Broadcasting in DMA-bound bounded degree graphs

AbstractBroadcasting is an information dissemination process in which a message is to be sent from a single originator to all members of a network by placing calls over the communication lines of the network. In [2], Bermond, Hell, Liestman and Peters studied the effect, on broadcasting capabilities, of placing an upper bound on the graph's degree. In this paper, we generalize their results allowing calls to involve more than two participants. We give lower bounds and construct bounded degree graphs which allow rapid broadcasting. Our constructions use the nation of compounding graphs in de Bruijin digraphs. We also obtain asymptotic upper and lower bounds for broadcast time, as the maximum degree increases

Time-relaxed broadcasting in communication networks

AbstractBroadcasting is the process of information dissemination in communication networks (modelled as graphs) whereby a message originating at one vertex becomes known to all members under the constraint that each call requires one unit of time and at every step any member can call at most one of its neighbours. A broadcast graph on n vertices is a network in which message can be broadcast in the minimum possible (=⌊log2n⌋) time regardless of the originator. Broadcast graphs having the smallest number of edges are called minimum broadcast graphs, and are subjects of intensive study. On the other hand, in Shastri (1995) we have considered how quickly broadcasting can be done in trees. In this paper, we study how the number of edges in a minimum broadcast graphs decrease, as we allow addition time over ⌊log2 n⌋, until we get a tree. In particular, the sparsest possible time-relaxed broadcast graphs are constructed for small n(⩽15) and very sparse time-relaxed broadcast graphs are given for larger n(⩽65). General constructions are also provided putting bounds which hold for all n. Some of these constructions make use of the techniques developed in Bermond et al. (1995, 1992) and Chau and Liestman (1985)

Parameterized Complexity of Broadcasting in Graphs

The task of the broadcast problem is, given a graph G and a source vertex s, to compute the minimum number of rounds required to disseminate a piece of information from s to all vertices in the graph. It is assumed that, at each round, an informed vertex can transmit the information to at most one of its neighbors. The broadcast problem is known to NP-hard. We show that the problem is FPT when parametrized by the size k of a feedback edge-set, or by the size k of a vertex-cover, or by k=n-t where t is the input deadline for the broadcast protocol to complete.Comment: Full version of WG 2023 pape

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

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)

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

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

Tight Bounds on Minimum Broadcast Networks

A broadcast graph is an n-vertex communication network that supports a broadcast from any one vertex to all other vertices in optimal time dlg ne, given that each message transmission takes one time unit and a vertex participates in at most one transmission per time step. This paper establishes tight bounds for B(n), the minimum number of edges of a broadcast graph, and D(n), the minimum maxdegree of a broadcast graph. Let L(n) denote the number of consecutive leading 1's in the binary representation of integer n \Gamma 1. We show B(n) = \Theta(L(n) \Delta n) and D(n) = \Theta(lg lg n+L(n)), and for every n we give a construction simultaneously within a constant factor of both lower bounds. For all n we also construct graphs with O(n) edges and O(lg lg n) maxdegree requiring at most dlg ne + 1 time units to broadcast. Our broadcast protocols may be implemented with local control and O(lg lg n) bits overhead per message