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

The SLOOP project: Simulations, Parallel Object-Oriented Languages, Interconnection Networks

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Lower bounds on systolic gossip

AbstractGossiping is an extensively investigated information dissemination process in which each processor has a distinct item of information and has to collect all the items possessed by the other processors. In this paper we provide an innovative and general lower bound technique relying on the novel notion of delay digraph of a gossiping protocol and on the use of matrix norm methods. Such a technique is very powerful and allows the determination of new and significantly improved lower bounds in many cases. In fact, we derive the first general lower bound on the gossiping time of systolic protocols, i.e., constituted by a periodic repetition of simple communication steps. In particular, given any network of n processors and any systolic period s, in the directed and the undirected half-duplex cases every s-systolic gossip protocol takes at least log(n)/log(1/λ)−O(loglog(n)) time steps, where λ is the unique solution between 0 and 1 of λ·p⌊s/2⌋(λ)·p⌈s/2⌉(λ)=1, with pi(λ)=1+λ2+⋯+λ2i−2 for any integer i>0. We then provide improved lower bounds in the directed and half-duplex cases for many well-known network topologies, such as Butterfly, de Bruijn, and Kautz graphs. All the results are extended also to the full-duplex case. Our technique is very general, as for s→∞ it allows the determination of improved results even for non-systolic protocols. In fact, for general networks, as a simple corollary it yields a lower bound only an O(loglog(n)) additive factor far from the general one independently proved in [Proc. 1st ACM Symposium on Parallel Algorithms and Architectures (SPAA), 1989, p. 318; Topics in Combinatorics and Graph Theory (1990) 451; SIAM Journal on Computing 21(1) (1992) 111; Discrete Applied Mathematics 42 (1993) 75] for all graphs and any (non-systolic) gossip protocol. Moreover, for specific networks, it significantly improves with respect to the previously known results, even in the full-duplex case. Correspondingly, better lower bounds on the gossiping time of non-systolic protocols are determined in the directed, half-duplex and full-duplex cases for Butterfly, de Bruijn, and Kautz graphs. Even if in this paper we give only a limited number of examples, our technique has wide applicability and gives a general framework that often allows to get improved lower bounds on the gossiping time of systolic and non-systolic protocols in the directed, half-duplex and full-duplex cases

New Heuristic for Message Broadcasting in Arbitrary Networks

Efficient information dissemination in interconnection networks is a key research area because of the major role it plays in the modern interconnected world. A vast number of topics ranging from distributed computing to Internet communication rely on efficient information dissemination. Broadcasting is one of the information dissemination primitives. The minimum broadcast time problem in arbitrary networks has been examined since the 1970s. Finding an optimal broadcasting scheme for any originator in an arbitrary network has been proved to be an NP-Hard problem. In the current thesis, a new heuristic that generates broadcast schemes in arbitrary networks is presented. The heuristic has O(|E|log|V|) time complexity, where V is the set of nodes and E is the set of the links of the network. Computer simulations in some commonly used topologies and network models show that compared to the existing heuristics the new heuristic shows better performance in some network models, and comparable performance in other network models, while having a low complexity similar to the best existing heuristics. Another advantage of the new heuristic is that approximately one half of the vertices receive the message via a shortest path from the broadcast originator, while the rest of the vertices receive the message via a path at most three hops longer

Heuristics 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, two new heuristics that generate broadcast schemes in arbitrary networks are presented. Both of them have O(|E|) time complexity. Moreover, in the broadcast schemes generated by the heuristics, each vertex in the network receives the message via a shortest path. Based on computer simulations of these heuristics in some commonly used topologies and network models, and comparing the results with the best existing heuristics, we conclude that the new heuristics show comparable performances while having lower complexity

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

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