## 1 Introduction  
Wireless sensor networks (WSNs) [1] have been widely used to collect sensing data in various applications, such as industrial control [2; 3], environment monitoring [4; 5] and transportation systems [6; 7]. Since wireless sensors are generally powered by batteries with limited energy, how to improve the efficiency of data collection has attracted considerable attention from different research communities [8; 9]. Dynamic clustering has been proven to be an efficient scheme for data collection in WSNs [10; 11; 12]. In dynamic clustering, sensor nodes are periodically grouped into clusters, which have a leader called the cluster head (CH) and a number of cluster members (CMs). The CH collects the data from the CMs in its cluster, and then, it forwards the aggregated data to the sink. The network operation is divided into rounds, and the cluster-based topology is completely reorganized at the beginning of every round in order to balance the energy consumption of nodes. Since the original dynamic clustering protocol was proposed in LEACH (Low-Energy Adaptive Clustering Hierarchy) [10], many research works have been proposed to improve the clustering algorithms. EECS (Energy Efficient Clustering Scheme) [13] proposes a novel algorithm for CMs to select reasonable CHs based on their locations. HEED (Hybrid, Energy-Efficient Distributed clustering) [14] proposes an iteration-based algorithm to select well-distributed CHs, and the random backoff strategy is adopted in [15; 16] to reduce the overhead. Some research works have studied the performance of dynamic clustering in multi-hop data collection. 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. The research work in [18] formulates an optimization problem to assign cluster ranges based on the hop count to the sink. However, to the best of our knowledge, all of the works stated above have not considered the limitation of the transmission range. In [19], the authors argue that the energy efficiency of these works can hardly be achieved with the constraint of network connectivity. Therefore, they propose a hybrid clustering and routing (HCR) protocol that combines the gradient routing 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 hence has poor performance in network lifetime. It motivates us to further exploit the imbalance of energy consumption in HCR and to propose an improved protocol based on HCR in this paper. In this paper, we provide a detailed analysis of the imbalanced energy consumption in HCR via simulations. The simulation results show that there is over \(50\%\) energy left in the network when the first dead node appears. It depicts that the imbalance of energy consumption is severe in HCR. Moreover,we exploit the distribution of the energy consumption in HCR and discover an interesting result: the imbalanced energy consumption generally appears in the nodes that can transmit the data to the sink directly, _i.e._, the nodes in gradient \(k=1\). This result motivates us to improve the performance of HCR by balancing the energy consumption in gradient \(k=1\). An intuitive idea is to select reasonable relays for the nodes with low energy in gradient \(k=1\). However, the realization of this idea is a non-trivial issue that has several challenges as follows:  
1. Identify the node that requires a relay: Intuitively, the node with low energy should select a relay to reduce its energy consumption. However, due to the periodical cluster reorganization, the residual energy of nodes is variable in different rounds. Thus, it is important to design an adaptive scheme to identify if a node requires the relay. 2. Relay selection: For the node deciding to select a relay, how to select the relay with energy efficiency is an important issue.