# A Jamming-Resilient and Scalable Broadcasting Algorithm for Multiple Access Channel Networks
## 4 Proposed Broadcast Algorithm
This unique pattern is represented by two consecutive silence feedback, which represents the end of a segment. When a node identifies this pattern, it can then synchronize its execution of the KTFW algorithm with existing nodes in the network. Therefore, new nodes can join without having their clocks synchronized with existing nodes. Any new node can start communicating on the network after completing the synchronization step. The second case for the synchronization step is when no network activity is present and the new node has at least one packet to transmit. In such cases, the new node waits for a maximum of \(k\) rounds before beginning the execution of the KTFW algorithm. The existing nodes synchronize their execution with the new node as soon as it starts processing a segment, represented by a packet being heard on the channel. When a collision is received as feedback, it implies that the existing nodes started processing a segment. In this scenario, the new node synchronizes its execution with the existing nodes and continues its execution of the KTFW algorithm accordingly. The synchronization step is performed exactly once by any new node, which allows them to communicate as long as they are connected. Because the synchronization step is performed once when a new node joins the network, it does not affect the network's performance. Next, we present an example of how the KTFW algorithm works, assuming \(k=8\). After 8 rounds elapsed, the queues of the nodes are \(\{1,1,2,0,0,0,0,1\}\), and their IDs based on the activation round are \(\{1,2,3,-,-,-,-,8\}\). The nodes are referred to as \(n_{1}\), \(n_{2}\),\(\ldots\),\(n_{8}\). The execution of the KTFW algorithm begins in round 9. All nodes with IDs in the range  
Figure 1: How the binary search works in the K-tuple full withholding (KTFW) when \(k=8\). (\(1-8\)) transmit their packets at this round. A collision is detected in the same round by all the nodes. Therefore, the range is divided into two halves: \((1-4)\) and \((5-8)\). The first half of nodes \((1-4)\) transmit their packets at round \(10\), which results in a collision. Similarly, the range is divided into two halves: \((1-2)\) and \((3-4)\). Then, \(n_{1}\) and \(n_{2}\) transmit their packets in round \(11\), resulting in a collision. Because the range \((1-2)\) consists of two nodes, the next rounds are used by both nodes to transmit their packets. Therefore, \(n_{1}\) transmits its packet in round \(12\), and then after detecting silence in round \(13\), \(n_{2}\) will transmit its packet. The nodes \(n_{1}\) and \(n_{2}\) become passive as they have transmitted all their packets. After detecting silence in round \(15\), the algorithm is reset by setting the search range to \((1-8)\) in all active nodes. Next, a collision is found in round \(16\) because the range \((1-8)\) contains two active nodes: \(n_{3}\) and \(n_{8}\). Then, node \(n_{3}\) transmits its two packets in rounds \(17\) and \(18\), and then it becomes passive. The algorithm is reset in round \(19\) because silence is detected. Afterward, node \(n_{8}\) transmits its packet at round \(20\) because it is the only active node in the range \((1-8)\). Finally, silence is detected in rounds \(21\) and \(22\), and the second silence implies the end of the segment since it is detected after the algorithm's reset. Next, we highlight the novelty of the algorithm and describe how it is fundamentally different from the existing tree-based algorithms [37; 38; 39]. Although they share a common methodology of processing nodes for broadcast using a tree-based approach, they are different in two major ways: how the tree is constructed and how the nodes are processed for broadcasting. Typically, a tree is formed based on nodes with packets that must be broadcast for these algorithms. For the existing tree-based algorithms, a tree is constructed with nodes that participated in a collision. Therefore, there is no bound on the number of nodes in a formed tree.