## 4 Conclusions  
In this paper, we presented a novel methodology for addressing the complex challenge of optimal sensor placement in two-dimensional and three-dimensional spaces. We have proposed and thoroughly investigated a new variant of exhaustive search, the recursive exhaustive search (RES). It provides the same or better coverage than traditional exhaustive search while significantly reducing the computational overhead, especially in three-dimensional contexts. This efficiency makes RES a robust benchmark for evaluating the performance of various stochastic optimization algorithms in sensor placement tasks.  
We then investigated the effects of two key factors--the number of evaluations of the optimization function and the rasterization of the environment--on the efficiency of the sensor placement strategies. Our comparative analysis of three prominent stochastic optimization algorithms--Artificial Bee Colony (ABC), Fireworks Algorithm and Particle Swarm Optimization (PSO)--with the benchmarks established by RES provided valuable insights into their effectiveness. In particular, the ABC and PSO algorithms performed commendably. They often achieved or came very close to the optimal coverage values determined by RES, while significantly reducing the computation time.  
The main limitation of the proposed approach is the increase in computational complexity with the size and topological complexity of the environment models. This limitation can be mitigated by a parallelized implementation of the computation of the optimization functions, which is executed on multiple CPU or GPU cores. Another limitation related to the proposed RES is the potentially suboptimal solution, as mentioned in Section 2.4.  
The insights gained emphasize the importance of fine-tuning parameters and balancing computational complexity and solution quality. These considerations can inform decision-making in real-world sensor networks and improve their efficiency and reliability. Future research directions could investigate advanced optimization techniques or hybrid approaches to further improve the efficiency of sensor placement. Furthermore, investigating the applicability of our findings to specific domains and considering dynamic environments could lead to more robust and adaptive sensor placement strategies. One of the possibilities is to train an agent (sensor) model using reinforcement learning, which could lead to the possibility of finding optimal results in an unknown environment.  
Conceptualization, D.S.; Methodology, M.B. and D.S.; Software, M.B. and D.S.; Validation, M.B., D.P. and K.L.; Formal analysis, M.B.; Data curation, D.P.; Writing--original draft, M.B., D.P., K.L. and D.S.; Writing--review & editing, M.B., D.P., K.L. and D.S.; Visualization, M.B. and D.S.; Supervision, D.S.; Project administration, K.L. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the University of Rijeka, Rijeka, Croatia under grant uniri-iskusni-tehnic-23-261.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Environment 1**} & \multicolumn{2}{c}{**Environment 2**} & \multicolumn{2}{c}{**Environment 3**} \\  & & **2D** & **3D** & **2D** & **3D** & **2D** & **3D** \\ \hline RES & \(C\) & 0.1446 & 0.0795 & 0.1191 & 0.0814 & 0.1359 & 0.0742 \\  & \(E\) & 1567 & 21,230 & 1533 & 17,051 & 1534 & 16,874 \\ \hline ABC & \(C\) & 0.1444 & **0.0795** & **0.1190** & **0.0814** & 0.1352 & 0.0739 \\ FWA & \(C\) & 0.1433 & 0.0781 & 0.1170 & 0.0806 & 0.1346 & 0.0732 \\ PSO & \(C\) & **0.1445** & **0.0795** & **0.1190** & **0.0814** & **0.1356** & **0.0741** \\  & \(E\) & 300 & 1200 & 300 & 1200 & 300 & 1200 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Coverage values \(C\) and the total number of evaluations \(E\) for RES and three selected optimization algorithms for single sensor placement problem in all three environments.