## 2 Related Work
### Research on Data Aggregation Routing  
Improving the energy efficiency is a valuable research field [11; 15; 21; 33; 35; 36; 37; 38]. Data aggregation is a method to effectively reduce the data volume in the data transmission process [54; 55] and improve the lifetime of the network. Also, other performances of the network, such as the probability of the collision, can be optimized [56]. Thus, plenty attention and effort have been put into this field [57]. There are different data aggregation methods for different data aggregation ratios. The methods based on the \(\max()\), \(\min()\), \(\mathrm{avg}()\) functions have the highest data aggregation degree. Such methods aggregate \(n\) data packets into 1 data packet, which can be abstracted as a convergecast problem [41; 58]. Convergecast are divided into two phases. In the first phase, what nodes do is to produce data packets. In the second phase, nodes determine whether to send data. If a node sends data, all data packets in the node are aggregated into one data packet. In [41], [PERSON] et al. propose an algorithm and prove the delay of their scheme is at most \(16R+\Delta-14\) time slots. In [58], [PERSON] et al. 's algorithm has an upper bound of \(23R+\Delta-18\) time slots in delay, where \(R\) is the network radius and \(\Delta\) is the maximum node degree. Convergecast refers to a particular case of data aggregation in which each node sends only one data packet during the data transmission process. The networks with an aggregation ratio between 0 to 1 are normal [59], the aggregation conforms to Equation (1):  
\[\mathcal{L}\big{(}d_{i},d_{j}\big{)}=Max\big{(}l(d_{i})+l\big{(}d_{j}\big{)} \big{)}+(1-C_{i,j})*Min\big{(}l(d_{i}),l\big{(}d_{j}\big{)}\big{)} \tag{1}\]  
\(d_{i},d_{j}\) is data packets queue \(i\) and \(j\). \(l(d_{i})\) refers to the length before aggregation. \(C_{i,j}\) is the correlation coefficient between nodes \(d_{i}\) and \(d_{j}\), which is between 0 to 1. The value of \(C_{i,j}\) is inversely proportional to distance of the two nodes [59]. \(\mathcal{L}\big{(}d_{i},d_{j}\big{)}\) is the length of data packets queue after the aggregation of \(d_{i}\) and \(d_{j}\). Unlike Convergecast, the generation of data in this kind of networks is not periodical. The time data packets received by a node is uncertain, this increases the difficulty of designing a data aggregation routing scheme. The nodes in this scenario keep a principle for data aggregation in advance, that is an upper bound of the length of the data queue and the time packets waits in a node. If the length exceeds or the time expires, the node performs aggregation. These two bounds are described as aggregation threshold \(N_{t}\) and aggregation timer \(T_{t}\). The average aggregation delay linearly increases as the aggregation timer or the aggregation threshold increases [42]. In AAR scheme proposed in [40], the waiting time in nodes is significantly reduced and the aggregating and transmitting load are balanced. The optimization strategies above are intended to optimize the routing path, not to optimize data aggregation. Data aggregation is efficient in the case that as many packets as possible perform aggregation. [PERSON] et al. [60] proposed a DRINA strategy which improves the probability of packets routing along the same path. Traditional routing methods (e.g., shortest routing method) makes data packets route along many routing paths scattered in the network, which reduces the number of data packets routing in the same path. DRINA constrains the routing path generation so that the data packets can only reach the sink along specific few paths. The number of data which routes along the same path can be significantly improved, the network delay can be reduced, and the lifetime can also be improved. A modified minimum hop routing strategy is adopted in DRINA. Each node selects the node with the minimum \"hop\" as next hop. In the initialization, \"hop\" is the number of hops to the sink.