Designing Distance-Preserving Fault-Tolerant Topologies

DESIGNING DISTANCE-PRESERVING FAULT-TOLERANT TOPOLOGIES By Sitarama Swamy Kocherlakota The objective of designing a fault-tolerant communication network (CN) is to maintain its functionality in the presence of failures. The approaches taken in designing fault-tolerant systems can be classified, in general, by the functionality criteria. There are two functionality criteria that have received much attention. First, preserving a topology, which considers a CN functional as long as a desired topology is contained in the system. Second, preserving a property, which considers a CN functional as long as a given topological property is satisfied by the underlying topology. In this dissertation, we study the second functionality criterion. Specifically, designing fault-tolerant topologies that preserve distance between every pair of non-adjacent nodes in presence of edge failures. We introduce and study a new family of fault-tolerant graphs called distance preserving graphs. A graph G is said to be k-edge distance preserving with respect to a spanning subgraph D, if there exist k edge-disjoint u-v paths in G of length at most dD(u; v) for every pair of non-adjacent vertices u; v of D. We present few properties of distance preserving graphs along with a theorem that characterizes the distance preserving graphs. We study two design models each with a different optimality criterion. In the first model, minimizing the overall redundancy is considered. The second model considers regular fault-tolerant topologies with minimum regularity. We present tight lower and upper bounds for both design models. The focus of this dissertation is on designs based on the second model. In particular, we construct regular distance preserving graphs with respect to cycles, crowns, chordal rings, and n-ary 2-cubes. We also prove that the given constructions are optimal when D = Cp and k p 2 . We also address the issue of the distance between adjacent vertices in D in presence of edge failures, and show that the distance between adjacent vertices is bounded by a constant. Finally, given a graph G and a spanning subgraph D, we have showen that there exist a polynomial time algorithm to determine if G is k-edge distance preserving with respect to D. This study will result in more cost-effective fault-tolerant systems where distance between communicating nodes is critical.