Combinatorial and algebraic methods in star and de bruijn networks

Star networks and de Bruijn networks have received considerable attention recently by researchers as graph models for interconnection networks. It has been shown that they are attractive alternatives to the widely used hypercube model. In this dissertation, we investigate these two networks from combinatorial and algebraic points of view. We study parallel routing problems in star networks and network embedding problems in de Bruijn networks. We start with a combinatorial problem called Maximum Partition Matching and develop powerful greedy techniques to construct an optimal solution to this problem. Sophisticated combinatorial analysis is performed to show the correctness of our greedy algorithms. We then show how the problem of finding a maximum number of node-disjoint shortest paths between any two nodes in a star network can be reduced to the Maximum Partition Matching problem. A sufficient and necessary condition is derived for two arbitrary nodes in a star network to have a given bulk distance. This analysis together with the efficient algorithm for Maximum Partition Matching results in an efficient algorithm that constructs optimal parallel routing between any two nodes in a star network. This significantly improves previous research results. In addition to studying the node-disjoint routing problem between two nodes, we also investigate the problem of finding $n-1$ node-disjoint paths between one node u and a set S of $n-1$ other nodes in the n-dimensional star network. We present an efficient algorithm that solves this problem in $O(n\sp3)$ time. Moreover, any one of the constructed paths connecting the node u and a node v in S has length no larger than 6 plus the minimum distance between u and v. This again significantly improves previous known results. Finally, we study the de Bruijn networks from algebraic point of view. We present a homomorphism for the de Bruijn networks and show that it is superior to the previously known one. We then apply this homomorphism to embed higher dimensional de Bruijn networks to smaller de Bruijn networks. Results of embeddings from other large-size networks to smaller de Bruijn networks are also presented.