## 1 Introduction  
Wireless sensor networks (WSNs) are composed of a lot of small, low cost sensor nodes that work together to measure various parameters of the environment and send the data to a unique or several sinks where they will be processed [1]. WSNs have a wide range of uses in military, medical, metropolitan and industrial venues. They are employed in many applications such as security surveillance, battlefield and habitat monitoring, intrusion detection, and target tracking purposes. Although reducing the size of sensors could make them cheaper, this also requires that all hardware equipment, specially the batteries, be extremely small. Since the sensor nodes should be functional for a long period of time and battery replacement in harsh environments like battlefields is usually impossible, nodes may lose their energy very fast, thus becoming nonfunctional in a short time. This situation can negatively affect the whole network connectivity, fault tolerance and lifetime. Therefore, optimization for energy consumption is an important issue, especially to prolong network lifetime in WSNs [2]. To address this problem, a variety of approaches are implemented in the area of routing strategies, which play a key role in network functionality and performance [3]. Routing in wireless sensor networks is very challenging. One of the problems that affect the network lifetime refers to nodes in the vicinity of the sink, whose activity imposes a high traffic on this series of sensor nodes. In this state, the nodes that are closer to the sink lose their energy very fast. These nodes are the neighbors located at one hop away from a single static sink. Not only do they utilize energy to relay the data from any other nodes through the network to the sink, but also for sending their own data. This problem is known as the \"sink neighborhood problem\" [4], which can lead to premature network disconnection. When most of the sink's neighbors' energy is fully depleted, this isolates the sink from the rest of the network, while there is still a huge potential for most of the sensor nodes to continue to perform their tasks and functionalities normally. One of the basic solutions for the sink neighborhood problem is to employ more than one static sink in the network. Using multiple sinks [5, 6, 7] that are statically distributed across the sensor field, it is possible to spread traffic load uniformly among sensor nodes. This can enhance the network lifetime and decrease the end-to-end delays significantly. Another solution for the sink neighborhood problem is to provide some of the network elements with mobile capability [4]. A good strategy to balance energy consumption for data transmission across the network could be replacing the neighbors of the sink. Since nodes' power is limited, a mobilizer unit in mobile nodes consumes the remaining energy faster than under static conditions. The key idea is to maintain the sensors stationary while moving the sink periodically to the parts of the network with sufficient energy. This can prevent network partitioning and consequently prolong the network lifetime. Many protocols [8, 9, 10, 11] are proposed for sink mobility, but they differ from each other in the aspect of mobility itself [4]. For instance, in some applications where the sink goes through the network to collect data by itself, an uncontrolled sink movement pattern is applied to the approaches. This means the network may be unable to control the sink movement by applying a specific trajectory based on the nodes' remaining energy or the amount of traffic at each sensor [4]. On the other hand, controlled sink mobility [10, 11] can efficiently improve the network lifetime without any negative effects on end-to-end delay. Although the sink neighborhood problem is one of the most important reasons for network partitioning, there is another problem that can affect the network lifetime. In fact, using a singleoptimal path [12, 13] for every communication may gradually drain the energy of nodes which are located on the route.