## 4 Analysis
### Complexity Analysis
#### DASA Scheme  
In the DASA approach, the clusterhead triggers two rounds of 1-hop flooding to initialize the sensor scheduling management process in the 2-hop network topology. Next, 1 round of local flooding is applied for updating the round ID of each node. Then, the clusterhead generates another two rounds of local flooding for broadcasting the sensor scheduling information. Finally, the gateway nodes are selected using two rounds of 1-hop flooding. Therefore, the time complexity is \(\mathcal{O}(7)\) rounds.  
Accordingly, the total energy consumption is \(E_{sch}=N_{T}\cdot E_{T}+N_{R}\cdot E_{R}\), where:  
\[N_{T}=2\cdot(1+|H_{i}^{(2)}|)+3\cdot N_{C}, \tag{36}\]  
\[N_{R}=2\cdot(N_{i}+\sum_{j\in H_{i}^{(2)}}N_{j})+3\cdot\sum_{j=1}^{N_{C}}N_{j}, \tag{37}\]  
where \(H_{i}^{(2)}\) denotes the index set of 1-hop cluster members of cluster \(i\) with neighboring 2-hop cluster members. Hence, the communication complexity due to scheduling management is \(\mathcal{O}(N_{C})\), \(N_{C}\) is the number of sensors in the cluster. Similar to the CASA scheme, when operating the sensing task, the energy consumption is related to the number of active sensors. Therefore, the communication complexity for collecting sensing data is \(\mathcal{O}(M_{RG})\), where \(M_{RG}\) is the number of members in each round in the cluster.  
Due to the operation of the DASA scheme, the computation burden is distributed among the sensors. Thus, the computation cost analysis is considered with respect to the clusterhead and cluster members, respectively. For the clusterhead, it arranges the sensing schedule based on the largest received round ID. Hence, the computation complexity for updating the round ID in a clusterhead is \(\mathcal{O}(N_{C})\). For the cluster members, they need to initialize a waiting time, to check if a claim of being a round member is received, to update the round ID, to extend the waiting time, and to check if the waiting timer expires. Therefore, the computation complexity for scheduling management in a cluster member is \(\mathcal{O}(N_{b}^{(j)})\), where \(N_{b}^{(j)}\) is the number of the neighboring cluster members of sensor \(j\).  
Suppose that each sensor node has a _L_-byte data packet to transmit. According to the operation of the DASA scheme, the memory usage \(M_{DASA}\) yields:  
\[M_{DASA} = M_{I}+M_{R}+M_{S}+M_{G} \tag{38}\] \[= (2\cdot\sum_{j=1}^{N_{C}}N_{j}+2|H_{1}|+\sum_{k\in H_{i}^{(2)}}N _{k}+2N_{G})\cdot L, \tag{39}\]  
where \(M_{I}\) is the memory usage for initializing the procedure of scheduling management, \(M_{R}\) is the memory usage for updating the round ID, \(M_{S}\) is the memory usage for broadcasting the scheduling information, and \(M_{G}\) is the memory usage for gateway selection, \(N_{C}\) is the number of sensors in the cluster, \(H_{1}\) is the set of 1-hop nodes, and \(N_{G}\) is the number of gateway sensors for inter-cluster communication in the cluster.  
Note that although the DASA scheme has a higher time complexity due to the round ID updating process, the DASA allows the cluster members to organize themselves into round groups and complete the scheduling assignment automatically with only local neighboring information.