## 2 Experimental Section
### Method
#### 2.2.2 Searching for Minimal Dominating Set  
In the previous section, the discretization of the region is elaborated and the next step is to discover the minimal dominating set from all the dominating sets. Therefore, this section mainly discusses the minimal dominating set. Two definitions are given first. **Definition 1**. : _Given two sets of sensors under coverage, \(S_{i}\) and \(S_{j}\), if \(S_{i}\subseteq S_{j}\), then \(S_{j}\) dominates \(S_{i}\)._  
**Definition 2**. : _If there is no other set dominating \(S_{j}\) in the region, then \(S_{j}\) will be determined to be the optimal dominating set in the region._  
In order to find the optimal dominating set, we first assume that a charger is placed on a fixed point in the region. It is then rotated through different directions in order to determine all the coverage dominating sets, after which the best dominating set is selected. The minimal dominating set in the region is the optimal dominating set of the directional chargers. There is no need to calculate all of the coverage sets, because the target area has already been separated, so the charging powers in the same area are the same. The above Algorithm 1 can be regarded as one kind of greedy algorithm. The coverage dominating sets are obtained through continuous rotations of the directional chargers, and then the optimal dominating set is selected from the calculated coverage dominating sets, as such, the region can be covered with the least amount of chargers. Steps 1 to 3 initialize the angles and Steps 4 to 7 rotate the angles and discover the angle that can cover the maximum number of sensors and label the sensors as a coverage dominating set. Step 8 removes the sensor with the smallest angle until the current set can be covered again. Figure 6 shows an example of finding the coverage dominating sets through rotations. The algorithm starts from the first coverage dominating set {s\({}_{2}\), s\({}_{3}\)} and then discovers three more dominating sets, {s\({}_{3}\), s\({}_{4}\)}, {s\({}_{5}\), s\({}_{6}\)}, and {s\({}_{5}\), s\({}_{6}\), s\({}_{7}\), s\({}_{8}\)}. Figure 5: The distribution of the charging demands of the sensors. ```
Input: Directional charger \(a_{i}\), angle \(\theta\) and the sensor set \(S_{i}\) with the locations of all the sensors Output: Coverage Dominating Sets \(CDSs\) Step 1. Calculate the included angle between the sensor and the charger; Step 2. Sort sensors using angles, from small to large. \(\theta_{1}\leq\theta_{2}\leq\ldots\leq\theta_{X}\) with respect to sensors \(s_{1}\), \(s_{2}\), \(\ldots\)\(s_{X}\); Step 3. Set a parameter \(S_{t}\) to record the current coverage set. Two parameters, \(\theta_{min}\) and \(\theta_{max}\), record the minimum and maximum of the current coverage set. Initialize \(\theta_{min}=\theta_{1}\), \(\theta_{max}=\theta_{1}\); \(S_{t}=\{s_{1}\}\); Step 4. Starting from second sensors, \(j=2\); Step 5. Rotate the chargers until no more sensors can be covered. While \((\theta_{j}-\theta_{min}\leq\theta)\){ \(\theta_{max}=\theta_{j}\); Add \(s_{j}\) to \(S_{t}\); \(j=(j+1)\bmod X\); } Step 6. Add the current \(S_{t}\) to the cover dominating sets \(CDSs\); Step 7. Let \(\theta_{max}=\theta_{j}\) and add \(s_{j}\) to \(S_{t}\); Step 8. Remove the sensor from the smallest angle until the current set can be covered again \(\text{While}(\theta_{max}-\theta_{min}>\theta)\){  From the \(S_{t}\) to remove the minimum angle sensor, and \(\theta_{min}\) update into the current minimum angle value; If(\(\theta_{min}\)=\(\theta_{1}\)){  End;} } Go to Step 5. ```  
**Algorithm 1**Calculate the Coverage Dominating Sets of Given Sensors  
Figure 6: A coverage dominating set finding example. After discovering all the coverage dominating sets \(CDSs\), the total power \(P\), charging efficiency \(W\) and charging demand \(H\) are then calculated. The coverage dominating set with minimum \(H\)-\(W\) is selected as the best dominating set. The following algorithm II illustrates the procedures of finding the best dominating set.