## 5 The Tri-Angular Nearest Vector-Based Energy Efficient Routing (TANVEER)
### Empty Regions Selection  
An optimal way to select the empty regions is based upon the number of resources used in the acoustic environment. The energy is one of the most fundamental resources where nodes are enabled to join the network and process the data communication towards the sonobuoys [47; 48; 49]. The section of empty regions '\(Ser\)' is directly based upon the node's energy '\(En\)', where '\(i\)' is the total number of empty regions selected (i varies from 0 to \(N\); where '\(N\)' is the total number of nodes inside the empty regions) every node, before forwarding the data packets computes angle three times for three directions like (horizontal '\(x\)', vertical '\(y\)', and diagonal '\(z\)'). Multiple neighbor's nodes with respect to angle are also selected when there is no empty region throughout the routing path from the anchored node to sonobuoys. Additionally, the related mathematical equation is used to selection of total number of empty regions as in Equation (1):  
\[S_{er}=(En/(Ni\;*\;A\;(x,\;y,\;z)+1)) \tag{1}\]  
This equation boosts the concept of smooth delivery of data (when there is no void nodes [50; 51; 52]) and trace out the empty region on the routing path. It fits for medium and large-scale networks because when the number of nodes increases, the ratio of occurrence of empty regions '\(Er\)' also increases [53; 54]. Selection of the multiple number of nodes increased the data packet ratio and throughput of the network. Another factor that needs to be discussed is the packet acknowledgement (\(P_{ACK}\)) that helps to select the next layer and node, once the layer will be selected, that is obviously select by \(P_{ACK}\), it is easily select the next data forwarder node. This selection leads the routing path through bypassing the empty regions [40] until the data are received to sonobuoys. Basically, the \(P_{ACK}\) is defined as and is the confirmation of the receiving of beacon or hello packet message. It is the practice of acoustic communication before sending the data to check which nodes are alive/part of the network [55]. If the packet is sent by the node of the first layer (\(N\) (\(l_{1}\)) to the node of the second layer (\(N\) (\(l_{2}\)) with the difference between layer (\(D\)), it first computes the angle with all coordinates to create an existing routing path between these layers with a possible direction. The selection of angle direction would be considered, where there is no '\(Er\)' throughout the path. If the consecutive layers are selected for forwarding the data, all layers are responsible for providing their number of nodes to make the route robust. For example, the path was found to be smooth from anchored node to the first layer, and '\(Er\)' occurs in the sink neighboring nodes area. The corresponding angle changes the direction of the vector for the flooding of data. The route vector can change at any stage by avoiding the '\(Er\)' occurrence, as this is only done to handle the exceptional case. The corresponding mathematical Equation (2) below describes the selection of layers and nodes and routing path with angle:  
\[Ni=\{\;(N\;(l_{1})\!(N\;(l_{2})\in P\_ACK\;:\;\exists\;N\;(Ax,\; Ay,\;Az)\to Sn \tag{2}\]  
where '\(Sn\)' is the closest optimized sink in which the total number of nodes with the difference of empty regions that selects for data delivery as in Equation (3):  
\[Sn*=opt\;\sum_{i}^{m}Min\{\;Tn-Er\} \tag{3}\]