## 2 Related Works  
Clustering structures have been proposed to balance energy consumption in WSNs. [PERSON] and [PERSON] [10] proposed an Unequal Clustering Size (UCS) model to balance energy consumption of cluster heads in multi-hop WSNs. This work focuses on a heterogeneous network where cluster heads are deterministically deployed at some pre-computed locations, making it easy to control the cluster size. [PERSON] _et al._[17] proposed an Unequal Cluster-based Routing (UCR) protocol to partition the sensor nodes into clusters with unequal sizes; those clusters closer to the sink node shall have smaller cluster sizes, will consume less energy during the intra-cluster data processing, and preserve more energy for the inter-cluster relay traffic. They also proposed a greedy geographic and energy-aware multi-hop routing protocol for inter-cluster communications. [PERSON] _et al._[5] further investigated the theoretical aspects of the energy hole problem in wireless sensor networks with clustering. They proposed to employ an unequal cluster-radius and alternate between dormancy and work to mitigate the energy hole problem. [PERSON] _et al._[19] discussed the problems related to energy unbalance among sensors in many-to-one sensor networks with uniform node distribution. When all the sensor nodes have fixed transmission ranges, those nodes closer to the sink tend to deplete energy faster. On the other hand, if all the sensors transmit to the sink directly, sensors farther away from the sink will deplete energy faster than those closer to the sink. They thereby proposed a general model to study the optimal transmission range distribution to maximize the network lifetime. They concluded that varying the transmission range among nodes at different distances to the sink can alleviate the energy hole problem only to a limited extent, as energy balance can only be achieved at the expense of using energy resources of some nodes inefficiently. [PERSON] _et al._[20] proposed an improved corona model with levels for analyzing sensors with adjustable transmission ranges in WSNs with circular multi-hop deployment. They considered that the right transmission ranges of sensors in each corona is the decision factor for optimizing the network lifetime after nodes deployment. They also proved that searching optimal transmission ranges of sensors among all coronas is a multi-objective optimization problem, which is NP hard. Therefore, the authors proposed a centralized algorithm and a distributed algorithm for assigning the transmission ranges of sensors in each corona for different node distributions. The two algorithms can not only reduce the searching complexity but also obtain results approximated to the optimal solution. [PERSON] and [PERSON] [11] developed a mathematical model to analyze the energy hole problem in a circular WSN and investigated the effect of several possible schemes that aim to mitigate the energy hole problem, such as deployment assistant, data compression and data aggregation. They assumed that nodes are uniformly and randomly distributed, and each node continuously generates constant bit rate data. Energy lost in data sensing, data transmission and reception is considered. The simulation results confirmed that hierarchical deployment, data aggragation and data compression can alleviate the energy hole problem, while under the same network diameter conditions, higher data rates will worsen the energy hole problem and higher node density cannot prolong the network lifetime. [PERSON] and [PERSON] [12] were the first to study how to avoid the energy hole problem in WSNs. They investigated the theoretical aspects of uneven energy depletion problem in sink-based WSNs with uniform node distribution and constant data reporting. They assumed an energy consumption model governed by \(E=d^{\alpha}+c\), where \(d\) is the transmission range and \(c\) is a positive constant parameter.