## 2 Related Works  
In this section, we will provide a brief introduction about the dynamic clustering and routing protocols that are related to the research works in this paper. Dynamic clustering is firstly proposed in LEACH [10]. The basic idea is to periodically rotate the cluster heads (CHs) and to reorganize the cluster-based topology based on the new set of CHs. In this case, the heavy energy burden of CHs can be dispersed all over the network. Moreover, the CHs can use data aggregation techniques [20, 21] to reduce the data volume and the energy consumption for transmitting the data to the sink. Due to the advantages of scalability and energy efficiency, dynamic clustering is considered a promising solution for large-scale data collection in WSNs. Therefore, it has attracted considerable attention from various research communities. Cluster head selection is a fundamental issue in dynamic clustering. In HEED [14], an iteration-based algorithm is proposed to consider both residual energy and communication cost in cluster head selection. It improves the CHs' distribution in the network and has better efficiency than LEACH. However, HEED has considerable message overhead due to the iteration in the algorithm. To solve this problem, BSC(Backoff Strategy Clustering) [16] adopts the random backoff scheme to control the process of CH selection. The node with a smaller backoff time has a higher probability to be the CH. BSC can generate well-distributed CHs with low overhead. Different from the distributed algorithms given above, in [22], the harmony search algorithm (HSA) is used to select the CHs with centralized optimization. It is expected to minimize the intra-cluster communication cost and to optimize the energy distribution of the network. Cluster formation is another important issue in dynamic clustering. EECS [13] proposes a novel cost function for CMs to select proper CHs by considering the difference of the distance between the CH and the sink. The energy consumption of inter-cluster communication is balanced by intra-cluster communication in EECS, and the network lifetime is significantly improved. Different from the design of cluster formation algorithms, some research works have provided mathematical analysis on how to determine the cluster ranges in different scenarios [17,18]. In [17], the authors provide a mathematical framework to determine the optimal number of clusters by minimizing the energy consumption in both single-hop and multi-hop scenarios. EC (Energy-Efficient Clustering) [18] considers the multi-hop data collection scenario and formulates an optimization problem that determines suitable cluster ranges depending on the hop distance to the sink. The inter-cluster communication has not been considered as an important problem in most research works on dynamic clustering. Many dynamic clustering protocols are developed based on the assumption that the CHs can communicate with the sink directly [10,13,16]. Some research works have considered the multi-hop routing among CHs. For example, HEED [14] uses the greedy routing algorithm for inter-cluster communication, and EC [18] uses a routing algorithm that considers the load balance among CHs. Nonetheless, the routing algorithm and the clustering algorithm are generally considered as two isolated problems in these works. On the other hand, the research work in [19] proposes an interesting result that the connectivity and efficiency of the inter-cluster topology have tight relations with the cluster head selection. Based on this result, they propose a hybrid clustering and routing (HCR) protocol that combines gradient routing [23,24] with dynamic clustering to generate a connected and efficient inter-cluster topology with limited transmission range. Nevertheless, HCR suffers from the imbalance of energy consumption and, thus, has poor performance in network lifetime.