Routing algorithms in networks with the topology of circulant graphs

The thesis "Routing algorithms in networks with the topology of circulant graphs" investigates topological properties of circulant graphs and their application to computer data exchange. Circulant graphs are undirected vertex transitive graphs with n vertices and edges of length hi, (i=1, 2, ..., k). When selecting an appropriate topology for a computer network, circulant graphs represent an intermediate choice between simple but unreliable ring topology and reliable but expensive (and sometimes technologically unfeasible) fully connected topology. Due to their favorable properties--among which are symmetry, scalability, reliability, small diameter, and small average node distance,--circulant graphs are widely studied as suitable topologies for local area networks and parallel computer architectures. They have been efficiently used in ILLIAC IV, FDDI-token, SILK and SONET rings, Intel Paragon, Cray T3D, MPP, and MICROS. Routing algorithms are used to solve data exchange problems in computer networks. Data (i.e. a message) is split into several packages. The task of a routing algorithm is to transfer the packages along the network edges to their final destination and then to combine them into original message. There are two types of routing algorithms. Static routing algorithms determine the path for each packet in the preprocessing phase (i.e. before the actual start of routing); dynamic routing algorithms perform all the necccessary routing calculations on-the-fly. After the preprocessing phase the routing algorithm makes several successive routing steps. In each routing step the algorithm can move a packet from its current node to one of the neighboring nodes. Routing algorithms can be used to solve various routing problems. A two-terminal routing is a problem where only one package has to be routed from its source to its final destination. Routing algorithms for this problem aim to route a package along one of its shortest paths. Much more complicated situation occurs when more than one package is present in the network. In this case, which is also known as the general routing problem, a routing algorithm tries to route all the packages to their final destination in as few as possible routing steps. In this thesis we first introduce the wide area of our work. We extensively explain the meaning of the following terms: network, routing model, routing problem, routing algorithm. We also present the criteria for measuring the quality of a given routing algorithm. In the following there are two main chapters. The first one is dedicated to the selected topology (i.e. circulant graphs) and the second one to the routing algorithms designed and tested for this topology. In the chapter dedicated to circulant graphs we first summarize the known results and the most important topological properties. Then we present a method for solving certain problems (such as calculating the diameter and calculating the shortest paths) by reduction to integer lattice in which a point represents a walk in a graph. We introduce the notion of minimal projection and give the corresponding algorithm to construct it. We present the integer lattice as a module, define the notion of packed basis, and present two algorithms for calculating it. Then we introduce and prove several important features concerning vectors of packed basis. We also introduce a new topology, the so called semi-directed circulant graphs, and present an O(log n) algorithm for calculating the shortest paths between nodes in these graphs. The proof of correctness for this algorithm, which calculates the shortest paths by using the minimal projection along the vectors of packed basis, is based on the properties of packed basis proved in earlier sections. We also present some k-generalizations of the algorithms presented for 2-circulant graphs. In the chapter on the routing in circulant graphs we first present two routing algorithms for the two-terminal routing problem (static and dynamic version). Then we present a parameterized model used to describe routing algorithms and give five parameter-sets of five static routing algorithms for the general routing problem. We also describe a dynamic algorithm for the general routing problem and compare the efficiency of this algorithm with the best static one. At the end of the chapter we present a generalization of a routing algorithm for general k-circulant graphs. In the last chapter of the thesis we give some open questions and ideas for future work