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

Picture theory: algorithms and software

This thesis is concerned with developing and implementing algorithms based upon the geometry of pictures. Spherical pictures have been used in many areas of combinatorial group theory, and particularly, they have shown to be a useful method when studying the second homotopy module, 1T2, of a presentation ([3],[4],[7],[12],[41] and [64]). Computational programs that implement picture theoretical and design algorithms could advance the areas in which picture theory can be used, due to the much faster time taken to derive results than that of manual calculations. A variety of algorithms are presented. A data structure has been devised to represent spherical pictures. A method is given that verifies that a given data structure represents a picture, or set of pictures, over a group presentation. This method includes a new planarity testing algorithm, which can be performed on any graph. A computational algorithm has been implemented that determines if a given presentation defines a group extension. This work is based upon the algorithm of Baik et al. [1] which has been developed using the theory of pictures. A 3-presentation for a group G is given by , where P is a presentation for G and s is a set of generators for 1T2. The set s can be described in a number of ways. An algorithm is given that produces a generating set of spherical pictures for 1T2 when s is given in the form of identity sequences. Conversely, if s is given in terms of spherical pictures, then the corresponding identity sequences that describe 1T2 can be determined. The above algorithms are contained in the Spherical PIcture Editor (SPICE). SPICE is a software package that enables a user to manually draw pictures over group presentations and, for these pictures, call the algorithms described above. It also contains a library of generating pictures for the non abelian groups of order at most 30. Furthermore, a method has been implemented that automatically draws a spherical picture from a corresponding identity sequence. Again, this new graph drawing technique can be performed on any arbitrary graph

The Complexity of Broadcasting in Planar and Decomposable Graphs

Broadcasting in processor networks means disseminating a single piece of information, which is originally known only at some nodes, to all members of the network. The goal is to inform everybody using as few rounds as possible, that is minimize the broadcasting time. Given a graph and a subset of nodes, the sources, the problem to determine its specific broadcast time, or more general to find a broadcast schedule of minimal length has shown to be NP - complete. In contrast to other optimization problems for graphs, like vertex cover or traveling salesman, little was known about restricted graph classes for which polynomial time algorithms exist, for example for graphs of bounded treewidth. The broadcasting problem is harder in this respect because it does not have the finite state property. Here, we will investigate this problem in detail and prove that it remains hard even if one restricts to planar graphs of bounded degree or constant broadcasting time. A simple consequence is that t..

The complexity of broadcasting in planar and decomposable graphs

AbstractBroadcasting in processor networks means disseminating a single piece of information, which is originally known only at some nodes, to all members of the network. The goal is to inform everybody using as few rounds as possible, that is minimize the broadcast time.Given a graph and a subset of nodes, the sources, the problem to determine its specific broadcast time, or more generally to find a broadcast schedule of minimal length has been shown to be NP-hard. In contrast to other optimization problems for graphs, like vertex cover or traveling salesman, little was known about restricted graph classes for which polynomial time algorithms exist, for example for graphs of bounded treewidth. The broadcasting problem is harder in this respect because it does not have the finite-state property. Here, we will investigate this problem in detail and prove that it remains hard even if one restricts to planar graphs of bounded degree or constant broadcasting time. A simple consequence is that the minimal broadcasting time cannot even be approximated with an error less than frbuil|1/8, unless P = NPOn the other hand, we will investigate for which classes of graphs this problem can be solved efficiently and show that broadcasting and even a more general version of this problem becomes easy for graphs with good decomposition properties. The solution strategy can efficiently be parallelized, too. Combining the negative and the positive results reveals the parameters that make broadcasting difficult. Depending on simple graph properties the complexity jumps from NB or P to NP