## 4 Cluster Sleep-Wake Scheduling Algorithm  
### Related Definitions  
**Definition 1** (Coverage rate \(C_{r}\)). : _The coverage rate of sensor networks refers to the ratio between the intersection of sensor nodes' sensing range and the volume of the monitoring area, namely \(C_{r}=\frac{V_{1}\bigcap V_{2}\bigcap...\bigcap V_{n}}{V_{A}}\)._  
Where \(V_{1},V_{2},...,V_{n}\) represents the sensing range of \(node_{1}\), \(node_{2}\),..., \(node_{n}\). \(V_{A}\) refers to the volume of monitoring area \(A\). The coverage rate of a sensor network is an important indicator in evaluating the coverage performance of nodes; it represents the network's sensing ability in the monitoring area but is not the only parameter by which to measure coverage performance. Therefore, coverage control for 3D UWSNs  
Figure 4: 3D coordinate system for the monitoring area. Figure 5: Topological model of three-dimensional dense network. does not necessarily pursue a coverage rate of 1; rather, it should also consider the manufacturing and deployment costs of nodes and determine coverage requirements according to specific applications. **Definition 2** (Partition unit). : _A special polyhedron that can closely fill a 3D space without overlap or vacancy. The target 3D region \(A\) can be divided into several identical polyhedral \(P_{u}\), as \(V_{A}=\sum V_{P_{u}}\)._  
The partition unit defined here is the virtual component unit mentioned above; therefore, the topology control of an underwater 3D network seeks to ensure an active node in each division unit at any time while maintaining high coverage. Cubes, hexagonal prisms, rhombic dodecahedrons, and truncated octahedrons can be used as segmentation units. Considering the convenience of subsequent algorithms, a cube is used as the partition unit in this paper. A regular 3D spatial segmentation scheme was adopted to divide the network, and the model is denoted as \(CB_{d}\). Figure 6 depicts a cube consisting of eight partition units. **Definition 3** (Cluster). : _Each cube partition unit has 26 first-level physically adjacent units. Among them, six overlap with a surface of the segmentation unit and are called Class 1 adjacency units. Twelve adjacency units overlap with an edge of the partition units and are called Class 2 adjacency units. The eight adjacent units are overlapped with a point of the segmentation unit and are called Class 3 adjacent units. The cube, composed of the cube partition units and its 26 first-level physical adjacent units, is called a cluster as shown in Figure 7._  
**Definition 4** (Partition unit radius \(R\)). : _The radius of the partition unit refers to the radius of the circumscribed sphere of the partition cube, as shown in Figure 8._  
Figure 6: Cube consisting of 8 partition units. Figure 7: A cluster. **Definition 5** (Partition unit side length \(l\)). : _The partition unit side length \(l\) refers to the length of the side of the partition cube._  
**Definition 6** (Cluster border length \(l\)). : _Similar to the definition of the partition unit length, it refers to the side length of a cluster. That is, \(L=3l\)._  
**Definition 7** (Node number \(1D_{0}\)). : _Set the coordinates of the nodes in the 3D coordinate system of the monitoring area as \((x,y,z)\). Where \(z\) represents the length of rope between the node and the anchor and is a fixed value known at the beginning of deployment. The coordinates are the relative coordinates of sensor nodes with respect to the origin of coordinates, which can be obtained by the localization algorithm. Therefore, this unique coordinate is used in this paper as the node number, denote as \(1D_{0}=(x,y,z)\)._  
**Definition 8** (Partition unit number \(1D_{1}\)). : _For the convenience of partition unit management. A unique identification number is set for each partition unit in this paper, that is, partition unit number \(1D_{1}\).