## 2 Related Work  
### Minimum Scheduling Time for Convergecast  
[PERSON] _et al._[13] proposed a scheduling algorithm for convergecast in WSNs and proved that it requires \(3n\) time slots, where \(n\) is the number of nodes. They demonstrated the validity of their claimsby simulation and showed that the actual time slot needed is \(1.5n\). [PERSON] _et al._[4] developed an optimal algorithm for line and tree topology to get the minimum scheduling time. The authors proved that the time slot needed is \(3(n-2)\), where \(n\) is the number of nodes. They also proved that finding the optimal scheduling time for a general graph topology is NP-hard. [PERSON] _et al._[5] developed a tree-based heuristic algorithm that includes both broadcasting and convergecasting to approach the optimal scheduling time for the convergecast problem. [PERSON] _et al._[14] performed a study to obtain the optimal scheduling and channel assignment and proved that the minimum time slot for convergecast in a wireless HART (Highway Address Remote Transducer Protocol) network is \(2n-1\) and that the minimum number of channels is \(\lceil n/2\rceil\). [PERSON] _et al._[8] performed a study to find the minimum number of time slots in a single frequency channel and combined transmission power and scheduling for convergecast. They showed that controlling the transmission power helps reduce the minimum number of time slots and gave a lower bound by max (\(2n\)-\(1\), \(N\)), where \(n\) is the number of nodes and \(N\) is the number of slots. They also showed that constructing a minimum spanning tree and a degree-constraint spanning tree enhanced the scheduling performance significantly. [PERSON] _et al._[15] constructed a fusion rate-based spanning tree for convergecast. They showed that their proposed method is better than the shortest path spanning tree or minimum spanning tree in any range of fusion rates, where the fusion rate is quantified from zero to one (\(0=\) no fusion; and \(1=\) full fusion). [PERSON] _et al._[1] proved two NP-hard problems: (1) minimizing the scheduling time for a random network with multiple frequencies; and (2) finding the minimum number of frequencies to remove all interference links in a random network. They developed an approximation algorithm to minimize the scheduling time and that gives an upper bound (\(\Delta\) (\(G_{c}\)) + 1), where \(\Delta\) (\(G_{c}\)) is the maximum node degree in constraint graph \(G_{c}\) for the maximum number of frequencies required. [PERSON] and [PERSON] [16] proposed a conflict free convergecast scheduling method and simulated their method with two structures--breadth first search tree and load balanced tree [17]--to show that it reduced the time slots effectively. [PERSON] _et al._[6] showed that the shortest path spanning tree approach to finding the minimum number of time slots is inappropriate. They proposed a new weighted shortest path tree, which is based on a weighted cost function that they suggested, and showed that the proposed method increased the network lifetime to 17% compared to the shortest path spanning tree approach. [PERSON] _et al._[18] studied the scheduling problem for aggregation convergecast. They proposed the balanced shortest path tree (BSPT) algorithm and found the lower bound to the scheduling problem. They showed that their algorithm saved 15% of the scheduling time.