Special issue on algorithms for hypercube computers : Guest editor's introduction

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28649/1/0000465.pd

An Improved Characterization of 1-Step Recoverable Embeddings: Rings in Hypercubes

An embedding is 1-step recoverable if any single fault occurs, the embedding can be reconfigured in one reconfiguration step to maintain the structure of the embedded graph. In this paper we present an efficient scheme to construct this type of 1-step recoverable ring embeddings in the hypercube. Our scheme will guarantee finding a 1-step recoverable embedding of a length-k (even) ring in a d-cube where 6 less than or equal to k less than or equal to (3/4)2/sup d/ and d greater than or equal to 3, provided such an embedding exists. Unlike previously proposed schemes, we solve the general problem of embedding rings of different lengths and the resulting embeddings are of smaller expansion than in previous proposals. A sufficient condition for the non-existence of 1-step recoverable embeddings of rings of length \u3e(3/4)2d in d-cubes is also give

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

Work-preserving real-time emulation of meshes on butterfly networks

The emulation of a guest network G on a host network H is work-preserving and real-time if the inefficiency, that is the ratio WG/WH of the amounts of work done in both networks, and the slowdown of the emulation are O(1). In this thesis we show that an infinite number of meshes can be emulated on a butterfly in a work-preserving real-time manner, despite the fact that any emulation of an s x s-node mesh in a butterfly with load 1 has a dilation of Ω(logs). The recursive embedding of a mesh in a butterfly presented by Koch et al. (STOC 1989), which forms the basis for our work, is corrected and generalized by relaxing unnecessary constraints. An algorithm determining the parameter for each stage of the recursion is described and a rigorous analysis of the resulting emulation shows that it is work-preserving and real-time for an infinite number of meshes. Data obtained from simulated embeddings suggests possible improvements to achieve a truly work-preserving emulation of the class of meshes on the class of butterflies

Optimal simulation of full binary trees on faulty hypercubes

The problem of operating full binary tree based algorithms on a hypercube with faulty nodes was investigated. Developing a method for embedding a full binary tree into the faulty hypercube is the solution to this problem. Two outcomes for embedding an (n-1)-tree into an n-cube with unit dilation and load, that were based on a new embedding technique, were presented. For the problem where the root can be mapped to any nonfaulty hypercube node, the optimum toleration of faults was shown. Moreover, it was demonstrated that the algorithm for the variable root embedding problem is maximal within a class algorithms called recursive embedding algorithms as far as the number of tolerable faults is concerned. Lastly, it was demonstrated that when an O(1/√n) fraction of nodes in the hypercube are faulty, a O(1)-load variable root embedding is not always possible regardless of the significance of the dilation.published_or_final_versio

Fault-Tolerant Ring Embeddings in Hypercubes -- A Reconfigurable Approach

We investigate the problem of designing reconfigurable embedding schemes for a fixed hypercube (without redundant processors and links). The fundamental idea for these schemes is to embed a basic network on the hypercube without fully utilizing the nodes on the hypercube. The remaining nodes can be used as spares to reconfigure the embeddings in case of faults. The result of this research shows that by carefully embedding the application graphs, the topological properties of the embedding can be preserved under fault conditions, and reconfiguration can be carried out efficiently. In this dissertation, we choose the ring as the basic network of interest, and propose several schemes for the design of reconfigurable embeddings with the aim of minimizing reconfiguration cost and performance degradation. The cost is measured by the number of node-state changes or reconfiguration steps needed for processing of the reconfiguration, and the performance degradation is characterized as the dilation of the new embedding after reconfiguration. Compared to the existing schemes, our schemes surpass the existing ones in terms of applicability of schemes and reconfiguration cost needed for the resulting embeddings