## 4 Simulation Results and Discussion
### Simulation for One-Hop Setup
#### 4.1.4 Performance Comparison to Related Approaches  
To further show the performance advantages of our proposed method, we compare the accuracy performance of our proposed solution to other solution that can be deployed to perform localization in the same environment without additional specialized localization hardware. For this performance comparison, we select the \(k\)-nearest neighbors (kNN) approach, where \(k\) neighbors are the \(k\) pre-mapped geographic points whose RF fingerprints are closet matching to that of a sensor, and each of the k neighbors is weighed inversely proportional to its Euclidean distance in signal space from the sensor whose location is being estimated [23]. We implement a script following [23], and \(k=3\) is used in the script for this performance comparison.  
One additional requirement for kNN approach is the database construction of RSS fingerprints through a site-survey prior to the localization. For the database construction, the more sampling points are performed, the better is the performance. With the same setup of the map size, the placement of beacons and the number of sensors, as well as the same path loss model and log-normal shadowing, we conduct performance studies for the kNN approach using different number of survey points via simulation, and report the results in Table 2. For each number of survey points, we execute kNN ten times and take average over ten times for the final mean estimation error. As can be seen, only for 25,600 (_i.e._, 160 by 160) survey points, the kNN approach achieves just below 16 m, which is of 2.3 times the localization error of our proposed method. We also observe from the table that further increasing of the survey points to 250,000 does not have significant improvement in the accuracy. This confirms that our solution is a good RSS-based localization candidate.  
Figure 8: The estimated locations from two phases comparing to the true locations. (a) phase 1 estimated locations; (b) phase 2 estimated locations.  
Furthermore, we compare our proposed solution to maximum-likelihood estimators (MLEs) for sensor location estimation [24] in the same one-hop setup. MLE works in a similar network setup that has a number of reference devices with known coordinates, and a number of blindfolded devices whose positions are to be estimated. With the combined range information between many pairs of devices and the known locations of reference devices, a Maximum-likelihood solution for the location of all of the blindfolded devices is determined. For this performance comparison, we used the MLE script implemented by [PERSON] which can be obtained in [25]. Using the same system setup and sensor deployment shown in Figure 8, we execute ten trials of MLE, take mean over all ten sets of estimated locations produced in different trials as the resultant location estimates, and plot these location estimates in Figure 9c. The estimated locations produced by our algorithm as well as by kNN are also plotted for comparison. As can be seen in the figure, almost none of the location estimates can reach their actual locations, nor can them form a shape similar to that of the actual sensor deployment. Numerically, the mean localization error for MLE is 35.27 m, _i.e._, 1.77_-R_, which is five times of our spring-relaxation solution.