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

Improved upper bounds and lower bounds on broadcast function

Given a graph G=(V,E) and an originator vertex v, broadcasting is an information disseminating process of transmitting a message from vertex v to all vertices of graph G as quickly as possible. A graph G on n vertices is called broadcast graph if the broadcasting from any vertex in the graph can be accomplished in \lceil log n\rceil time. A broadcast graph with the minimum number of edges is called minimum broadcast graph. The number of edges in a minimum broadcast graph on n vertices is denoted by B(n). A long sequence of papers present different techniques to construct broadcast graphs and to obtain upper bounds on B(n). In this thesis, we study the compounding and the vertex addition broadcast graph constructions, which improve the upper bound on B(n). We also present the first nontrivial general lower bound on B(n)

Approximation Algorithms for Broadcasting in Flower Graphs

Over the last century, telecommunication networks have become the nervous system of our society. As data is generated and stored on varied nodes, effective communication is imperative to ensure efficient use of the network. Our ever-growing reliance on these increasingly large and complex networks make ineffective communication strategies evermore apparent. Broadcasting is a fundamental information-dissemination problem which models communication across a connected graph in the following manner: a single vertex, the originator, seeks to pass some message along to all other vertices in the graph. In general, research on broadcasting can be grouped in roughly two categories: Firstly, given some particular graph and some particular vertex chosen to be originator, what is a broadcast scheme that informs the entire graph in the minimum time possible? Secondly, given some number of nodes, how can we arrange them in a particular network topology such that we can achieve minimal broadcast time from any vertex? This thesis focuses on problems of the first category. Finding the minimum broadcast time of any vertex in an arbitrary graph is NP-Complete, but efficient algorithms have been found for particular graph families. In particular, polynomial time algorithms have been found for trees and some tree-like graphs: unicyclic graphs, tree of cycles. Such algorithms have also been found for some graphs with no intersecting cliques, such as fully connected trees and trees of cliques. Finally, graphs containing cycles with particular restrictions were also studied, and efficient algorithms for necklace graphs and k-restricted cactus graphs were also found. The question still stands however, of whether these restrictions may be too conservative, and that efficient algorithms exist on broader classes of graphs. In particular, significant research has been made towards finding an efficient broadcasting algorithm on cactus graphs, which has not been found so far. This thesis studies the broadcasting problem on Flower graphs, which capture the difficulty of cactus graphs in a simple graph family. Flower graphs, or k-cycle graphs, are graphs composed of k cycles all joined on a single central vertex v_c. The contributions of this thesis for broadcasting on flower graphs is two-fold: it first improves the approximation ratio for broadcasting on flower graphs. It then provides a heuristic which performs significantly better in practice than the current best heuristic. We also demonstrate that our heuristic finds the optimal broadcast time for particular subcases of flower graphs