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Broadcasting in Grid Graphs Iwona WojciechowskaThis work consists of two separate parts. The first part deals with the problem of multiple message broadcasting, and the topic of the second part is line broadcasting. Broadcasting is a process in which one vertex in a graph knows one or more messages. The goal is to inform all remaining vertices as fast as possible. In this work we consider a special kind of graphs, grids.In 1980 A. M. Farley showed that the minimum time required to broadcast a set of M messages in any connected graph with diameter d is d + 2(M − 1). This work presents an approach to broadcasting multiple messages from the corner vertex of a 2-dimensional grid. This approach gives us a broadcasting scheme that differs only by 2 (and in the case of an even × even grid by only 1) from the above lower bound.Line broadcasting describes a different variant of the broadcasting process. A. M. Farley showed that line broadcasting can always be completed in log n time units in any connected graph on n vertices. He defined three different cost measures for line broadcasting. This work presents strategies for minimizing those costs for various grid sizes.

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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 > 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. Committee. Finally, I wish to thank my parents, James and Dorothy Kovalchick and my sister, Amanda Kovalchick for their support and encouragement.