## 3 Results
### Complexity of the AUAS Swarm Routing Algorithm  
In this section, we will discuss the arithmetic and computational complexity of the proposed AUAS swarm routing algorithm. Let us start with the arithmetic complexity of the proposed algorithm.  
**Proposition 1**.: _The arithmetic complexity of the algorithm corresponding to the AUAS swarm model (1) having \(M\) number of drones, and each drone consisting of a uniform linear array with \(N\)-elements is \(\mathcal{O}(NM^{2})\)._  
Proof.: We have \(M\)-sets of \(N\)-beamformed signals of \(M\)-drones, i.e., \(|\mathbf{y}^{(1)},\mathbf{y}^{(2)},\cdots,\mathbf{y}^{(M)}|\) as the columns of a throughput matrix to navigate the AUAS swarm. As each \(N\)-beamformed signal \(\mathbf{y}^{(n)}\) correspondences to each drone described via (1), the computation of the throughput matrix cost \(\mathcal{O}(NM^{2})\).  
Figures 4 and 5 show the proportionality between the arithmetic and computational complexities of the proposed AUAS swarm routing algorithm. Figure 4 depicts the execution time, algorithmic additions, and algorithmic multiplications of the proposed algorithm based on the code written in Python. The x-axis and y-axis show the number of drones and the complexity of the graph, respectively. Each line in the graphs shows the number of elements in the antenna arrays which are 64, 32, 16, 8, and 4 from top to bottom. Thus, it is evident that the execution time is directly proportional to the arithmetic complexity as in Proposition 1. For a better visual purpose in showing the algorithmic complexity variation based on the elements of the antenna array, we present Figure 5 on a logarithmic scale. Based on this figure, the computational complexity, algorithmic additions, and algorithmic multiplications increase as the number of drones and/or elements of the antenna arrays increases, and hence the computational complexity is proportional to the arithmetic complexity in Proposition 1.  
Figure 4: Using the code written in Python: (**a**) shows the number of algorithmic multiplications involved in computing the non-optimized AUAS algorithm; (**b**) shows the number of algorithmic  
additions; (**c**) shows the number of algorithmic multiplications on a log-scaled graph; and (**d**) shows the number of algorithmic additions on a log-scaled graph of the proposed AUAS swarm routing algorithm. Each line in the subfigures (**a**-**d**) shows the number of elements in the antenna arrays which are 64, 32, 16, 8, and 4, and varies from top to bottom, respectively.  
**Remark 1**.: _Although particle swarm optimization is a popular stochastic optimization method, it may not always provide low-complexity routing algorithms [45; 46]. Authors in [47] compared the performance of the particle swarm optimization and differential evolution techniques to find delivery routings with minimum travel distances with quadratic complexity in time and space. This complexity is calculated based on a spatial distance matrix from the deployment to the landing. Thus, once the AUAS swarm is deployed based on the proposed model, the proposed algorithm also has quadratic complexity in time as in Proposition 1 because \(N\) remains a constant after deployment._