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

Approximation Algorithms for Broadcasting in Simple Graphs with Intersecting Cycles

Broadcasting is an information dissemination problem in a connected network in which one node, called the originator, must distribute a message to all other nodes by placing a series of calls along the communication lines of the network. Every time the informed nodes aid the originator in distributing the message. Finding the minimum broadcast time of any vertex in an arbitrary graph is NP-Complete. The problem remains NP-Complete even for planar graphs of degree 3 and for a graph whose vertex set can be partitioned into a clique and an independent set. The best theoretical upper bound gives logarithmic approximation. It has been shown that the broadcasting problem is NP-Hard to approximate within a factor of 3-ɛ. The polynomial time solvability is shown only for tree-like graphs; trees, unicyclic graphs, tree of cycles, necklace graphs and some graphs where the underlying graph is a clique; such as fully connected trees and tree of cliques. In this thesis we study the broadcast problem in different classes of graphs where cycles intersect in at least one vertex. First we consider broadcasting in a simple graph where several cycles have common paths and two intersecting vertices, called a k-path graph. We present a constant approximation algorithm to find the broadcast time of an arbitrary k-path graph. We also study the broadcast problem in a simple cactus graph called k-cycle graph where several cycles of arbitrary lengths are connected by a central vertex on one end. We design a constant approximation algorithm to find the broadcast time of an arbitrary k-cycle graph. Next we study the broadcast problem in a hypercube of trees for which we present a 2-approximation algorithm for any originator. We provide a linear algorithm to find the broadcast time in hypercube of trees with one tree. We extend the result for any arbitrary graph whose nodes contain trees and design a linear time constant approximation algorithm where the broadcast scheme in the arbitrary graph is already known. In Chapter 6 we study broadcasting in Harary graph for which we present an additive approximation which gives 2-approximation in the worst case to find the broadcast time in an arbitrary Harary graph. Next for even values of n, we introduce a new graph, called modified-Harary graph and present a 1-additive approximation algorithm to find the broadcast time. We also show that a modified-Harary graph is a broadcast graph when k is logarithmic of n. Finally we consider a diameter broadcast problem where we obtain a lower bound on the broadcast time of the graph which has at least (d+k-1 choose d) + 1 vertices that are at a distance d from the originator, where k >= 1

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