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.

In classical network information theory, information packets are treated as commodities, and the nodes of the network are only allowed to duplicate and forward the packets. The new paradigm of network coding, which was introduced by Ahlswede et al., states that if the nodes are permitted to combine the information packets and forward a function of them, the throughput of the network can dramatically increase. In this dissertation we focused on the design and analysis of low complexity network coding schemes for different topologies of wired and wireless networks. In the first part we studied the routing capacity of wired networks. We provided a description of the routing capacity region in terms of a finite set of linear inequalities. We next used this result to study the routing capacity region of undirected ring networks for two multimessage scenarios. Finally, we used new network coding bounds to prove the optimality of routing schemes in these two scenarios. In the second part, we studied node-constrained line and star networks. We derived the multiple multicast capacity region of node-constrained line networks based on a low complexity binary linear coding scheme. For star networks, we examined the multiple unicast problem and offered a linear coding scheme. Then we made a connection between the network coding in a node-constrained star network and the problem of index coding with side information. In the third part, we studied the linear deterministic model of relay networks (LDRN). We focused on a unicast session and derived a simple capacity-achieving transmission scheme. We obtained our scheme by a connection to the submodular flow problem through the application of tools from matroid theory and submodular optimization theory. We also offered polynomial-time algorithms for calculating the capacity of the network and the optimal coding scheme. In the final part, we considered the multicasting problem in an LDRN and proposed a new way to construct a coding scheme. Our construction is based on the notion of flow for a unicast session in the third part of this dissertation. We presented randomized and deterministic polynomial-time versions of our algorithm.