Improving bounds on link failure tolerance of the star graph

Determination of the minimum number of faulty links, f(n,k), that make every n-k-dimensional sub-star graph S_n-k faulty in an n-dimensional star network S_n, has been the subject of several studies. Bounds on f(n,k) have already been derived, and it is known that f(n,1)=n+2. Here, we improve the bounds on f(n,k). Specifically, it is shown that f(n,k)?(k+1)F(n,k), where F(n,k) is the minimum number of faulty nodes that make every S_n-k faulty in S_n. The complexity of f(n,k) is shown to be O(n²k) which is an improvement over the previously known upper bound of O(n³); this result in a special case leads to f(n,2)=O(n^↑2), settling a conjecture introduced in an earlier paper. A systematic method to derive the labels of the faulty links in case of f(n,1) is also introduced.     

Robustness of star graph network under link failure

The star graph is an attractive underlying topology for distributed systems. Robustness of the star graph under link failure model is addressed. Specifically, the minimum number of faulty links, f(n, k), that make every (n − k)-dimensional substar S_n−k faulty in an n-dimensional star network S_n, is studied. It is shown that f(n,1)=n+2. Furthermore, an upper bound is given for f(n, 2) with complexity of O(n³) which is an improvement over the straightforward upper bound of O(n⁴) derived in this paper.     

Properties of a hierarchical network based on the star graph

This paper introduces a new class of interconnection networks named star-pyramid. An n-level star-pyramid is formed by piling up star graphs of dimensions 1 to n in a hierarchy, connecting any node in each i-dimensional star, 1 < i ? n, to a node in the (i − 1)-dimensional star whose index is reached by removing the i symbol from the index of the former node in the i-dimensional star graph. Having extracted the properties of the new topology, featuring topological properties, a minimal routing algorithm, a simple but efficient broadcast algorithm, Hamiltonicity and pancyclicity, we then compare the network properties of the proposed topology and the well-known pyramid topology. We show that the star-pyramid has some more attractive properties than its equivalent pyramid. Finally, we propose two variants of star-pyramid, namely the generic star-pyramid and wrapped star-pyramid, as topologies with improved scalability, fault-tolerance, and diameter.     

Disjoint Hamilton cycles in the star graph

In 1987, Akers, Harel and Krishnamurthy proposed the star graph Σ(n) as a new topology for interconnection networks. Hamiltonian properties of these graphs have been investigated by several authors. In this paper, we prove that Σ(n) contains ⌊n/8⌋ pairwise edge-disjoint Hamilton cycles when n is prime, and Ω(n/loglogn) such cycles for arbitrary n.     

Probabilistic analysis on mesh network fault tolerance

Mesh networks are among the most important interconnection network topologies for large multicomputer systems. Mesh networks perform poorly in tolerating faults in the view of worst-case analysis. On the other hand, such worst cases occur very rarely in realistic situations. In this paper, we study the fault tolerance of 2-D and 3-D mesh networks under a more realistic model in which each network node has an independent failure probability. We first observe that if the node failure probability is fixed, then the connectivity probability of these mesh networks can be arbitrarily small when the network size is sufficiently large. Thus, it is practically important for multicomputer system manufacture to determine the upper bound for node failure probability when the probability of network connectivity and the network size are given. We develop a novel technique to formally derive lower bounds on the connectivity probability for 2-D and 3-D mesh networks. Our study shows that these mesh networks of practical size can tolerate a large number of faulty nodes thus are reliable enough for multicomputer systems. For example, it is formally proved that as long as the node failure probability is bounded by 0.5%$0.5\%$, a 3-D mesh network of up to a million nodes remains connected with a probability larger than 99%$99\%$.     

Diagnosability of star graphs under the comparison diagnosis model

In this paper, the diagnosability of n-dimensional star graph Sn under the comparison diagnosis model has been studied. It is proved that Sn is (n−1)-diagnosable under the comparison diagnosis model when n?4.     

An accurate mathematical performance model of adaptive routing in the star graph

Analytical modelling is indeed the most cost-effective method to evaluate the performance of a system. Several analytical models have been proposed in the literature for different interconnection network systems. This paper proposes an accurate analytical model to predict message latency in wormhole-switched star graphs with fully adaptive routing. Although the focus of this research is on the star graph but the approach used for modelling can be, however, used for modelling some other regular and irregular interconnection networks. The results obtained from simulation experiments confirm that the proposed model exhibits a good accuracy for various network sizes and under different operating conditions.     

A kind of conditional fault tolerance of alternating group graphs

A vertex subset F is a k-restricted vertex-cut of a connected graph G if G−F is disconnected and every vertex in G−F has at least k good neighbors in G−F. The cardinality of the minimum k-restricted vertex-cut of G is the k-restricted connectivity of G, denoted by κk(G). This parameter measures a kind of conditional fault tolerance of networks. In this paper, we show that for the n-dimensional alternating group graph AGn, κ²(AG₄)=4 and κ²(AGn)=6n−18 for n?5.     

