## 4 Sub-Optimal Data Gathering  
According to Definition 4, the values of \(a\) and \(b\) are sub-optimized to form a sub-optimal elliptical path of the AUV in order to maximize the total amount of data. As we have two types of nodes/vertices, such as V\({}_{CDS}\) and V\({}_{non-CDS}\)., V\({}_{CDS}\) nodes and E\({}_{CDS}\) edges form the elliptical path of the AUV, whereas, V\({}_{non-CDS}\) node and E\({}_{non-CDS}\) edges are used for data forwarding to the GN. In this case, we divide the total number of hops of all of the network into two categories:  
* CDS hops (h\({}_{CDS}\))
* Non-CDS hops (h\({}_{non-CDS}\))  
In other words, we can say that CDS hops are equal to \(h_{CDS}=N(E_{CDS})\), and non-CDS hops are equal to \(h_{non-CDS}=N(E_{non-CDS})\), where \(N\) represent total number of CDS edges or non-CDS edges. Therefore, according to Definition 4, our objective function to maximize _d\({}_{total}\)_ = \(\sum_{i=1}^{T_{GN}}\left(kr_{i}\text{ + }d_{i}\right)\) for efficient data gathering is transformed into minimizing \(\sum_{i=1}^{non-CDS}hi\), which ultimately depends on the elliptical trajectory of the AUV and also on the values of \(a\) and \(b\). Minimize:  
\[\sum_{i=1}^{non-CDS}h_{i} \tag{12}\]  
Subject to:  
\[\sum_{j=1}^{CDS}h_{j}\ +\sum_{k=1}^{non-CDS}h_{k}\ \leq\sum_{i=1}^{n}h_{i} \tag{13}\]  
In other words, we can say that in order to minimize \(\sum_{i=1}^{non-CDS}h_{i}\), we have to maximize the \(\sum_{i=1}^{CDS}h_{i}\). As we have discussed earlier, that random or elliptical movement of the AUV is basically a combination of all traversed links of the CDS-MST or the number of hops in the CDS-MST. Therefore,  
Figure 7: Impact of \(\beta\). (**a**) The number of dead nodes of the AEDG for different values of \(\beta\); (**b**) the number of dead nodes of the AEDG for different values of \(\beta\) at a specific time instant. the elliptical movement of the AUV is approximately equal to the number of CDS hops traversed by the AUV as given:  
\[P=2\times Number\;\;of\;\;CDS-hops \tag{14}\]  
\[P\cong 2\times\sum_{i=1}^{CDS}h_{i} \tag{15}\]  
where \(P\) is the perimeter of the elliptical path traversed by the AUV. In other words, we can say that the perimeter is sub-optimized to maximize the total number of CDS hopes, which leads to minimized non-CDS hops. When the number of hops between the AUV trajectory and the end node is minimized, the multiple transmission and receptions are avoided in the harsh underwater environment. This leads to efficient data gathering with minimum energy consumption. According to Equation (15), the CDS hops depend on the perimeter, \(P\), which depends on the eccentricity (_e_) of the ellipse, as well as the values of \(a\) and \(b\). Therefore, the sub-optimal range of \(a\), \(b\) and \(e\) leads to the sub-optimized perimeter or elliptical path of the AUV, which further leads to efficient data gathering. Starting from the equation of the perimeter and solving for \(h_{i}\), we reach the following equation (see Appendix C for details). \[ \begin{split}\sum_{i=1}^{CDS-hops}h_{i}&=\left( \frac{\sum_{\forall\;e(i,j)\in EMST}d\left(i,j\right)+\alpha}{\pi}+\frac{\sum_ {\forall\;e(i,j)\in EMST}d\left(i,j\right)+\alpha}{\pi}\;\beta\;\right)\\ &\times\pi[1-\sum_{i=1}^{\infty}\frac{(2i)!^{2}}{(2^{i}.i)!^{4}} \;\frac{(\frac{\sqrt{\beta^{2}+2\beta}}{(1+\beta)})^{2i}}{2i-1}]\end{split}  \tag{16}\]  
After further simplification, we have the following expression (the detailed steps are given in Appendix D). \[\sum_{i=1}^{CDS-hops}h_{i}=b\pi(1+\beta)[1-\sum_{i=1}^{\infty} \frac{(2i)!^{2}}{(2^{i}.i)!^{4}}\;\frac{(\beta^{2}+2\beta)^{i}}{(2i-1)(1+ \beta)}] \tag{17}\] \[\sum_{i=1}^{CDS-hops}h_{i}=[\sum_{\forall\;e(i,j)\in EMST}d\left( i,j\right)+\alpha](1+\beta)[1-\sum_{i=1}^{\infty}\frac{(2i)!^{2}}{(2^{i}.i)!^{4}} \;\frac{(\beta^{2}+2\beta)^{i}}{(2i-1)(1+\beta)}] \tag{18}\]  
From the above equation, we may conclude that \(\sum_{i=1}^{CDS-hops}h_{i}\;\) directly relates to the value of \(\beta\).