## 1 Introduction  
Recently, wireless sensor networks (WSNs) have gained enormous attention for their wide range of applications [1]. WSNs consist of many tiny sensor nodes and one or multiple sinks, sensor nodes gather data from the sensing environment to sinks by communicating with each other. Energy efficiency is the most important issue in WSNs due to the limited battery capacity of sensor nodes. In WSNs with static sinks, the nodes close to the sinks would become hotspots and die earlier than others, because they are more likely to be the intersection of multihop routes and need to deplete their battery supplies to transmit huge amounts of data for other sensor nodes to the sinks [2]. Node death will result in a series of problems, such as disruptions in the topology, reduction of sensing coverage and packets loss, and so on. Therefore, routing protocols designed for WSNs have to incorporate load-balancing in order to achieve uniformity of energy consumption throughout the network. Mobile sinks are a good solution to avoid the hotspots [3; 4]. In mobile based routing protocols, the hotspots around the sink change as the sink moves, which means that each node in the network has the chance to become the neighbor of the sink, so the high energy drainage around the sink is spread through the network, which helps achieving uniform energy consumption, and the network lifetime can be prolonged. However, due to the sink mobility, the paths between source nodes and sinks are frequently updated, which introduces significant energy overhead. Therefore, mobile sink routing protocols should minimize the energy overhead of such operations while avoiding an extremely high increase of the sensor data delivery latency. A lot of distributed mobile sink protocols are proposed. Among these, the hierarchical protocols [2; 5; 6] are the most important and the most widely adopted, which aim to decrease the load of advertising the sink's position to the network by constructing a virtual multitier (two or more) hierarchy role among the nodes. The high-tier nodes that can be easily reachable obtain and store the fresh sink position, while the remaining nodes query them to acquire the sink position whenever needed. Such a scheme does not need the network-wide sink advertisement, and thus significantly decreases the communication overhead and enhances the energy-efficiency of the network. However, the high-tier nodes may become hotspots. Yet the non-hierarchical protocols utilize mechanisms like flooding, overhearing or exploit geometric properties to advertise the sink's position to the source nodes [7; 8]. Thus, they can eliminate the overhead of constructing the virtual hierarchical structure and the possibility of hotspots on such a structure. However, they need the network-wide sink advertisement. The review of prominent hierarchical and non-hierarchical mobile sink routing protocols are presented in [9]. Recently, centralized nature-inspired algorithms have been studied to design routing protocol for MWSNs and have been proven to be good a solutions [10; 11; 12; 13]. PSO is now prevailing due to its simple concept, easy implementation and effectiveness [14]; therefore, in this study, we employ PSO to build the optimal routing tree. In PSO-based protocols, routing establishment is formulated as an optimization problem; mobile sink masters the topology of the entire network by using network-wide flooding and runs various improved PSO to build the optimal routing paths for all source nodes. However, there are some shortages in existing schemes: (1) PSO is originally designed for continuous optimization problem, thus, they cannot deal with the discrete optimization problem as well as continuous optimization problems. Yet the routing optimization problem in MWSNs is discrete, and none of the existing solutions attempt to solve this problem.