Deep Heuristic: A Heuristic for Message Broadcasting in Arbitrary Networks

With the increasing popularity of interconnection networks, efficient information dissemination has become a popular research area. Broadcasting is one of the information dissemination primitives. Finding the optimal broadcasting scheme for any originator in an arbitrary network has been proved to be an NP-Hard problem. In this thesis, a new heuristic that generates broadcast schemes in arbitrary networks is presented, which has O(|E| + |V | log |V |) time complexity. Based on computer simulations of this heuristic in some commonly used topologies and network models, and comparing the results with the best existing heuristics, we conclude that the new heuristic show comparable performances while having lower complexity

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

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)

Broadcasting in highly connected graphs

Throughout history, spreading information has been an important task. With computer networks expanding, fast and reliable dissemination of messages became a problem of interest for computer scientists. Broadcasting is one category of information dissemination that transmits a message from a single originator to all members of the network. In the past five decades the problem has been studied by many researchers and all have come to demonstrate that despite its easy definition, the problem of broadcasting does not have trivial properties and symmetries. For general graphs, and even for some very restricted classes of graphs, the question of finding the broadcast time and scheme remains NP-hard. This work uses graph theoretical concepts to explore mathematical bounds on how fast information can be broadcast in a network. The connectivity of a graph is a measure to assess how separable the graph is, or in other words how many machines in a network will have to fail to disrupt communication between all machines in the network. We initiate the study of finding upper bounds on broadcast time b(G) in highly connected graphs. In particular, we give upper bounds on b(G) for k-connected graphs and graphs with a large minimum degree. We explore 2-connected (biconnected) graphs and broadcasting in them. Using Whitney's open ear decomposition in an inductive proof we propose broadcast schemes that achieve an upper bound of ceil(n/2) for classical broadcasting as well as similar bounds for multiple originators. Exploring further, we use a matching-based approach to prove an upper bound of ceil(log(k)) + ceil(n/k) - 1 for all k-connected graphs. For many infinite families of graphs, these bounds are tight. Discussion of broadcasting in highly connected graphs leads to an exploration of dependence between the minimum degree in the graph and the broadcast time of the latter. By using similar techniques and arguments we show that if all vertices of the graph are neighboring linear numbers of vertices, then information dissemination in the graph can be achieved in ceil(log(n)) + C time. To the best of our knowledge, the bounds presented in our work are a novelty. Methods and questions proposed in this thesis open new pathways for research in broadcasting

A general upper bound on broadcast function B(n) using Knodel graph

Broadcasting in a graph is the process of transmitting a message from one vertex, the originator, to all other vertices of the graph. We will consider the classical model in which an informed vertex can only inform one of its uninformed neighbours during each time unit. A broadcast graph on n vertices is a graph in which broadcasting can be completed in ceiling of log n to the base 2 time units from any originator. A minimum broadcast graph on n vertices is a broadcast graph that has the least possible number of edges, B(n), over all broadcast graphs on n vertices. This thesis enhances studies about broadcasting by applying a vertex deletion method to a specific graph topology, namely Knodel graph, in order to construct broadcast graphs on odd number of vertices. This construction provides an improved general upper bound on B(n) for all odd n except when n=2^k−1

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

Problems Related to Classical and Universal List Broadcasting

Broadcasting is a fundamental problem in the information dissemination area. In classical broadcasting, a message must be sent from one network member to all other members as rapidly as feasible. Although it has been demonstrated that this problem is NP-Hard for arbitrary graphs, it has several applications in various fields. As a result, the universal lists model, replicating real-world restrictions like the memory limits of nodes in large networks, is introduced as a branch of this problem in the literature. In the universal lists model, each node is equipped with a fixed list and has to follow the list regardless of the originator. In this study, we focus on both classical and universal lists broadcasting. Classical broadcasting is solvable for a few families of networks, such as trees, unicyclic graphs, tree of cycles, and tree of cliques. In this study, we begin by presenting an optimal algorithm that finds the broadcast time of any vertex in a Fully Connected Tree (FCT_n) in O(|V | log log n) time. An FCT_n is formed by attaching arbitrary trees to vertices of a complete graph of size n where |V| is the total number of vertices in the graph. Then, we replace the complete graph with a Hypercube H_k and propose a new heuristic for the Hypercube of Trees (HT_k). Not only does this heuristic have the same approximation ratio as the best-known algorithm, but our numerical results also show its superiority in most experiments. Our heuristic is able to outperform the current upper bound in up to 90% of the situations, resulting in an average speedup of 30%. Most importantly, our results illustrate that it can maintain its performance even if the network size grows, making the proposed heuristic practically useful. Afterward, we focus on broadcasting with universal lists, in which once a vertex is informed, it must follow its corresponding list, regardless of the originator and the neighbor from which it received the message. The problem of broadcasting with universal lists could be categorized into two sub-models: non-adaptive and adaptive. In the latter model, a sender will skip the vertices on its list from which it has received the message, while those vertices will not be skipped in the first model. In this study, we will present another sub-model called fully adaptive. Not only does this model benefit from a significantly better space complexity compared to the classical model, but, as will be proved, it is faster than the two other sub-models. Since the suggested model fits real-world network architectures, we will design optimal broadcast algorithms for well-known interconnection networks such as trees, grids, and cube-connected cycles. We also present an upper bound for tori under the same model. Then we focus on designing broadcast graphs (bg)’s under this model. A bg is a graph with minimum possible broadcast time from any originator. Additionally, a minimum broadcast graph (mbg) is a bg with the minimum possible number of edges. We propose mbg’s on n vertices for n ≤ 10 and sparse bg’s for 11 ≤ n ≤ 14 under the fully-adaptive model. Afterward, we introduce the first infinite families of bg’s under this model, and we prove that hypercubes are mbg under this model. Later, we establish the optimal broadcast time of k−ary trees and binomial trees under the nonadaptive model and provide an upper bound for complete bipartite graphs. We also improved a general upper bound for trees under the same model. We then suggest several general upper bounds for the universal lists by comparing them with the messy broadcasting model. Finally, we propose the first heuristic for this problem, namely HUB-GA: a Heuristic for Universal lists Broadcasting with Genetic Algorithm. We undertake various numerical experiments on frequently used interconnection networks in the literature, graphs with clique-like structures, and synthetic instances in order to cover many possibilities of industrial topologies. We also compare our results with state-of-the-art methods for classical broadcasting, which is proved to be the fastest model among all. Although the universal list model utilizes less memory than the classical model, our algorithm finds the same broadcast time as the classical model in diverse situations