Fault-free longest paths in star networks with conditional link faults

AbstractThe star network, which belongs to the class of Cayley graphs, is one of the most versatile interconnection networks for parallel and distributed computing. In this paper, adopting the conditional fault model in which each node is assumed to be incident with two or more fault-free links, we show that an n-dimensional star network can tolerate up to 2n−7 link faults, and be strongly (fault-free) Hamiltonian laceable, where n≥4. In other words, we can embed a fault-free linear array of length n!−1 (n!−2) in an n-dimensional star network with up to 2n−7 link faults, if the two end nodes belong to different partite sets (the same partite set). The result is optimal with respect to the number of link faults tolerated. It is already known that under the random fault model, an n-dimensional star network can tolerate up to n−3 faulty links and be strongly Hamiltonian laceable, for n≥3

Random induced subgraphs of Cayley graphs induced by transpositions

In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations with independent probability, $\lambda_n$. Our main result is that for any minimal generating set of transpositions, for probabilities $\lambda_n=\frac{1+\epsilon_n}{n-1}$ where $n^{-{1/3}+\delta}\le \epsilon_n0$, a random induced subgraph has a.s. a unique largest component of size $\wp(\epsilon_n)\frac{1+\epsilon_n}{n-1}n!$, where $\wp(\epsilon_n)$ is the survival probability of a specific branching process.Comment: 18 pages, 1 figur

Interconnection networks for parallel and distributed computing

Parallel computers are generally either shared-memory machines or distributed- memory machines. There are currently technological limitations on shared-memory architectures and so parallel computers utilizing a large number of processors tend tube distributed-memory machines. We are concerned solely with distributed-memory multiprocessors. In such machines, the dominant factor inhibiting faster global computations is inter-processor communication. Communication is dependent upon the topology of the interconnection network, the routing mechanism, the flow control policy, and the method of switching. We are concerned with issues relating to the topology of the interconnection network. The choice of how we connect processors in a distributed-memory multiprocessor is a fundamental design decision. There are numerous, often conflicting, considerations to bear in mind. However, there does not exist an interconnection network that is optimal on all counts and trade-offs have to be made. A multitude of interconnection networks have been proposed with each of these networks having some good (topological) properties and some not so good. Existing noteworthy networks include trees, fat-trees, meshes, cube-connected cycles, butterflies, Möbius cubes, hypercubes, augmented cubes, k-ary n-cubes, twisted cubes, n-star graphs, (n, k)-star graphs, alternating group graphs, de Bruijn networks, and bubble-sort graphs, to name but a few. We will mainly focus on k-ary n-cubes and (n, k)-star graphs in this thesis. Meanwhile, we propose a new interconnection network called augmented k-ary n- cubes. The following results are given in the thesis.1. Let k ≥ 4 be even and let n ≥ 2. Consider a faulty k-ary n-cube Q(^k_n) in which the number of node faults f(_n) and the number of link faults f(_e) are such that f(_n) + f(_e) ≤ 2n - 2. We prove that given any two healthy nodes s and e of Q(^k_n), there is a path from s to e of length at least k(^n) - 2f(_n) - 1 (resp. k(^n) - 2f(_n) - 2) if the nodes s and e have different (resp. the same) parities (the parity of a node Q(^k_n) in is the sum modulo 2 of the elements in the n-tuple over 0, 1, ∙∙∙ , k - 1 representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan and Hsu, and by Fu. Furthermore, we extend known results, obtained by Kim and Park, for the case when n = 2.2. We give precise solutions to problems posed by Wang, An, Pan, Wang and Qu and by Hsieh, Lin and Huang. In particular, we show that Q(^k_n) is bi-panconnected and edge-bipancyclic, when k ≥ 3 and n ≥ 2, and we also show that when k is odd, Q(^k_n) is m-panconnected, for m = (^n(k - 1) + 2k - 6’ / ‘_2), and (k -1) pancyclic (these bounds are optimal). We introduce a path-shortening technique, called progressive shortening, and strengthen existing results, showing that when paths are formed using progressive shortening then these paths can be efficiently constructed and used to solve a problem relating to the distributed simulation of linear arrays and cycles in a parallel machine whose interconnection network is Q(^k_n) even in the presence of a faulty processor.3. We define an interconnection network AQ(^k_n) which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube Q(^k_n) has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube Q(^k_n) - is a Cayley graph (and so is vertex-symmetric); has connectivity 4n - 2, and is such that we can build a set of 4n - 2 mutually disjoint paths joining any two distinct vertices so that the path of maximal length has length at most max{{n- l)k- (n-2), k + 7}; has diameter [(^k) / (_3)] + [(^k - 1) /( _3)], when n = 2; and has diameter at most (^k) / (_4) (n+ 1), for n ≥ 3 and k even, and at most [(^k)/ (_4) (n + 1) + (^n) / (_4), for n ^, for n ≥ 3 and k odd.4. We present an algorithm which given a source node and a set of n - 1 target nodes in the (n, k)-star graph S(_n,k) where all nodes are distinct, builds a collection of n - 1 node-disjoint paths, one from each target node to the source. The collection of paths output from the algorithm is such that each path has length at most 6k - 7, and the algorithm has time complexity O(k(^3)n(^4))

Fault-tolerant Hamiltonian laceability of Cayley graphs generated by transposition trees

AbstractA bipartite graph is Hamiltonian laceable if there exists a Hamiltonian path joining every pair of vertices that are in different parts of the graph. It is well known that Cay(Sn,B) is Hamiltonian laceable, where Sn is the symmetric group on {1,2,…,n} and B is a generating set consisting of transpositions of Sn. In this paper, we show that for any F⊆E(Cay(Sn,B)), if |F|≤n−3 and n≥4, then there exists a Hamiltonian path in Cay(Sn,B)−F joining every pair of vertices that are in different parts of the graph. The result is optimal with respect to the number of edge faults

