## 1 Introduction  
With many large-scale Low Earth Orbit (LEO) constellation projects launched, such as Iridium, SpaceX, OneWeb, and Hongyan, LEO satellites have recently received attention concerning their prospects of augmenting the Global Navigation Satellite System (GNSS). The large-scale LEO constellation greatly increases the number of visible satellites, which offers both opportunities and challenges for the GNSS system. More satellites enable more favorable geometric configurations for positioning, thus reducing the Geometric Dilution of Precision (GDOP) and improving the positioning results under occluded conditions. However, they also impose new burdens and challenges on the receivers. Therefore, how the optimal number of satellites can be selected from a large pool of candidates and how the receiver computation load can be balanced with the positioning accuracy is an urgent problem to be solved. As GNSS satellite constellations continue to expand, the investigation into selecting appropriate satellites from among the visible ones has gained momentum. Several distinct technical approaches have emerged from this research. The initial approach involves determining the optimal satellite combination by assessing the influence of currently observable satellites on the overall GDOP. Subsequently, a recursive strategy is employed to finalize the optimal satellite constellation [1; 2; 3; 4; 5; 6; 7; 8; 9]. Contrastingly, the second technical approach acquires satellite selection outcomes directly by leveraging the geometric interactions among satellites. The principal strength of this method lies in its rapid and convenient nature. However, it does exhibit a certain deficiency in theoretical underpinning, relying more on empirical algorithms. Notably,within this category, the Quasi-Optimal algorithm, which eliminates satellites sharing a common line of sight [10; 11], and the Fast algorithm [12], which achieves the optimal configuration of 4-8 satellites through exhaustive computer enumeration, stand out as the most prominent examples. These algorithms have subsequently formed the basis for a series of investigations into optimization selection methods [13; 14; 15]. The third approach involves delving into the theoretical examination of the GDOP. Prior research encompasses scrutinizing the closed-form formulation of the GDOP for multi-GNSS systems [16; 17], investigating the correlation between the GDOP and user-to-satellite unit vectors [18] and refining the outcomes of approximate closed-form formulations using the PDOP (Position Dilution of Precision) [19]. Notably, optimal GDOP configurations have been explored as well. For instance, [PERSON] et al. conducted an analysis of optimal GDOP configurations in underwater scenarios devoid of observation constraints [20]. Their findings underscored the pivotal roles that symmetry and uniformity play in achieving accurate positioning results. Moreover, [PERSON] et al. and [PERSON] et al. extended this line of inquiry by scrutinizing the optimal positioning configuration of a dual GNSS system under a 5-satellite scenario [21; 22]. As technology continues to advance in the realm of artificial intelligence, there has been a progressive surge in studies exploring the application of intelligent algorithms to tackle GDOP challenges. For example, this surge includes the utilization of modified genetic algorithms [23], the implementation of end-to-end neural networks employing point cloud segmentation [24; 25], the integration of adaptive simulated annealing particle swarm optimization algorithms [26], the application of improved particle swarm optimization algorithms [27], the utilization of the easy binary particle swarm optimization with a queen informant (EPSOq) algorithm to address GDOP optimization distribution [28], and the combination of unsupervised learning with the Fast algorithm to tackle satellite distribution challenges [29].