## 2 Related Works
### Clustering in LoRa Networks  
Clustering is one of the useful methods for enhancing network connectivity and prolonging the network lifetime in wireless networks by managing nodes efficiently. In clustering for IoT LoRa networks, LoRa devices are grouped into clusters, and the cluster head (CH) in each cluster acts as the mediator for data transmissions between LoRa devices and a GW. Clustering not only reduces packet redundancy but also manages efficient data routing, resulting in improved efficiency, scalability, and the avoidance of redundant message exchange. Clustering protocols enable end devices to reduce data packets on networks through data aggregation, which improves network lifetime and energy consumption. As a result, several studies based on clustering algorithms have been proposed for IoT LoRa networks. In [29], the authors proposed a two-stage energy-efficient cluster-based solution for data collection by a mobile sink in a robot network by considering the first stage as a travelling salesman problem and the second stage as the removal of some CH robots from the path to reduce the UAV's energy consumption. In [30], the cluster-based layering approach for uplink multihop communication was analysed. In that paper, the authors presented a multihop uplink communication scheme that took advantage of LoRa's ability to customize its transmission parameters and combines them with a novel routing protocol to solve network coverage and high energy consumption problems. Clusters were used in that scheme to associate an end device with exactly one GW, and once an end device was associated with a GW, the end device communicated through the GW. This type of scheme, however, limited the ability of end devices closer to the GW to transmit directly to the main GW. In [21], an SF-based clustering scheme was proposed to improve the multihop capacity of the LoRa network. Based on network clustering, each subnet rooted at a sink node was assigned a specific SF. The procedure focused on balancing airtime between subnets while maintaining connectivity.  
However, end device extraction from a higher SF subnet has an impact on network maintenance and appears to be an unrealistic scenario. Based on this clustering, Ref. [31] described a method for logically partitioning end devices into several clusters. End devices in nearby clusters served as relay devices for end devices in the next more distant zone.  
As a result, clustering with relaying provided an appealing solution to this problem by focusing on distant end devices one at a time. Clusters formed in that case were either the same or different widths. The clustering style in that approach considered clusters of the entire network but ignored the transmission capability of end devices closer to the GW.  
As examined above, the existing studies concentrate on clustering techniques that partition the entire network in order to reduce energy consumption and improve connectivity between LoRa devices. However, these approaches deprive LoRa devices with a high link quality of directly connecting to the GW and acting as relay LoRa devices of distant devices. As a result of only forming subspace clusters in the designated HSFZ, among the benefits of the proposed density-based subspace clustering are a reduced energy consumption, communication overhead, and increased connectivity.