Conditional fault hamiltonian connectivity of the complete graph

A path in G is a hamiltonian path if it contains all vertices of G. A graph G is hamiltonian connected if there exists a hamiltonian path between any two distinct vertices of G. The degree of a vertex u in G is the number of vertices of G adjacent to u. We denote by δ(G) the minimum degree of vertices of G. A graph G is conditional k edge-fault tolerant hamiltonian connected if G−F is hamiltonian connected for every F⊂E(G) with |F|?k and δ(G−F)?3. The conditional edge-fault tolerant hamiltonian connectivity is defined as the maximum integer k such that G is k edge-fault tolerant conditional hamiltonian connected if G is hamiltonian connected and is undefined otherwise. Let n?4. We use Kn to denote the complete graph with n vertices. In this paper, we show that for n∉{4,5,8,10}, , , , and .     

Characterization of spatial fault patterns in interconnection networks

Parallel computers, such as multiprocessors system-on-chip (Mp-SoCs), multicomputers and cluster computers, are consisting of hundreds or thousands multiple processing units and components (such as routers, channels and connectors) connected via some interconnection network that collectively may undergo high failure rates. Therefore, these systems are required to be equipped with fault-tolerant mechanisms to ensure that the system will keep running in a degraded mode. Normally, the faulty components are coalesced into fault regions, which are classified into two major categories: convex and concave regions. In this paper, we propose the first solution to calculate the probability of occurrences of common fault patterns in torus and mesh interconnection networks which includes both convex (-shaped, □-shaped) and concave (L-shaped, T-shaped, +-shaped, H-shaped) regions. These results play a key role when studying, particularly, the performance analysis of routing algorithms proposed for interconnection networks under faulty conditions.     

Perfect load balancing on the star interconnection network

In this paper, we use the regular distribution method to design a perfect load balancing algorithm for an n-star with a maximum error of 1 and a time complexity of 3n(n+1). This algorithm is based on the novel notion of leader trees. A second algorithm proposed in this paper as an enhancement to our first algorithm and uses an arbitrary spanning tree as the leader tree and has a worst time complexity of 2.25n ²−3n+0.75. We also discuss the issue of dynamically selecting the leader tree and hybrid load balancing algorithms in general. Furthermore, we present a hybrid algorithm for load balancing on the star interconnection network which benefits from a diffusion load balancing preprocessing phase and shows a smaller mean time complexity than our two first algorithms.     

Edge-bipancyclicity of star graphs with faulty elements

In this paper, we investigate the fault-tolerant edge-bipancyclicity of an n-dimensional star graph S_n. Given a set F comprised of faulty vertices and/or edges in S_n with |F|≤n−3 and any fault-free edge e in S_n−F, we show that there exist cycles of every even length from 6 to n!−2|F_v| in S_n−F containing e, where n≥3. Our result is optimal because the star graph is both bipartite and regular with the common degree n−1. The length of the longest fault-free cycle in the bipartite S_n is n!−2|F_v| in the worst case, where all faulty vertices are in the same partite set. We also provide some sufficient conditions from which longer cycles with length from n!−2|F_v|+2 to n!−2|F_v| can be embedded.     

Feedback vertex sets in star graphs

In a graph G=(V,E), a subset F⊂V(G) is a feedback vertex set of G if the subgraph induced by V(G)?F is acyclic. In this paper, we propose an algorithm for finding a small feedback vertex set of a star graph. Indeed, our algorithm can derive an upper bound to the size of the feedback vertex set for star graphs. Also by applying the properties of regular graphs, a lower bound can easily be achieved for star graphs.     

A fault-tolerant wormhole routing scheme for torus networks with nonconvex faults

In this paper, we present a fault-tolerant routing algorithm for torus networks by using only 4 virtual channels. The proposed algorithm is based on the solid fault model, which includes rectangular faults and many practical nonconvex faults. Previous works need at least 6 virtual channels to achieve the same fault-tolerant ability.     

星型网络的3-限制边连通性

星型互连网络是并行与分布式处理领域中最流行的互连网络之一,它以n维星图作为拓扑结构。k-限制边连通度是衡量网络的可靠性的参数之一。一般来说,一个网络的k-限制边连通度越大,其连通性就越好。研究了星型互连网络的k限制边连通度;证明了当n≥3时,n维星型互连网络的3-限制连通度为3n－7。     

End-to-end fault tolerance

In this review we survey both standard fault tolerance theory and Kitaev’s model for quantum computation, and demonstrate how they can be combined to yield quantitative results that reveal the interplay between the two. This analysis establishes a methodology allowing one to quantitatively determine design parameters for quantum computers, the values of which ensure that an overall computation yields a correct final result with some prescribed probability of success, as opposed to merely ensuring that the desired final quantum state is obtained. As an example, we explicitly calculate the number of levels of error correction concatenation needed to achieve a correct final result with some prescribed success probability. This methodology allows one to determine parameters required in order to achieve the correct final result for the quantum computation, as opposed to merely ensuring that the desired final quantum state is produced.      

Fault-diameter of Cartesian graph bundles

Cartesian graph bundles is a class of graphs that is a generalization of the Cartesian graph products. Let G be a kG-connected graph and Dc(G) denote the diameter of G after deleting any of its c<kG vertices. We prove that Da+b+1(G)?Da(F)+Db(B)+1 if G is a graph bundle with fibre F over base B, a<kF, and b<kB.