## 2 Preliminaries
### Broadcast Algorithms
In contrast, our algorithm does not require the assumption of lock-step synchrony to provide the propagation 1-safety and propagation reliability properties. In our proposed algorithm, the legitimate broadcast is isolated from the arbitrarily initialized processes to prevent perturbation of the states of the processes already involved in the broadcast by those that are arbitrary initialized and those whose state changes are caused directly or indirectly by the arbitrarily initialized processes. In addition, our proposed algorithm ensures that no process joins the broadcast tree prior to resetting its state in a separate step. Since the algorithm proposed in [2] assumes lock-step synchrony, its implementation requires separate implementations of the algorithm and the lock-step synchrony along with their interfacing. The complexity and the overhead of the implementation of [2] are eliminated by our proposed simpler algorithm by not requiring a separate mechanism to implement lock-step synchrony. In addition, the time complexity of [2] and our proposed algorithm are the same, i.e., our algorithm does not introduce any overhead to eliminate the assumption of lock-step synchrony. Table 1 compares the abovementioned broadcast approaches with our proposed algorithm. The comparison shows that most existing algorithms are asynchronous (do not assume lock-step synchrony), do not support arbitrary initialization, and offer limited scalability. They also do not guarantee 1-safety (handles spurious messages) and premature feedback safety, which can lead to safety violations. In contrast, our algorithm is asynchronous and handles arbitrary initialization, premature feedback safety, 1-safety (handles spurious messages) while being scalable with low complexity, as shown in the table. Numerous broadcast algorithms have been developed with varying properties such as reliability, safety, efficiency, and network overhead for a number of application areas. In wireless communication networks, broadcast is commonly used to disseminate information about network topology and available resources, enabling efficient coordination and communication among nodes. Military operations rely on broadcast to provide troops in the field with real-time updates on the battlefield, tactical movements, and changes in the tactical situation [51]. Financial transactions leverage broadcast to distribute transaction data and updates to all involved parties, ensuring timely and accurate information sharing [52]. Emergency response systems utilize broadcast to communicate critical information about the location and status of emergency responders, facilitating coordinated \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Approach** & \begin{tabular}{c} **Handling of** \\ **Arbitrary** \\ **Initialization** \\ \end{tabular} & **Scalability** & \begin{tabular}{c} **1-Safety (Handling** \\ **of Spurious** \\ **Message)** \\ \end{tabular} & \begin{tabular}{c} **Requires** \\ **Lock-Step** \\ **Synchrony** \\ \end{tabular} & \
\begin{tabular}{c} **Premature** \\ **Feedback** \\ **Safety** \\ \end{tabular} &
\begin{tabular}{c} **Time** \\ **Complexity** \\ \end{tabular} \\ \hline Floodog & No & Low & No & No & No & \(O(D)\) \\ Broadcast [27] & No & Low & No & No & No & \(O(h)\) \\ \hline PIF [31; 32; 50] & No & Low & No & No & No & \(O(h)\) \\ \hline PFC [37; 38] & No & Low & No & No & No & \(O(h)\) \\ \hline Distributed & No & Low & No & No & No & \(O(N^{2})\) \\ Reset [44; 45] & No & Low & No & No & No & \(O(N^{2})\) \\ \hline Al-Jady \& Karaata [2] & Yes & Moderate & Yes & Yes & Yes & \(O(D^{2})\) \\ \hline Proposed Algorithm & Yes & High & Yes & No & Yes & \(O(D^{2})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of broadcast approaches. response efforts [53]. In inventory management systems, broadcast is utilized to distribute information about inventory levels, restock requests, and order updates, optimizing supply chain operations [54].