Longest paths and cycles in faulty star graphs

We investigate hamiltonian properties of Pm × Cn, m ≥ 2 and even n ≥ 4, which is bipartite, in the presence of faulty vertices and/or edges. We show that Pm × Cn with n even is strongly hamiltonian-laceable if the number of faulty elements is one or less. When the number of faulty elements is two, it has a fault-free cycle of length at least mn - 2 unless both faulty elements are contained in the same partite vertex set; otherwise, it has a fault-free cycle of length mn - 4. A sufficient condition is derived for the graph with two faulty edges to have a hamiltonian cycle. By applying fault-hamiltonicity of Pm × Cn to a two-dimensional torus Cm × Cn, we obtain interesting hamiltonian properties of a faulty Cm × Cn.

A generalized p-cycle is a digraph whose set of vertices is partitioned into p parts that can be ordered in such a way that a vertex is adjacent only to vertices in the next part. The families of BGC(p,d,d/sup k/) (de Bruijn Generalized Cycles) and KGC(p,d,d/sup p+k/+d/sup k/) (Kautz Generalized Cycles) are the largest known p-cyles for their degree and diameter. In this paper, we present routing algorithms for both families. Such algorithms route over paths of length at most the value of the diameter plus two units. Moreover, this bound is attained only in the case that the number of faulty elements (nodes or links) is maximum.

Edge-bipancyclicity of star graphs with faulty elements

In this paper, we propose a scheme to solve the problem of adaptive fault diagnosis in n-dimensional twisted cubes in at most three test rounds, provided that the number of faulty vertices is at most n for n ≥ 9. First, each vertex tests one specific neighbour and is tested by another specific neighbour to provide a basic syndrome in two rounds in our scheme. Then, we assign other necessary tests to diagnose the vertices that cannot be identified according to the previous syndrome in one more round. The scheme is optimal for at most three rounds since the adaptive diagnosis needs at least three rounds to complete.KeywordsInterconnection networkstwisted cubeshypercubesadaptive diagnosisHamiltoniandistributed systemsmultiprocessors

Modeling and simulation of high redundancy actuator for fault-tolerance

Let fv (respectively, fe) denote the number of faulty vertices (respectively, edges) in an n-dimensional hypercube. In this paper, we show that a fault-free cycle of length of at least 2n–2fv can be embedded in an n-dimensional hypercube with fe ≤ n – 2 and fv + fe ≤ 2n – 4. Our result not only improves the previously best known result of Sengupta (1998) where fv > 0 or fe ≤ n – 2 and fv + fe ≤ n – 1 were assumed, but also extends the result of Fu (2003) where only the faulty vertices are considered.

Constructing vertex–disjoint paths in ( n, k)-star graphs

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

Long paths in hypercubes with a quadratic number of faults

We determine numerically the probability distribution for the longest self-avoiding path lengths connecting two distant points on a diluted hierarchical lattice at the percolation threshold. The evolution of this distribution with the system size is studied and the distribution is observed to approach a universal scale-invariant form under proper rescaling of its argument. The longest path length scales as |Δp ζmax| and our estimate for ζmax=1.816±0.013 is clearly different from the previously estimated ζmin=1.531+0.002 for the shortest path lengths on the same hierarchical lattice. This gives support to the multifractal behavior of SAWs on percolating clusters.

A note on path embedding in crossed cubes with faulty vertices

The star graph proposed by [1] has many advantages over the n-cube. However it suffers from having large gaps in the number of possible vertices. The (n,k)-star graph was proposed in [18] to address this issue. Since it is a generalization of the star graph, it retains many of the nice properties of the star graph. There are many different measures of structural integrity of interconnection networks. In this paper, we prove results of the following type for the (n,k)-star graph. If n + (r - 1)k - g(r) vertices are deleted from an (n,k)-star graph, the resulting graph will either be connected or has a large component and small components having at most r - 1 vertices in total. Additional results on conditional vertex connectivity and cycle connectivity will also be given.

Given an undirected graph G and two vertex subsets H 1 and H 2, the smallest bi-level augmentation problem is that of adding to G the smallest number of edges such that G contains two internally vertex-disjoint paths between every pair of vertices in H 1 and two edgedisjoint paths between every pair of vertices in H 2. We solve the bi-level augmentation problem in O(n + m) time, where n and m are the numbers of vertices and edges in G. Our algorithm can be parallelized to run in O(log2 n) time using n + m processors on an EREW PRAM.KeywordsCritical NodeGraph ProblemVertex SubsetConnectivity RequirementAugmentation ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We introduce a model for differentially private analysis of weighted graphs in which the graph topology (υ,e) is assumed to be public and the private information consists only of the edge weights ω : e → R+. This can express hiding congestion patterns in a known system of roads. Differential privacy requires that the output of an algorithm provides little advantage, measured by privacy parameters e and δ, for distinguishing between neighboring inputs, which are thought of as inputs that differ on the contribution of one individual. In our model, two weight functions w,w' are considered to be neighboring if they have l1 distance at most one.We study the problems of privately releasing a short path between a pair of vertices and of privately releasing approximate distances between all pairs of vertices. We are concerned with the approximation error, the difference between the length of the released path or released distance and the length of the shortest path or actual distance.For the problem of privately releasing a short path between a pair of vertices, we prove a lower bound of Ω(|υ|) on the additive approximation error for fixed privacy parameters e,δ. We provide a differentially private algorithm that matches this error bound up to a logarithmic factor and releases paths between all pairs of vertices, not just a single pair. The approximation error achieved by our algorithm can be bounded by the number of edges on the shortest path, so we achieve better accuracy than the worst-case bound for pairs of vertices that are connected by a low-weight path consisting of o(|υ|) vertices.For the problem of privately releasing all-pairs distances, we show that for trees we can release all-pairs distances with approximation error $O(log2.5|υ|) for fixed privacy parameters. For arbitrary bounded-weight graphs with edge weights in [0,M] we can brelease all distances with approximation error O(√>(|υ|M).

The $t/k$ -diagnosis strategy can significantly enhance the system’s self-diagnosing capability at the expense of no more than $k$ fault-free processors (vertices) being mistakenly diagnosed as faulty under the PMC model. It is a generalization of the precise and pessimistic diagnosis strategies of system-level diagnosis on multiprocessor systems. It can detect up to $t$ faulty processors (vertices) which might include at most $k$ misdiagnosed processors (vertices), where $k$ is typically a small number. In the case $k\ge 1$ , to our knowledge, there is no known $t/k$ -diagnosis algorithm for general regular networks. In this paper, we first propose a general $t/k$ -diagnosis ( $k\ge 1$ ) algorithm for some $m$ -regular networks. These $m$ -regular networks satisfying some conditions could establish the $t/k$ -diagnosis algorithm, say $t/k$ - $G$ - $DIAG$ , to determine the $t/k$ -diagnosability. The complexity of this algorithm is only $O(N\log N)$ (when $N\ge 2^m$ ) or $O(Nm)$ (when $N ) where $N$ is the number of vertices in the network. Second, we present a complete proof that the network $G$ is actually $t/k$ -diagnosable. Finally, we establish the $t/k$ -diagnosability ( $1\le k\le 3$ ) of some regular networks, including an $n$ -dimensional alternating group graph , an $n$ -dimensional Split-Star Network , a $l^n$ -hypermesh and an $(n,l)$ -star graph , which are well-known interconnection networks proposed for multiprocessor systems.