Investigation of the robustness of star graph networks

The star interconnection network has been known as an attractive alternative to n-cube for interconnecting a large number of processors. It possesses many nice properties, such as vertex/edge symmetry, recursiveness, sublogarithmic degree and diameter, and maximal fault tolerance, which are all desirable when building an interconnection topology for a parallel and distributed system. Investigation of the robustness of the star network architecture is essential since the star network has the potential of use in critical applications. In this study, three different reliability measures are proposed to investigate the robustness of the star network. First, a constrained two-terminal reliability measure referred to as Distance Reliability (DR) between the source node u and the destination node I with the shortest distance, in an n-dimensional star network, Sn, is introduced to assess the robustness of the star network. A combinatorial analysis on DR especially for u having a single cycle is performed under different failure models (node, link, combined node/link failure). Lower bounds on the special case of the DR: antipode reliability, are derived, compared with n-cube, and shown to be more fault-tolerant than n-cube. The degradation of a container in a Sn having at least one operational optimal path between u and I is also examined to measure the system effectiveness in the presence of failures under different failure models. The values of MTTF to each transition state are calculated and compared with similar size containers in n-cube. Meanwhile, an upper bound under the probability fault model and an approximation under the fixed partitioning approach on the ( n-1)-star reliability are derived, and proved to be similarly accurate and close to the simulations results. Conservative comparisons between similar size star networks and n-cubes show that the star network is more robust than n-cube in terms of ( n-1)-network reliability

Network Survivability Analysis: Coarse-Graining And Graph-Theoretic Strategies

In this dissertation, the interplay between geographic information about the network and the principal properties and structure of the underlying graph are used to quantify the structural and functional survivability of the network. This work focuses on the local aspect of survivability by studying the propagation of loss in the network as a function of the distance of the fault from a given origin-destination node pair

Pulse propagation, graph cover, and packet forwarding

We study distributed systems, with a particular focus on graph problems and fault tolerance. Fault-tolerance in a microprocessor or even System-on-Chip can be improved by using a fault-tolerant pulse propagation design. The existing design TRIX achieves this goal by being a distributed system consisting of very simple nodes. We show that even in the typical mode of operation without faults, TRIX performs significantly better than a regular wire or clock tree: Statistical evaluation of our simulated experiments show that we achieve a skew with standard deviation of O(log log H), where H is the height of the TRIX grid. The distance-r generalization of classic graph problems can give us insights on how distance affects hardness of a problem. For the distance-r dominating set problem, we present both an algorithmic upper and unconditional lower bound for any graph class with certain high-girth and sparseness criteria. In particular, our algorithm achieves a O(r·f(r))-approximation in time O(r), where f is the expansion function, which correlates with density. For constant r, this implies a constant approximation factor, in constant time. We also show that no algorithm can achieve a (2r + 1 − δ)-approximation for any δ > 0 in time O(r), not even on the class of cycles of girth at least 5r. Furthermore, we extend the algorithm to related graph cover problems and even to a different execution model. Furthermore, we investigate the problem of packet forwarding, which addresses the question of how and when best to forward packets in a distributed system. These packets are injected by an adversary. We build on the existing algorithm OED to handle more than a single destination. In particular, we show that buffers of size O(log n) are sufficient for this algorithm, in contrast to O(n) for the naive approach.Wir untersuchen verteilte Systeme, mit besonderem Augenmerk auf Graphenprobleme und Fehlertoleranz. Fehlertoleranz auf einem System-on-Chip (SoC) kann durch eine fehlertolerante Puls- Weiterleitung verbessert werden. Das bestehende Puls-Weiterleitungs-System TRIX toleriert Fehler indem es ein verteiltes System ist das nur aus sehr einfachen Knoten besteht. Wir zeigen dass selbst im typischen, fehlerfreien Fall TRIX sich weitaus besser verhält als man naiverweise erwarten würde: Statistische Analysen unserer simulierten Experimente zeigen, dass der Verzögerungs-Unterschied eine Standardabweichung von lediglich O(log logH) erreicht, wobei H die Höhe des TRIX-Netzes ist. Das Generalisieren einiger klassischer Graphen-Probleme auf Distanz r kann uns neue Erkenntnisse bescheren über den Zusammenhang zwischen Distanz und Komplexität eines Problems. Für das Problem der dominierenden Mengen auf Distanz r zeigen wir sowohl eine algorithmische obere Schranke als auch eine bedingungsfreie untere Schranke für jede Klasse von Graphen, die bestimmte Eigenschaften an Umfang und Dichte erfüllt. Konkret erreicht unser Algorithmus in Zeit O(r) eine Annäherungsgüte von O(r · f(r)). Für konstante r bedeutet das, dass der Algorithmus in konstanter Zeit eine Annäherung konstanter Güte erreicht. Weiterhin zeigen wir, dass kein Algorithmus in Zeit O(r) eine Annäherungsgüte besser als 2r + 1 erreichen kann, nicht einmal in der Klasse der Kreis-Graphen von Umfang mindestens 5r. Weiterhin haben wir das Paketweiterleitungs-Problem untersucht, welches sich mit der Frage beschäftigt, wann genau Pakete in einem verteilten System idealerweise weitergeleitetwerden sollten. Die Paketewerden dabei von einem Gegenspieler eingefügt. Wir bauen auf dem existierenden Algorithmus OED auf, um mehr als ein Paket-Ziel beliefern zu können. Dadurch zeigen wir, dass Paket-Speicher der Größe O(log n) für dieses Problem ausreichen, im Gegensatz zu den Paket-Speichern der Größe O(n) die für einen naiven Ansatz nötig wären