## 1 Introduction
Data aggregation can reduce data traffic significantly by removing data redundancy [27]. Since different sensor nodes partially monitor the same spatial region, sensing information is generally correlated [28]. To saving power, data should already be processed as it flows from the information source to the sink [29], which will decrease the accuracy of information around the monitoring area due to some relative packets being removed [30]. Therefore, we should let the sink collect more data only if its energy can support the action [31]. However, more packets indicate that more energy will be depleted for generating and forwarding packets. Hence, data aggregation technology can be utilized to deal with packets compression prudently [32]. More packets collected at the sink means more  
Figure 1: Components of a battery-powered sensor node. accuracy of the information and more energy needed. However, data aggregation causes less packet transmission, energy depletion and accuracy. Energy expenditure and data accuracy will be traded off before utilizing data aggregation for multiple-hop transmission in WSNs and R-WSNs. Topology control is used for energy conservation of nodes or energy utilization reasonably, by shifting the roles of nodes from active to sleeping state and active state [33], which is significantly different to traditional WSNs. In T-WSNs, the topology control technique also is used for handling the limited energy of sensor [34], for example, if a sensor node has not been assigned new tasks, it will shut down the wireless transceiver device and stay in an idle state to save energy [35]. When a task coming, the sensor node wakes up itself and shifts into an active state for dealing with the issues [36]. However, it is inevitable to cause serious data transmission latency due to frequent shifts into the idle state in T-WSNs. In contrast, in R-WSNs, a sensor node does not shift into sleeping state when no task is assigned, but only if it has enough energy to support its operation. Therefore, in R-WSNs, topology control is utilized flexibly compared to the use of energy in T-WSNs. In summary, we propose a maximum data collection rate routing protocol based on data aggregation for packets communications in R-WSNs. To the best of our knowledge, this is the first generic work that studies the use of distribute algorithm for maximizing data collection rate based on data aggregation technology in R-WSNs. In the work, we compute an upper data generation rate by maximizing it as an optimization problem and energy definition as constraint condition for a network, which is formulated as a linear programming problem. Subsequently, a dual problem by introducing Lagrange multipliers is constructed, and subgradient algorithms are used to solve it in a distributed manner. At the same time, a topology controlling scheme is adopted for improving network performance. The remainder of this paper is organized as follows. In Section 2 we present a number of existing routing protocol based on data aggregation, while in Section 3 specify the network model and assumption. In Section 4 we present our method and design. Section 5 contains experimental results. Conclusions are presented in Section 6.