Title:

Embeddings, fault tolerance and communication strategies in kary ncube interconnection networks

The kary ncube interconnection network Qkn, for k 3 and n 2, is ndimensional network with k processors in each dimension. A kary ncube parallel computer consists of kn identical processors, each provided with its own sizeable memory and interconnected with 2n other processors. The kary ncube has some attractive features like symmetry, high level of concurrency and efficiency, regularity and high potential for the parallel execution of various algorithms. It can efficiently simulate other network topologies. The kary ncube has a smaller degree than that of its equivalent hypercube (the one with at least as many nodes) and it has a smaller diameter than its equivalent mesh of processors. In this thesis, we review some topological properties of the kary ncube Qkn and show how a Hamiltonian cycle can be embedded in Qkn using the Gray codes strategy. We also completely classify when a Qkn contains a cycle of some given length. The problem of embedding a large cycle in a Qkn with both faulty nodes and faulty links is considered. We describe a technique for embedding a large cycle in a kary ncube Qkn with at most n faults and show how this result can be extended to obtain embeddings of meshes and tori in such a faulty kary ncube. Embeddings of Hamiltonian cycles in faulty kary ncubes is also studied. We develop a technique for embedding a Hamiltonian cycle in a kary ncube with at most 4n5 faulty links where every node is incident with at least two healthy links. Our result is optimal as there exist kary ncubes with 4n  4 faults (and where every node is incident with at least two healthy links) not containing a Hamiltonian cycle. We show that the same technique can be easily applied to the hypercube. We also show that the general problem of deciding whether a faulty kary ncube contains a Hamiltonian cycle is NPcomplete, for all (fixed) k 3.
