Embedding cube-connected cycles graphs into faulty hypercubes

We consider the problem of embedding a cube-connected cycles graph (CCC) into a hypercube with edge faults. Our main result is an algorithm that, given a list of faulty edges, computes an embedding of the CCC that spans all of the nodes and avoids all of the faulty edges. The algorithm has optimal running time and tolerates the maximum number of faults (in a worst-case setting). Because ascend-descend algorithms can be implemented efficiently on a CCC, this embedding enables the implementation of ascend-descend algorithms, such as bitonic sort, on hypercubes with edge faults. We also present a number of related results, including an algorithm for embedding a CCC into a hypercube with edge and node faults and an algorithm for embedding a spanning torus into a hypercube with edge faults

Partitioning and broadcasting in hypercubes in the presence of faulty communication links

The problem of broadcasting in faulty hypercubes is considered, based upon a strategy of partitioning the faulty hypercube into subcubes in which currently known algorithms can be implemented. Three similar partitioning and broadcasting algorithms for an n-dimensional hypercube in the presence of up to (n² + 2n - c) / 4 faulty communication links are presented, where c = 4 if n is an even number or c = 3 if n is an odd number. The most efficient algorithm is implemented in 1.3n + 6log(n) + 9 time units. To the best of our knowledge, this algorithm is the most efficient one for an n-dimensional hypercube in the presence of O(n²) faults

Constructing Two Edge-Disjoint Hamiltonian Cycles in Locally Twisted Cubes

The $n$-dimensional hypercube network $Q_n$ is one of the most popular interconnection networks since it has simple structure and is easy to implement. The $n$-dimensional locally twisted cube, denoted by $LTQ_n$, an important variation of the hypercube, has the same number of nodes and the same number of connections per node as $Q_n$. One advantage of $LTQ_n$ is that the diameter is only about half of the diameter of $Q_n$. Recently, some interesting properties of $LTQ_n$ were investigated. In this paper, we construct two edge-disjoint Hamiltonian cycles in the locally twisted cube $LTQ_n$, for any integer $n\geqslant 4$. The presence of two edge-disjoint Hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the locally twisted cube.Comment: 7 pages, 4 figure

Reliability Analysis of the Hypercube Architecture.

This dissertation presents improved techniques for analyzing network-connected (NCF), 2-connected (2CF), task-based (TBF), and subcube (SF) functionality measures in a hypercube multiprocessor with faulty processing elements (PE) and/or communication elements (CE). These measures help study system-level fault tolerance issues and relate to various application modes in the hypercube. Solutions discussed in the text fall into probabilistic and deterministic models. The probabilistic measure assumes a stochastic graph of the hypercube where PE\u27s and/or CE\u27s may fail with certain probabilities, while the deterministic model considers that some system components are already failed and aims to determine the system functionality. For probabilistic model, MIL-HDBK-217F is used to predict PE and CE failure rates for an Intel iPSC system. First, a technique called CAREL is presented. A proof of its correctness is included in an appendix. Using the shelling ordering concept, CAREL is shown to solve the exact probabilistic NCF measure for a hypercube in time polynomial in the number of spanning trees. However, this number increases exponentially in the hypercube dimension. This dissertation, then, aims to more efficiently obtain lower and upper bounds on the measures. Algorithms, presented in the text, generate tighter bounds than had been obtained previously and run in time polynomial in the cube dimension. The proposed algorithms for probabilistic 2CF measure consider PE and/or CE failures. In attempting to evaluate deterministic measures, a hybrid method for fault tolerant broadcasting in the hypercube is proposed. This method combines the favorable features of redundant and non-redundant techniques. A generalized result on the deterministic TBF measure for the hypercube is then described. Two distributed algorithms are proposed to identify the largest operational subcubes in a hypercube C\sb{n} with faulty PE\u27s. Method 1, called LOS1, requires a list of faulty components and utilizes the CMB operator of CAREL to solve the problem. In case the number of unavailable nodes (faulty or busy) increases, an alternative distributed approach, called LOS2, processes m available nodes in O(mn) time. The proposed techniques are simple and efficient

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