## 4 Simulation Results and Analysis
### Computational Complexity and Runtime Evaluation  
We analyze the computational complexity of each sub-algorithm as follows. For Algorithm A, in the beam generation stage, K-means clustering is applied. Each iteration requires computing the distances between \(U\) users and \(MN\) beam centers. If the number of iterations is denoted by \(N_{\mathrm{k}}\), the complexity of this process is \(\mathrm{O}(N_{\mathrm{k}}UMN)\). In beam-user matching, an ILP formulation is adopted. Based on empirical results from commercial solvers, the complexity of the ILP grows approximately with the number of variables to the power of 1.3 to 1.6. Therefore, the complexity can be estimated as \(\mathrm{O}((UMN)^{1.5})\). Hence, the total complexity of Algorithm A is \(\mathrm{O}(N_{\mathrm{k}}UMN+(UMN)^{1.5})\). This complexity scales polynomially with both the number of users \(U\) and the number of candidate beams \(MN\), making it sensitive to large user populations. However, the modular structure ensures that beam generation and assignment are handled efficiently and converge quickly in practice. Therefore, even for large-scale LEO constellations, Algorithm A remains computationally feasible. For Algorithm 1, during greedy initialization, each candidate beam must be evaluated for placement within a time slot by checking the minimum distance constraint with up to \(N\) existing beams. This results in a complexity of \(\mathrm{O}(MN^{2})\). In the simulated annealing phase, assume the total number of random perturbations is \(N_{\mathrm{SA}}\). Each perturbation involves randomly selecting a beam and checking distances against up to \(N\) other beams in the new slot, yielding a complexity of \(\mathrm{O}(N_{\mathrm{SA}}N)\). Therefore, the overall complexity of Algorithm 1 is \(\mathrm{O}(MN^{2})+\mathrm{O}(N_{\mathrm{SA}}N)=\mathrm{O}(MN^{2})\). Since Algorithm 1 only depends on the number of beams and time slots, but not on the number of users \(U\), it exhibits excellent scalability in scenarios with dense user deployments. Its complexity increases quadratically with the number of beams per slot but remains tractable for typical satellite system parameters. This makes it suitable for real-time or near-real-time scheduling in large constellations. For Algorithm C, this algorithm involves sub-band assignment, intra-beam power allocation, and inter-beam power coordination. Sub-band assignment is performed using the Hungarian algorithm. For each beam, a reward matrix of dimension \(L\times L\) is constructed, where \(L\) is the maximum number of users per beam. The complexity of Hungarian matching per beam is \(\mathrm{O}(L^{3})\), and the total complexity is \(\mathrm{O}(MNL^{3})\). Intra-beam power allocation using the water-filling method requires ordering the channel gain values. The per-beam  
Figure 4: Performance of joint frequency–power allocation algorithm. (**a**) Throughput comparison of different power allocation algorithms for each satellite in time slot 1; (**b**) throughput comparison of different power allocation algorithms for each satellite in time slot 2.  
complexity is \(\mathsf{O}(L\log L)\), thus a total complexity of \(\mathsf{O}(MNL\log L)\). Inter-beam power control is formulated as a convex optimization problem, resulting in a total of \(\mathsf{O}(MN)\). Therefore, the total complexity of Algorithm C is \(\mathsf{O}(MNL^{3})+\mathsf{O}(MNL\log L)+\mathsf{O}(MN)\). The complexity of Algorithm C grows with the number of beams \(MN\) and the number of users per beam \(L\), but remains polynomial overall. This ensures tractability even in high-throughput systems where each beam serves many users. The use of efficient algorithms like Hungarian matching and water-filling further improves practicality, making this approach well-suited for dynamic resource management in large-scale satellite systems. The experiments were conducted under a hardware environment consisting of a Windows 11 operating system, an Intel i9-13900H processor (Intel Corporation, Santa Clara, CA, USA), and 16 GB of RAM.