## 4 Simulation
### Numerical Results
Long communication ranges help the mobile anchor to find more neighbors while traveling along a line segment. With the assistance of neighboring-position-aware nodes contributed by the mobile anchor, a mobile anchor with a long communication range will be able to find more neighbors when traveling along a line segment of its trajectory. With the assistance of neighboring-position-aware nodes contributed by the ISP algorithm, the gap in the mobile anchor's traveling distance between the i-SCAN and SCAN methods increases with increasing communication range. The reduction in traveling distance compared with when using the SCAN method is about 10% to 25%. The projection mechanism in the MARS method is able to further reduce the traveling distance of the mobile anchor. The reduction in traveling distance compared to that required when using the SCAN method is about 24% to 34%. When using the MARS method, the benefits of the projection mechanism can be best observed when using a short communication range. This advantage becomes minor with increasing communication range, as evidenced by the improved traveling distance compared with that when using i-SCAN. Compared to the traveling distance when using i-SCAN, there is about a 17% reduction in traveling distance with a communication range of 50 m, and 11.6% with a communication range of 150 m.  
Next, the percentage of unknown nodes successfully positioned under the constraint of a limited traveling distance of the mobile anchor was evaluated, as shown in Figure 7. This simulation is similar to the evaluation in which the mobile anchor had finite energy. In this experiment, the maximum traveling distances of the mobile anchor evaluated are [500, 1000, 1500, 2000, 2500]. The number of unknown nodes is 40, and the communication range is 100 m.  
Figure 6: Traveling distances with different communication ranges (40 nodes). Other than SCAN-ET and NHexCurve, none of the methods are able to position unknown nodes when the maximum traveling distance is 500 m. By applying the straight line as the initial path for scanning the deployment area from left to right, the mobile anchor is able to detect unknown nodes but is not able to position them because of the collinear problem. SCAN-ET and NHexCurve do not use a straight line for the mobile anchor's initial path, as in the SCAN method. When the mobile anchor detects any unknown nodes, the mobile anchor in these two methods is able to quickly locate the positions of the unknown nodes. When the maximum traveling distance is extended to 1500 m, the mobile anchor is able to quickly locate the discovered unknown nodes when using the i-SCAN and MARS methods, which use the ISP algorithm. Their positioning ratios for the unknown nodes are 28% and 45%, respectively. The ratios obtained when using the SCAN-ET and NHexCurve methods, which exhibit better positioning ratios than the other methods at 500 m and 1000 m, increase moderately, with ratios of 34% and 13%. When the maximum traveling distance is 2000 m, the advantage of using the ISP algorithm becomes obvious. The positioning ratios when using i-SCAN and MARS are 65% and 83%, respectively. The mobile anchor when using MARS is able to scan more areas by reducing its traveling distance compared to the i-SCAN method. However, the ratios obtained when using SCAN-ET and NHexCurve are still lower than 50%. At the maximum traveling distance of 2500 m, the ratios of successfully positioned unknown nodes when using the SCAN-ET, NHexCurve, i-SCAN, and MARS methods are 56%, 35%, 80%, and 94%. Here, the ratio of NHexCurve is 45%, which is worse than the results obtained when using the SCAN method. The following experiments evaluate the dense and sparse deployment of unknown nodes. In the dense deployment case, the topology of the deployed unknown nodes is a strongly connected graph. In the sparse deployment case, the unknown nodes are not able to find any neighbor nodes within their communication range. The unknown nodes in these two cases are still randomly deployed.