## 4 Results and Discussion
### Comparison with the Existing Approaches by Varying Source and Target Locations  
This scenario was tested using a \(W\) at 150 m \(\times\) 150 m \(\times\) 400-1000 m \(\times\) 1000 m \(\times\) 400 m. It encompassed five maps with obstacle counts of up to 25. We analyzed our method's performance through seven runs on every map with different coordinates for \(s\) and \(t\) in each run. By varying the positions of \(s\) and \(t\), the number of obstacles to be modeled in each run/test can be distinct, and, accordingly, comparison metrics can vary with \(W\) complexity. The proposed method's averages, obtained from the seven runs on each map, are given in Table 2. The results indicate that the proposed method yielded comparable performance in all tests. The proposed method yielded an average computing time of 0.71 s, compared to the ApVL and RRT*-AB algorithms, which had mean computing times of 1.01 s and 1.37 s, respectively. In addition, the proposed method lowered the path length, compared to both prior algorithms, by 5.05%. Although the proposed method gave better results, a relatively higher number of initial \(P\) nodes can degrade its performance in complex environments. The worst-case complexity with our method was \(O(n^{3})\). However, the test results revealed that time complexity did not accelerate like \(O(n^{3})\) in all test scenarios for finding good quality paths. The results obtained from all these scenarios showed that our method performed consistently better than the ApVL and the RRT*-AB algorithms. Aside from the path lengths and computing times, we analyzed and compared its performance against prior algorithms with respect to graph/tree nodes and path node counts. In Figure 11, the results of the proposed method in terms of average path nodes and graph/tree nodes for the above experiments are presented. As shown in Figure 11, both the graph nodes and path points of our ESWG method were lessened, compared to the prior methods. Figure 10: (**Left**) computing time: proposed method versus existing algorithms; (**Right**) path lengths: proposed method versus existing algorithms. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Maps Sizes (m)/Obstacles’ Count**} & \multicolumn{2}{c}{**ApVL Algorithm [60]**} & \multicolumn{2}{c}{**RRT”-AB Algorithm [70]**} & \multicolumn{2}{c}{**Proposed PP Method**} \\ \cline{2-7}  & **Avg. Time (s)** & **Avg. Path Length (m)** & **Avg. Time (s)** & **Avg. Path Length (m)** & **Avg. Time (s)** & **Avg. Path Length (m)** \\ \hline \(150\times 150\times 400/5\) & 0.35 & 305.4 & 0.61 & 295.91 & 0.26 & 300.14 \\ \(400\times 400\times 400/10\) & 0.55 & 620.12 & 0.75 & 650.75 & 0.47 & 599.27 \\ \(600\times 600\times 400/15\) & 0.74 & 860.51 & 0.96 & 900.90 & 0.59 & 800.05 \\ \(800\times 800\times 400/20\) & 0.98 & 1020.01 & 1.81 & 1050.05 & 0.75 & 990.5 \\ \(1000\times 1000\times 400/25\) & 2.45 & 1100.05 & 2.75 & 1120.75 & 1.52 & 1070.21 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Proposed method pathfinding results’ comparison with the ApVL and the RRT-AB algorithms. Analysis of the memory requirementsAs shown in Figure 11a, the proposed PP method generates a WG with fewer vertices. In addition, it does not register visibility of the edges that contribute minimally in an optimal/quasi-optimal path due to less coverage in terms of distance or they are not in the same direction as the target location while generating an ESWG. Therefore, the memory requirements of proposed method are not high compared to the existing methods and a complete graph. However, most global PP methods keep all visible edges in the memory that significantly increase the memory requirements. Furthermore, the visibility check function is called a substantial number of times in visibility graph-based PP methods, thereby space complexity drastically increases.