# Messy Broadcasting in Grid
## 4 Discussion  
This paper investigated the efficiency and performance of messy broadcasting in grid structures, offering valuable insights into the bottlenecks that contribute to broadcast delays in grid graphs. By analyzing messy models \(M_{1}\) and \(M_{3}\), we developed an optimal broadcast scheme that identified the final informed vertex and accurately calculated the broadcast time. This provided a tight upper bound on the broadcast duration in grid structures for these specific models. Our findings also indirectly established a bound for the \(M_{2}\) model. Corollary 1 offers an approximation for broadcast time and latently provides intuition of the positioning of the last-informed vertices in a manner that is both easier to calculate and more intuitive. Table 2 summarizes the found messy broadcast times in a grid structure when the originator is one of the corner vertices.  
Our work provides a more comprehensive analysis and a much tighter bound on broadcast times compared to the previous studies' upper bounds, thereby enhancing the theoretical framework within which messy broadcasting is understood. The provided insights on the broadcasting bottlenecks in grids and the exact upper bound could be beneficial in the design of efficient communication protocols in distributed systems, wireless networks, and other applications reliant on rapid information dissemination that require broadcasting.  
Future research directions could explore extending these models to more complex and heterogeneous network structures. Our work could serve as the foundation for determining the broadcast time for torus and higher-dimensional grid structures, both of which are prevalent in computer networking system designs. Since a torus can be viewed as a grid  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Model \(M_{1}\)** & **Model \(M_{2}\)** & **Model \(M_{3}\)** \\ \hline \(n+m\) & \(n+m\leq t_{2}\leq 3m+3\frac{1}{8}n+0.68\) & \(3m+3\frac{1}{8}n-7.33\leq t_{3}\leq 3m+3\frac{1}{8}n+0.68\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of the messy broadcast times in \(G[n+1,m+1]\), where \(n\leq m\). The exact result for model \(M_{3}\) is detailed in Theorem 3.  
with \"loop-back\" edges, our findings on the infinite grid may be applicable in determining the exact broadcast time for a torus. For higher-dimensional grids, the techniques employed in this paper, particularly our novel approach of using a continuous view, could also be utilized. In this approach, the borders of regions in higher dimensions would be represented by higher-dimensional planes described by a set of linear equations. This continuous perspective, possibly aided by linear programming, could help identify the optimum candidates. However, such analysis involves significant details and nuances, which is one reason this paper's results were limited to two-dimensional grids. Another intriguing topology for future exploration is hyper-cylinder graphs, as introduced in [1].  
Further research could also investigate obtaining a tighter result for model \(M_{2}\). Examining the impact of dynamic changes in the network during broadcasting and developing adaptive schemes to optimize performance under varying conditions is another valuable area to explore. Additionally, empirical studies could be conducted to validate our theoretical findings in real-world network scenarios, providing a more comprehensive understanding of the practical applications of our work.  
Conceptualization, A.A.; formal analysis, A.A.; supervision, H.A.H.; validation, H.A.H. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
No new data were created or analyzed in this study.  
The authors declare no conflicts of interest.