Adaptive fault-tolerant routing in hypercube multicomputers

A connected hypercube with faulty links and/or nodes is called an injured hypercube. To enable any non-faulty node to communicate with any other non-faulty node in an injured hypercube, the information on component failures has to be made available to non-faulty nodes so as to route messages around the faulty components. A distributed adaptive fault tolerant routing scheme is proposed for an injured hypercube in which each node is required to know only the condition of its own links. Despite its simplicity, this scheme is shown to be capable of routing messages successfully in an injured hypercube as long as the number of faulty components is less than n. Moreover, it is proved that this scheme routes messages via shortest paths with a rather high probabiltiy and the expected length of a resulting path is very close to that of a shortest path. Since the assumption that the number of faulty components is less than n in an n-dimensional hypercube might limit the usefulness of the above scheme, a routing scheme is introduced based on depth-first search which works in the presence of an arbitrary number of faulty components. Due to the insufficient information on faulty components, the paths chosen by the above scheme may not always be the shortest. To guarantee that all messages be routed via shortest paths, it is proposed that every mode be equipped with more information than that on its own links. The effects of this additional information on routing efficiency are analyzed, and the additional information to be kept at each node for the shortest path routing is determined. Several examples and remarks are also given to illustrate the results

From Hall's Matching Theorem to Optimal Routing on Hypercubes

AbstractWe introduce a concept of so-called disjoint ordering for any collection of finite sets. It can be viewed as a generalization of a system of distinctive representatives for the sets. It is shown that disjoint ordering is useful for network routing. More precisely, we show that Hall's “marriage” condition for a collection of finite sets guarantees the existence of a disjoint ordering for the sets. We next use this result to solve a problem in optimal routing on hypercubes. We give a necessary and sufficient condition under which there are internally node-disjoint paths each shortest from a source node to any others(s⩽n) target nodes on ann-dimensional hypercube. When this condition is not necessarily met, we show that there are always internally node-disjoint paths each being either shortest or near shortest, and the total length is minimum. An efficient algorithm is also given for constructing disjoint orderings and thus disjoint short paths. As a consequence, Rabin's information disposal algorithm may be improved

Near-optimal broadcast in all-port wormhole-routed hypercubes using error-correcting codes

A new broadcasting method is presented for hypercubes with wormhole routing mechanism. The communication model assumed allows an n-dimensional hypercube to have at most n concurrent I/O communication along its ports. It assumes a distance insensitivity of (n + 1) with no intermediate reception capability for the nodes. The approach is based on determination of the set of nodes called stations in the hypercube. Once stations are identified, node disjoint paths are formed from the source to all stations. The broadcasting is accomplished first by sending the message to all stations, which will inform the rest of the nodes. To establish node-disjoint paths between the source node and all stations, we introduce a new routing strategy. We prove that multicasting can be done in one routing step as long as the number of destination nodes are at most n in an n-dimensional hypercube. The number of broadcasting steps using our routing is equal to or smaller than that obtained in an earlier work; this number is optimal for all hypercube dimensions n ≤ 12, except for n = 10

On strong Menger-connectivity of star graphs

AbstractMotivated by parallel routing in networks with faults, we study the following graph theoretical problem. Let G be a graph of minimum vertex degree d. We say that G is strongly Menger-connected if for any copy Gf of G with at most d−2 nodes removed, every pair of nodes u and v in Gf are connected by min{degf(u),degf(v)} node-disjoint paths in Gf, where degf(u) and degf(v) are the degrees of the nodes u and v in Gf, respectively. We show that the star graphs, which are a recently proposed attractive alternative to the widely used hypercubes as network models, are strongly Menger-connected. An algorithm of optimal running time is developed that constructs the maximum number of node-disjoint paths of nearly optimal length in star graphs with faults

Efficient embedding of virtual hypercubes in irregular WDM optical networks

This thesis addresses one of the important issues in designing future WDM optical networks. Such networks are expected to employ an all-optical control plane for dissemination of network state information. It has recently been suggested that an efficient control plane will require non-blocking communication infrastructure and routing techniques. However, the irregular nature of most WDM networks does not lend itself to efficient non-blocking communications. It has been recently shown that hypercubes offer some very efficient non-blocking solutions for, all-to-all broadcast operations, which would be very attractive for control plane implementation. Such results can be utilized by embedding virtual structures in the physical network and doing the routing using properties of a virtual architecture. We will emphasize the hypercube due to its proven usefulness. In this thesis we propose three efficient heuristic methods for embedding a virtual hypercube in an irregular host network such that each node in the host network is either a hypercube node or a neighbor of a hypercube node. The latter will be called a “satellite” or “secondary” node. These schemes follow a step-by-step procedure for the embedding and for finding the physical path implementation of the virtual links while attempting to optimize certain metrics such as the number of wavelengths on each link and the average length of virtual link mappings. We have designed software that takes the adjacency list of an irregular topology as input and provides the adjacency list of a hypercube embedded in the original network. We executed this software on a number of irregular networks with different connectivities and compared the behavior of each of the three algorithms. The algorithms are compared with respect to their performance in trying to optimize several metrics. We also compare our algorithms to an already existing algorithm in the literature

I/O embedding and broadcasting in star interconnection networks

The issues of communication between a host or central controller and processors, in large interconnection networks are very important and have been studied in the past by several researchers. There is a plethora of problems that arise when processors are asked to exchange information on parallel computers on which processors are interconnected according to a specific topology. In robust networks, it is desirable at times to send (receive) data/control information to (from) all the processors in minimal time. This type of communication is commonly referred to as broadcasting. To speed up broadcasting in a given network without modifying its topology, certain processors called stations can be specified to act as relay agents. In this thesis, broadcasting issues in a star-based interconnection network are studied. The model adopted assumes all-port communication and wormhole switching mechanism. Initially, the problem treated is one of finding the minimum number of stations required to cover all the nodes in the star graph with i-adjacency. We consider 1-, 2-, and 3-adjacencies and determine the upper bound on the number of stations required to cover the nodes for each case. After deriving the number of stations, two algorithms are designed to broadcast the messages first from the host to stations, and then from stations to remaining nodes; In addition, a Binary-based Algorithm is designed to allow routing in the network by directly working on the binary labels assigned to the star graph. No look-up table is consulted during routing and minimum number of bits are used to represent a node label. At the end, the thesis sheds light on another algorithm for routing using parallel paths in the star network

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))