Messy Broadcasting in Grid

In classical broadcast models, information is disseminated in synchronous rounds under the constant communication time model, wherein a node may only inform one of its neighbors in each time-unit—also known as the processor-bound model. These models assume either a coordinating leader or that each node has a set of coordinated actions optimized for each originator, which may require nodes to have sufficient storage, processing power, and the ability to determine the originator. This assumption is not always ideal, and a broadcast model based on the node’s local knowledge can sometimes be more effective. Messy models address these issues by eliminating the need for a leader, knowledge of the starting time, and the identity of the originator, relying solely on local knowledge available to each node. This paper investigates the messy broadcast time and optimal scheme in a grid graph, a structure widely used in computer networking systems, particularly in parallel computing, due to its robustness, scalability, fault tolerance, and simplicity. The focus is on scenarios where the originator is located at one of the corner vertices, aiming to understand the efficiency and performance of messy models in such grid structures.