. showed that for any set of faulty edges F of an n-dimensional hypercube Q n with | F | ⩽ n − 1 , each edge of Q n − F lies on a cycle of every even length from 6 to 2 n , n ⩾ 4 , provided not all edges in F are incident with the same vertex. In this paper, we find that under similar condition, the number of faulty edges can be much greater and the same result still holds. More precisely, we show that, for up to | F | = 2 n − 5 faulty edges, each edge of the faulty hypercube Q n − F lies on a cycle of every even length from 6 to 2 n with each vertex having at least two healthy edges adjacent to it, for n ⩾ 3 . Moreover, this result is optimal in the sense that there is a set F of 2 n − 4 conditional faulty edges in Q n such that Q n − F contains no hamiltonian cycle.

A Local Diagnosability Measure for Multiprocessor Systems

The problem of fault diagnosis has been discussed widely and the diagnosability of many well-known networks has been explored. Under the PMC model, we introduce a new measure of diagnosability, called local diagnosability, and derive some structures for determining whether a vertex of a system is locally t-diagnosable. For a hypercube, we prove that the local diagnosability of each vertex is equal to its degree under the PMC model. Then, we propose a concept for system diagnosis, called the strong local diagnosability property. A system G(V, E) is said to have a strong local diagnosability property if the local diagnosability of each vertex is equal to its degree. We show that an n-dimensional hypercube Qn has this strong property, nges3. Next, we study the local diagnosability of a faulty hypercube. We prove that Qn keeps this strong property even if it has up to n-2 faulty edges. Assuming that each vertex of a faulty hypercube Qn is incident with at least two fault-free edges, we prove Qn keeps this strong property even if it has up to 3(n-2)-1 faulty edges. Furthermore, we prove that Qn keeps this strong property no matter how many edges are faulty, provided that each vertex of a faulty hypercube Qn is incident with at least three fault-free edges. Our bounds on the number of faulty edges are all tight

Path embedding in faulty hypercubes

There are some interesting results concerning longest paths or even cycles embedding in faulty hypercubes. This paper considers the embeddings of paths of all possible lengths between any two fault-free vertices in faulty hypercubes. Let f v (respectively, f e ) denote the number of faulty vertices (respectively, edges) in an n-dimensional hypercube Q n . We prove that there exists a fault-free path of length l between any two distinct fault-free vertices u and v in Q n with f v + f e ⩽ n - 2 for each l satisfying d Q n ( u , v ) + 2 ⩽ l ⩽ 2 n - 2 f v - 1 and 2 | ( l - d Q n ( u , v ) ) . The bounds on path length l and faulty set size f v + f e for a successful embedding are tight. That is, the result does not hold if l < d Q n ( u , v ) + 2 or l > 2 n - 2 f v - 1 or f v + f e > n - 2 . Moreover, our result improves some known results.

A (4 n − 9)/3 diagnosis algorithm on n-dimensional cube network

As a generalization of the precise and pessimistic diagnosis strategies of system-level diagnosis of multicomputers, the t/ k diagnosis strategy can significantly improve the self-diagnosing capability of a system at the expense of no more than k fault-free processors (nodes) being mistakenly diagnosed as faulty. In the case k ⩾ 2, to our knowledge, there is no known t/ k diagnosis algorithm for general diagnosable system or for any specific system. Hypercube is a popular topology for interconnecting processors of multicomputers. It is known that an n-dimensional cube is (4 n − 9)/3-diagnosable. This paper addresses the (4 n − 9)/3 diagnosis of n-dimensional cube. By exploring the relationship between a largest connected component of the 0-test subgraph of a faulty hypercube and the distribution of the faulty nodes over the network, the fault diagnosis of an n-dimensional cube can be reduced to those of two constituent ( n − 1)-dimensional cubes. On this basis, a diagnosis algorithm is presented. Given that there are no more than 4 n − 9 faulty nodes, this algorithm can isolate all faulty nodes to within a set in which at most three nodes are fault-free. The proposed algorithm can operate in O( N log 2 N) time, where N = 2 n is the total number of nodes of the hypercube. The work of this paper provides insight into developing efficient t/ k diagnosis algorithms for larger k value and for other types of interconnection networks.