On strong Menger-connectivity of star graphs

AbstractMotivated by parallel routing in networks with faults, we study the following graph theoretical problem. Let G be a graph of minimum vertex degree d. We say that G is strongly Menger-connected if for any copy Gf of G with at most d−2 nodes removed, every pair of nodes u and v in Gf are connected by min{degf(u),degf(v)} node-disjoint paths in Gf, where degf(u) and degf(v) are the degrees of the nodes u and v in Gf, respectively. We show that the star graphs, which are a recently proposed attractive alternative to the widely used hypercubes as network models, are strongly Menger-connected. An algorithm of optimal running time is developed that constructs the maximum number of node-disjoint paths of nearly optimal length in star graphs with faults

On strong fault tolerance (or strong Menger-connectivity) of multicomputer networks

As the size of networks increases continuously, dealing with networks with faulty nodes becomes unavoidable. In this dissertation, we introduce a new measure for network fault tolerance, the strong fault tolerance (or strong Menger-connectivity)in multicomputer networks, and study the strong fault tolerance for popular multicomputer network structures. Let G be a network in which all nodes have degree d. We say that G is strongly fault tolerant if it has the following property: Let Gf be a copy of G with at most d - 2 faulty nodes. Then for any pair of non-faulty nodes u and v in Gf , there are min{degf (u), degf (v)} node-disjoint paths in Gf from u to v, where degf (u) and degf (v) are the degrees of the nodes u and v in Gf, respectively. First we study the strong fault tolerance for the popular network structures such as star networks and hypercube networks. We show that the star networks and the hypercube networks are strongly fault tolerant and develop efficient algorithms that construct the maximum number of node-disjoint paths of nearly optimal or optimal length in these networks when they contain faulty nodes. Our algorithms are optimal in terms of their time complexity. In addition to studying the strong fault tolerance, we also investigate a more realistic concept to describe the ability of networks for tolerating faults. The traditional definition of fault tolerance, sustaining at most d - 1faulty nodes for a regular graph G of degree d, reflects a very rare situation. In many cases, there is a chance that a routing path between two given nodes can be constructed though the network may have more faulty nodes than its degree. In this dissertation, we study the fault tolerance of hypercube networks under a probability model. When each node of the n-dimensional hypercube network has an independent failure probability p, we develop algorithms that, with very high probability, can construct a fault-free path when the hypercube network can sustain up to 2np faulty nodes