Problems related to broadcasting in graphs

The data transmission delays become the bottleneck on modern high speed interconnection networks utilized by high performance computing or enterprise data centers. This motivates the study directed towards finding more efficient interconnection topologies as well as more efficient algorithms for information exchange between the nodes of the given network. Broadcasting is the process of distributing a message from a node, called the originator, to all other nodes of a communication network. Broadcasting is used as a basic communication primitive by many higher level network operations, which involve a set of nodes in distributed systems. Therefore, it is one the most important operations, which can determine the total efficiency of a given distributed system. We study interconnection networks via modeling them as graphs. The results described in this work can be used for efficient message routing algorithms in switch based interconnection networks as well as in the choice of the interconnection topologies of such networks. This thesis is divided into six chapters. Chapter 1 gives a general introduction to the research area and literature overview. Chapter 2 studies the family of graphs for which the broadcast time is equal to the diameter. Chapter 3 studies the routing and broadcasting problem in the Knodel graph. Chapter 4 studies the possible vertex degrees and the possible connections between vertices of different degrees in a broadcast graph. Using this, a new lower bound is obtained on broadcast function. Chapter 5 presents some miscellaneous results. Chapter 6 summarizes the thesis

Diameter and Broadcast Time of the Knödel graph

Efficient dissemination of information remains a central challenge for all types of networks. There are two ways to handle this issue. One way is to compress the amount of data being transferred and the second way is to minimize the delay of information distribution. Well-received approaches used in the second way either design efficient algorithms or implement reliable network architectures with optimal dissemination time. Among the well-known network architectures, the Knödel graph can be considered a suitable candidate for the problem of information dissemination. The Knödel graph W_(d, n) is a regular graph, of an even order n and degree d, 1 ≤ d ≤ floor(log n). The Knödel graph was introduced by W. Knödel almost four decades ago as network architecture with good properties in terms of broadcasting and gossiping in interconnected networks. Although the Knödel graph has a highly symmetric structure, its diameter is only known for W_(d, 2^d). Recently, the general upper and lower bounds on diameter and broadcast time of the Knödel graph have been presented. In this thesis, our motivation is to find the diameter, the number of vertices at a particular distance and the broadcast time of the Knödel graph. Theoretically, we succeed to prove the diameter and the broadcast time of the Knödel graph W_(3, n). We also claim that the Knödel graph W_(3, n) for n = 4 mod 4 and n > 16, is a diametral broadcast graph. We present that W_(3, 22) is a broadcast graph. Experimentally, however, we obtain the following results; (a) the diameter of some specific Knödel graphs, and (b) the propositions on the number of vertices at a particular distance. We also construct a new graph, denoted as HW_(d,2^d), by connecting Knödel graph W_(d-1,2^(d-1)) to hypercube H_(d-1) and experimentally show that HW_(d,2^d) has even a smaller diameter than Knödel graph W_(d,2^d)

Broadcasting in Harary Graphs

With the increasing popularity of interconnection networks, efficient information dissemination has become a popular research area. Broadcasting is one of the information dissemination primitives. Broadcasting in a graph is the process of transmitting a message from one vertex, the originator, to all other vertices of the graph. We follow the classical model for broadcasting. This thesis studies the Harary graph in depth. First, we find the diameter of Harary graph. We present an additive approximation algorithm for the broadcast problem in Harary graph. We also provide some properties for the graph like vertex transitivity, circulant graph and regularity. In the next part we introduce modified harary graph. We calculate the diameter and broadcast time for the graph. We will also provide 1-additive approximation algorithm to find the broadcast time in the modified harary graph