# A Type of Low-Latency Data Gathering Method with Multi-Sink for Sensor Networks
## 4 Data Gathering with the Help of Multi-Sink
### Energy Efficient Path for Data Transmission in DGA  
Compared with the leader selection and active nodes selection strategy mentioned above, finding the optimized paths for data uploading in each DGA is more important for balancing energy consumption and prolonging network lifetime. As shown in Figure 10, the circular region (length  
Figure 9: Sensing radius adjustment.  
of its diameter is just the straight line distance between \(S_{i}\) and the leader) is regarded as \"Region for Next-hop Node Selection\" (\"_NS Region\"_ for short). Therefore, if \(S_{i}\) meets Formula (13), it is obvious that there is no active node in \"_NS Region_\" and \(S_{i}\) transmits its data to the leader by one hop. In Formula (13), \(d(S_{i}\),_leader_) is the diameter length of the \"_NS Region_\" and and \(\rho_{A}\) is just the density of active nodes in this region.  
\[d\left(S_{i},leader\right)<\sqrt{4/\pi\rho_{A}} \tag{13}\]  
Otherwise, when an active node \(S_{i}\) in the \"_NS Region_\" of \(S_{i}\), it is regarded as the candidate next-hop node for \(S_{i}\) if and only if it meets all the following requirements.  
Requirement 1: Residual energy of \(S_{j}\) should be higher than the average value of active nodes in this \"_NS Region_\", just as Formula (14) shows. This ensures that \(S_{j}\) has enough energy to transmit the data to the next receiver. \(E_{r}(S_{j})\) and \(E_{r}(S_{k})\) in Formula (14) are defined as the residual energy of \(S_{j}\) and any active node (e.g., \(S_{k}\)) in this \"_NS Region_\" respectively. \(Num(NS)\) is the number of active nodes in this region and \(\lambda_{4}\) is an adjustable parameter whose value is between 0.5 to 1.  
\[E_{r}\left(S_{j}\right)\geq\frac{\lambda_{4}}{Num\left(NS\right)}\sum_{k=1}^{ Num(NS)}E_{r}\left(S_{k}\right) \tag{14}\]  
Requirement 2: Hop distance between \(S_{j}\) and \(S_{k}\) should be shorter than the length of the diameter, while both \(S_{j}\) and \(S_{k}\) are neighbors of \(S_{i}\) in its \"_NS Region_\". Thus,  
\[d\left(S_{j},S_{k}\right)\leq\lambda_{5}d\left(S_{i},leader\right) \tag{15}\]  
\(\lambda_{5}\) is an adjustable parameter whose value is also between 0.5 to 1.  
Requirement 3: The \"step distance\" form \(S_{j}\) to \(S_{k}\) should be longer than 0. As shown in Figure 10, it is just the projection of segment \(S_{j}S_{k}\) and its length is defined as,  
\[d^{\prime}\left(\overset{\rightarrow}{S_{j}},S_{k}\right)=d\left(S_{j},S_{k} \right)\text{cos}\theta \tag{16}\]  
As for all the candidate next-hop nodes (e.g., \(S_{j}\), \(S_{k}\),...), the weight of each path \(S_{j}S_{k}\) is defined as,  
\[W\left(S_{j},S_{k}\right)=2E_{dec}+\varepsilon_{fs}d^{2}\left(S_{j},S_{k}\right) \tag{17}\]  
\(W(S_{j},S_{k})\) is just the energy consumption on transmitting one bit of data from \(S_{j}\) to \(S_{k}\). Path \(S_{j}S_{k}\) is more likely to be one of the uploading paths from \(S_{i}\) to the leader when this weight is relatively small. Thus, a weighted directed graph could be constructed in the \"_NS Region_\", as shown in Figure 11. And the Single-Source Shortest Path form \(S_{i}\) to the leader could also be found (as the black solid line in Figure 11) with the help of the Dijkstra algorithm. This is the energy efficient data uploading path for \(S_{i}\).  
Figure 10: Next-hop node selection in DGA.