## 4 Problem Analysis and Algorithm Description
### Algorithm Description
#### 4.2.1 Optimal Activation Strategy Based on 2D Convex Hull  
In minimizing network coverage overlaps, nodes should be as far away from one another as possible, but they should not interrupt the communication links between source nodes and their neighboring nodes. The sink node broadcasts a message to all nodes, according to the received signal strength from node \(S_{i}\), to calculate the distance between \(S_{i}\) and the sink node. In a similar manner, the 2D level coordinates of all nodes can be determined based on the sink node. The sink node sends a control signal to randomly activate three sleep nodes, which then form the simplest 2D convex hull, as shown in Figure 5. In the figure, the active nodes form a simplest 2D convex hull (a triangle) through a solid line. The set of currently active nodes is denoted by Sa, and the set of sleep nodes is denoted by \(Ss\). Sleep nodes inside the 2D convex hull are referred to as _sleep_in_, and those outside are denoted by _sleep_out_, where _sleep_in_\(\cap\)_sleep_out_ = \(Ss\). The inside and outside of the 2D convex hull have many coverage holes and monitoring blanks. Obviously, the current regional network coverage area is determined by two parts: the coverage holes  
Figure 4: Algorithm framework. Figure 5: Schematic diagram of the 2D convex hull activation strategy. of the convex hull interior and the monitoring blanks of the convex hull exterior. Therefore, we obtain the following formula:  
\[S_{c}=S_{CH}-S_{Hole} \tag{10}\]  
where \(S_{c}\) denotes the 2D network coverage area, \(S_{CH}\) denotes the area of the 2D convex hull, and \(S_{Hole}\) denotes the area of the coverage holes. Thus, we can improve the value of \(S_{c}\) by increasing \(S_{CH}\) or decreasing \(S_{Hole}\). Furthermore, the coverage area of the network can be expressed:  
\[S_{c}=\bigcup_{i=1}^{m}C_{i} \tag{11}\]  
where \(C_{i}\) denotes the coverage area of the active node \(S_{i}\). \(m\) nodes are activated and \(S_{i}\in Sa\). A sleep node \(C_{j}\) is chosen from \(S\)s to be activated, and then the area of the 2D network coverage is maximized. At the current moment, the coverage area is presented by \(S_{c}^{\prime}\), and our goal is to maximize \(S_{c}^{\prime}\). \[\max{S_{c}}^{\prime}=S_{c}\bigcup C_{j}=\left(\bigcup_{i=1}^{m}C_{i}\right) \bigcup C_{j} \tag{12}\]  
where \(C_{j}\) denotes the coverage area of sleep node \(S_{j}\), and \(S_{j}\in S\)s.  
To maximize \(S_{c}^{\prime}\), by comparing the different coverage gains from selecting the different activated sleep nodes derived from _sleep_in_ or _sleep_out_, we determine the node that can obtain the greatest coverage gains. Therefore, the optimal activation strategy is transformed into the maximum coverage gains through the selection of activated sleep nodes in \(S\)s. Before the optimal activation strategy is solved, the following theorem is proposed, and we will prove it subsequently. **Theorem 1**. : _If the radius Rh of sn empty circle (Section 3.2.5) is greater than the sensing radius Rs, then the interior of the empty circle will certainly have coverage holes._  
Proof of Theorem 1.: By the definition in Section 3.2.5, the interior of the empty circle must not have additional active nodes, which means that the interior of the empty circle maybe has some sleeping nodes. Obviously, when \(Rh\) is greater than \(Rs\), the center of the empty circle must not be covered, so we can infer that coverage holes exist in the empty circle. When \(Rh\) is larger, the active nodes are relatively more sparse, and the probability of occurrence of coverage holes is greater. Therefore, we choose to activate a sleeping node from the interior of the empty circle with a maximum \(Rh\), and this activation strategy can effectively increase the network coverage area. At this moment, in selecting to activate the sleep node nearest to the center of this empty circle, the node becomes a candidate active node and is denoted by \(S_{Cand1}\).