approved: Bella Bose A fault-tolerant embedding of a ring in an interconnection network involves finding a cycle in the network that avoids a set of faulty processors or links. This thesis deals primarily with fault-tolerant ring embedding in a d-ary De Bruijn network with dn processors. It is shown that (a) A fault-free cycle of length at least dnfn can always be found in the event of f d-2 processor failures. (b) A fault-free Hamiltonian cycle can always be found if there are at most d-2 link

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... failures and d is a prime power. Both of these results are optimal when a worst-case fault distribution is assumed. The results on ring embedding in the presence of link failures can be extended, in some cases, to butterfly networks. It is also shown that the d-ary De Bruijn digraph admits d-1 disjoint Hamiltonian cycles when d is a power of 2 and at least (d-1)/2 disjoint Hamiltonian cycles when d is an odd prime power. We assume that an interconnection network is modeled by a graph, the nodes being the processors and the edges being the physical links between processors. Henceforth we will not make a distinction between a network and its underlying graph, e.g., node and processor will be used interchangeably. Component failures are assumed to be total, i.e., faulty nodes can neither perform computations nor route messages, and are modeled by removing the faulty nodes and/or edges from the graph. Let Rk denote a cycle (or ring) of length k. An embedding of Rk into graph G is a one-to-one mapping ti that takes the nodes of Rk to the nodes of G and the edges of Rk to paths in G. The principal measures of an embedding are its dilation and congestion. The dilation of ti is the length of the longest path t(e) taken over all edges e in Rk. The congestion of T is the largest number of paths t(e) using a single edge in G. This thesis addresses the problem of embedding the largest possible ring Rk in a De Bruijn graph so that nodes in Rk are mapped to nonfaulty nodes and edges in Rk are mapped to fault-free paths. The proposed embeddings have unit dilation and congestion implying that the embedded ring is a subgraph of the faulty graph. 1.2 The De Bruijn Network The De Bruijn interconnection network is modeled by either the directed or undirected De Bruijn graph. The d-ary directed De Bruijn graph B(d,n) has nodes corresponding to n-tuples over a d-letter alphabet A, and directed edges from each node xi ...xn to nodes {x2...xna I a E A} . Each node has indegree and outdegree d, and nodes of the form an have loops. The undirected De Bruijn graph, denoted UB(d,n), is obtained from B(d,n) by deleting loops, removing the orientation of the edges and merging any resulting parallel edges. This results in a graph possessing d nodes of degree 2d-2, d(d-1) nodes of degree 2d-1 and do d2 nodes of degree 2d [PR82]. The 8-node and 16-node De Bruijn graphs on the alphabet {OM are shown in Figure 1.1, and the undirected De Bruijn graph UB(2,3) is shown in Figure 1.2. \ / \010 / 101 / 000 001 011 Figure 1.2. Undirected binary De Bruijn graph UB(2,3).