Lih-Hsing Hsu

Conditional diagnosability measures for large multiprocessor systems

Diagnosability has played an important role in the reliability of an interconnection network. The classical problem of fault diagnosis is discussed widely and the diagnosability of many well-known networks have been explored. We introduce a new measure of diagnosability, called conditional diagnosability, by restricting that any faulty set cannot contain all the neighbors of any vertex in the graph. Based on this requirement, the conditional diagnosability of the n-dimensional hypercube is shown to be 4(n - 2) +1, which is about four times as large as the classical diagnosability. Besides, we propose some useful conditions for verifying if a system is t-diagnosable and introduce a new concept, called a strongly t-diagnosable system, under the PMC model. Applying these concepts and conditions, we investigate some t-diagnosable networks which are also strongly t-diagnosable.

Fault hamiltonicity and fault hamiltonian connectivity of the arrangement graphs

The arrangement graph A/sub n,k/ is a generalization of the star graph. There are some results concerning fault Hamiltonicity and fault Hamiltonian connectivity of the arrangement graph. However, these results are restricted in some particular cases and, thus, are less completed. We improve these results and obtain a stronger and simpler statement. Let n-k/spl ges/2 and F/spl sube/V(A/sub n,k/)/spl cup/E(A/sub n,k/). We prove that A/sub n,k/-F is Hamiltonian if |F|/spl les/k(n-k)-2 and A/sub n,k/-F is Hamiltonian connected if |F|/spl les/k(n-k)-3. These results are optimal.

Fault-Tolerant Hamiltonicity of Twisted Cubes

The twisted cube TQn, is derived by changing some connection of hypercube Qn according to specific rules. Recently, many topological properties of this variation cube are studied. In this paper, we consider a faulty twisted n-cube with both edge and/or node faults. Let F be a subset of V(TQn)?E(TQn), we prove that TQn?F remains hamiltonian if |F|?n?2. Moreover, we prove that there exists a hamiltonian path in TQn?F joining any two vertices u, v in V(TQn)?F if |F|?n?3. The result is optimum in the sense that the fault-tolerant hamiltonicity (fault-tolerant hamiltonian connectivity respectively) of TQn is at most n?2 (n?3 respectively).

Fault-tolerant hamiltonian laceability of hypercubes

It is known that every hypercube Qn is a bipartite graph. Assume that n ≥ 2 and F is a subset of edges with |F| ≤ n - 2. We prove that there exists a hamiltonian path in Qn - F between any two vertices of different partite sets. Moreover, there exists a path of length 2n - 2 between any two vertices of the same partite set. Assume that n ≥ 3 and F is a subset of edges with |F| ≤ n - 3. We prove that there exists a hamiltonian path in Qn - {υ} - F between any two vertices in the partite set without υ. Furthermore, all bounds are tight.

Hyper hamiltonian laceability on edge fault star graph

The star graph possess many nice topological properties. Edge fault tolerance is an important issue for a network since the edges in the network may fail sometimes. In this paper, we show that the n-dimensional star graph is (n - 3)-edge fault tolerant hamiltonian laceable, (n - 3)-edge fault tolerant strongly hamiltonian laceable, and (n - 4)- edge fault tolerant hyper hamiltonian laceable. All these results are optimal in a sense described in this paper.

Fault-tolerant cycle-emebedding of crossed cubes

The crossed cube CQn introduced by Efe has many properties similar to those of the popular hypercube. However, the diameter of CQn is about one half of that of the hypercube. Failures of links and nodes in an interconnection network are inevitable. Hence, in this paper, we consider the hybrid fault-tolerant capability of the crossed cube. Letting fe and fv be the numbers of faulty edges and vertices in CQn, we show that a cycle of length l, for any 4 ≤ l ≤ |V(CQn)| - fv, can be embedded into a wounded crossed cube as long as the total number of faults (fv + fv) is no more than n - 2, and we say that CQn is (n - 2)-fault-tolerant pancyclic. This result is optimal in the sense that if there are n - 1 faults, there is no guarantee of having a cycle of a certain length in it.

CONDITIONAL DIAGNOSABILITY OF CAYLEY GRAPHS GENERATED BY TRANSPOSITION TREES UNDER THE COMPARISON DIAGNOSIS MODEL

The diagnosis of faulty processors plays an important role in multiprocessor systems for reliable computing, and the diagnosability of many well-known networks has been explored. Zheng et al. showed that the diagnosability of the n-dimensional star graph Sn is n - 1. Lai et al. introduced a restricted diagnosability of multiprocessor systems called conditional diagnosability. They consider the situation when no faulty set can contain all the neighbors of any vertex in the system. In this paper, we study the conditional diagnosability of Cayley graphs generated by transposition trees (which include the star graphs) under the comparison model, and show that it is 3n - 8 for n ≥ 4, except for the n-dimensional star graph, for which it is 3n - 7. Hence the conditional diagnosability of these graphs is about three times larger than their classical diagnosability.

The diagnosability of the matching composition network under the comparison diagnosis model

The classical problem of diagnosability is discussed widely and the diagnosability of many well-known networks has been explored. We consider the diagnosability of a family of networks, called the matching composition network (MCN); a perfect matching connects two components. The diagnosability of MCN under the comparison model is shown to be one larger than that of the component, provided some connectivity constraints are satisfied. Applying our result, the diagnosability of the hypercube Qn, the crossed cube CQ/sub n/, the twisted cube TQ/sub n/, and the Mobius cube MQ/sub n/ can all proven to be n, for n/spl ges/4. In particular, we show that the diagnosability of the four-dimensional hypercube Q/sub 4/ is 4, which is not previously known.

Hamiltonian circuit and linear array embeddings in faulty k-ary n-cubes

In this paper, we investigate the fault-tolerant capabilities of the k-ary n-cubes for even integer k with respect to the hamiltonian and hamiltonian-connected properties. The k-ary n-cube is a bipartite graph if and only if k is an even integer. Let F be a faulty set with nodes and/or links, and let k>=3 be an odd integer. When |F|=<2n-2, we show that there exists a hamiltonian cycle in a wounded k-ary n-cube. In addition, when |F|=<2n-3, we prove that, for two arbitrary nodes, there exists a hamiltonian path connecting these two nodes in a wounded k-ary n-cube. Since the k-ary n-cube is regular of degree 2n, the degrees of fault-tolerance 2n-3 and 2n-2 respectively, are optimal in the worst case.

Long paths in hypercubes with conditional node-faults

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.