## 2 Preliminaries
### Broadcast Algorithms  
Broadcast, a fundamental building block in many computing systems, refers to sending messages from a process to all other processes in the system. The first broadcast algorithm was introduced as a flooding algorithm in [27], which, despite its conceptual simplicity, causes significant message duplication and network congestion. To overcome this, alternative broadcast algorithms were proposed [28; 29; 30], and research continues as new distributed systems such as blockchain, IoT, and cloud computing demand increased safety, reliability, and adaptability. Most broadcast algorithms operate through three core phases: broadcast (B), feedback (F), and cleaning (C). A classical model capturing these phases is the Propagation of Information with Feedback (PIF), introduced in [31], where a root initiates the broadcast, and feedback is received once all processes acknowledge message receipt. PIF has been used in several foundational distributed computing problems such as synchronization, termination detection, and mutual exclusion [32; 33; 34; 35; 36]. Propagation of Information with Feedback and Cleaning (PFC), a variant of PIF, introduces a cleaning phase to erase traces of completed broadcasts, thus enabling new broadcasts to proceed safely and efficiently [37; 38]. Both PIF and PFC have demonstrated efficiency and reliability in distributed systems, including wireless sensor networks [39; 40; 41], peer-to-peer networks [42], and cloud environments [43]. Distributed reset is a broadcast mechanism to bring all system processes to the same distinguished state [44; 45]. Distributed reset is similar in nature to a distributed broadcast and can be implemented using a distributed broadcast. It is used in performing system reconfigurations upon new processes or channels being added [45]. Since arbitrarily initialized and unpredictable states do not always align seamlessly with the reset mechanism, arbitrarily initialized processes present challenges. In addition, arbitrarily initialized processes may cause interferences with the actions of distributed reset algorithms, which may prevent reset from reaching all system processes or cause improper reset of some of the processes. These then may breach the safety, security, and reliability of the applications that use distributed reset. Apache ZooKeeper provides a distributed coordination service that includes a distributed lock service and a reliable broadcast protocol called ZooKeeper Atomic Broadcast (ZAB) [46]. ZAB is a consensus algorithm which ensures that all nodes in the system receive the same set of updates in the same order, even upon crash failures [47]. This ensures that the distributed system is always in a consistent state, which is important for applications that require high levels of reliability such as Skype, Azure Cosmos DB, Azure SQL Database, Azure IoT Hub, and Xbox Live [47; 48; 49]. In [2], [PERSON] and [PERSON] proposed the first self- and snap-stabilizing broadcast algorithm with the propagation 1-safety and the propagation reliability properties. Their proposed algorithm uses two waves, namely, a primary wave and an authentication wave. The first wave is employed to implement the broadcast and the authentication wave is employed to facilitate the 1-safety of the first wave by verifying the legitimacy of the broadcast tree prior to broadcasting further. They assume _lock-step synchronization_ where, in each lock-step, all the enabled system processes execute all their enabled actions exactly once. The objective of the lock-step synchrony assumption in their algorithm is to allow clearing the effects of arbitrary initialization to precede the actual broadcast. Without the assumption of lock-step synchrony, if a process in some state due to arbitrary initialization is reached by the broadcast before clearing the effect of the arbitrary initialization, the arbitrary initial state of the reached process may cause a premature initiation of a feedback phase, which leads to loss of the propagation reliability property.