The multiple originator broadcasting problem in graphs

AbstractGiven a graph G and a vertex subset S of V(G), the broadcasting time with respect to S, denoted by b(G,S), is the minimum broadcasting time when using S as the broadcasting set. And the k-broadcasting number, denoted by bk(G), is defined by bk(G)=min{b(G,S)|S⊆V(G),|S|=k}.Given a graph G and two vertex subsets S, S′ of V(G), define d(v,S)=minu∈Sd(v,u), d(S,S′)=min{d(u,v)|u∈S, v∈S′}, and d(G,S)=maxv∈V(G)d(v,S) for all v∈V(G). For all k, 1⩽k⩽|V(G)|, the k-radius of G, denoted by rk(G), is defined as rk(G)=min{d(G,S)|S⊆V(G), |S|=k}.In this paper, we study the relation between the k-radius and the k-broadcasting numbers of graphs. We also give the 2-radius and the 2-broadcasting numbers of the grid graphs, and the k-broadcasting numbers of the complete n-partite graphs and the hypercubes

Broadcasting in cycles with chords

Broadcasting is the process of information dissemination in which one node, the originator, knows a single piece of information and using a series of calls must inform every other node in the network of this information. We assume that at any given time, a node can communicate the message to another node, with which it shares an edge, by acting as either a sender or receiver, but not both. Multiple message broadcasting considers the case when the originator has m messages, where m \u3e 1, to disseminate. Whereas broadcasting limits the communication of a message from one node to another node via a single edge, line broadcasting allows one node to send a message to any other node in the network as long as a simple path exists between the sending node and the receiving node and every edge along the path is not in use.;In this dissertation, we consider the problem of broadcasting in a cycle with chords and we develop broadcast schemes for this type of network.;We begin by investigating the problem of broadcasting in a cycle with one and two chords, respectively. Then, we consider the problem of multiple message broadcasting in cycles with one and two chords. Finally, we consider the problem of line broadcasting in cycles with chords.;Through our investigations, we develop two algorithms for the problem of broadcasting in a cycle with one and two chords, respectively and we analyze the correctness and complexity of these algorithms. Then, we discuss problems associated with multiple message broadcasting in cycles with one and two chords. Finally, we use techniques developed for line broadcasting in cycles to create minimum time broadcast schemes for cycles through the addition of chords.;Using techniques developed in this dissertation, we are able to broadcast in minimum time in cycles with chords. In cycles whose size is a power of 2, we have proved that the number of chords that we add to the cycle is the minimum number of chords required to broadcast in minimum time in such a cycle

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

Minimum-time multidrop broadcast

AbstractThe multidrop communication model assumes that a message originated by a sender is sent along a path in a network and is communicated to each site along that path. In the presence of several concurrent senders, we require that the transmission paths be vertex-disjoint. The time analysis of such communication includes both start-up time and drop-off time terms. We determine the minimum time required to broadcast a message under this communication model in several classes of graphs

Line broadcasting in cycles

AbstractBroadcasting is the process of transmitting information from an originating node (processor) in a network to all other nodes in the network. A local broadcast scheme only allows a node to send information along single communication links to adjacent nodes, while a line broadcast scheme allows nodes to use paths of several communication links to call distant nodes. The minimum time possible for broadcasting in a network of n nodes when no node is involved in more than one communication at any given time is ⌊ log n⌋ phases. Local broadcasting is not sufficient, in general, for broadcasting to be completed in minimum time; line broadcasting is always sufficient. An optimal line broadcast is a minimum-time broadcast that uses the smallest possible total number of communication links. In this paper, we give a complete characterization of optimal line broadcasting in cycles, and we develop efficient methods for constructing optimal line broadcast schemes

Statistical privacy-preserving message dissemination for peer-to-peer networks

Concerns for the privacy of communication is widely discussed in research and overall society. For the public financial infrastructure of blockchains, this discussion encompasses the privacy of transaction data and its broadcasting throughout the network. To tackle this problem, we transform a discrete-time protocol for contact networks over infinite trees into a computer network protocol for peer-to-peer networks. Peer-to-peer networks are modeled as organically growing graphs. We show that the distribution of shortest paths in such a network can be modeled using a normal distribution $\mathcal{N}(\mu,\sigma^2).$ We determine statistical estimators for $\mu,\sigma$ via multivariate models. The model behaves logarithmic over the number of nodes n and proportional to an inverse exponential over the number of added edges k. These results facilitate the computation of optimal forwarding probabilities during the dissemination phase for optimal privacy in a limited information environment.Comment: 6 figures, 19 pages, single colum

Optimal broadcasting in treelike graphs

Broadcasting is an information dissemination problem in a connected network, in which one node, called the originator , disseminates a message to all other nodes by placing a series of calls along the communication lines of the network. Once informed, the nodes aid the originator in distributing the message. Finding the broadcast time of a vertex in an arbitrary graph is NP-complete. The problem is solved polynomially only for a few classes of graphs. In this thesis we study the broadcast problem in different classes of graphs which have various similarities to trees. The unicyclic graph is the simplest graph family after trees, it is a connected graph with only one cycle in it. We provide a linear time solution for the broadcast problem in unicyclic graphs. We also studied graphs with increasing number of cycles and complexity and provide again polynomial time solutions. These graph families are: tree of cycles, necklace graphs, and 2-restricted cactus graphs. We also define the fully connected tree graphs and provide a polynomial solution and use these results to obtain polynomial solution for the broadcast problem in tree of cliques and a constant approximation algorithm for the hierarchical tree cluster networks