Algorithm 1 W-GUN: Whale-centric Optimization for Green Underwater Networks

1.Initialize the population,$M_x\left(x=1,2,\dots ,y\right)$, // $M_x$ represents the set of random solutions based on the size and position of relay nodes 2. Initialize WOA parameters(x, X, G, Q and s) and GWO parameters(C, V and c) 3. Calculate the fitness($FF=max({\mathcal{P}}_n)$) of each search agent using Equation (1) // Fitness represents the maximum number of paths covered with minimal relay nodes 4. Selection based on the adapted search algorithm 5. For WOA-based search 6. Find the best solution $M^{*}$ 7. For GWO-based search 8. Do 9. Separate the solutions based on the fitness 10. $M_{\alpha }$ = the first best search solution 11. $M_{\beta }$ = the second-best search solution 12. $M_{\delta }$ = the third best search solution 13.Compare $M^{*}$ and $M_{\alpha }$ //Select the best solution among $M^{*}$ and $M_{\alpha }$ pass $M\left(final\_best\right)$ to WOA- and GWO-based updating procedures 14.While ($k$ < maximum number of iterations) 15. Do update based on WOA and GWO 16. // WOA-based update 17. For each search agent 18.  Update $\overrightarrow{x},\overrightarrow{X},\overrightarrow{G}$, $Q$ and $s$ 19.  if1($Q<0.5$) 20.   if2($|\overrightarrow{X}|<1$) 21.    Update the position of the current search agent by, 22.    $\overrightarrow{W}=|\overrightarrow{G}\cdot {\overrightarrow{M}}^{*}\left(k\right)-\overrightarrow{M}\left(k\right)|$ 23.    else if2($|\overrightarrow{X}|\ge 1$) 24.    Select a random search agent (${\overrightarrow{M}}^{rand}$) 25.    Update the position of the current search agent by, 26.    $\overrightarrow{M}\left(k+1\right)={\overrightarrow{M}}^{rand}-\overrightarrow{X}\cdot \overrightarrow{W}$ 27.   end if2 28.  else if1($Q\ge 0.5$) 29.  Update the position of the current search by, 30.  $\overrightarrow{M}\left(k+1\right)={\overrightarrow{W}}_{dist}.{exp}^{{log}_{spiral}s}\cdot cos\left(2{\displaystyle \prod }s\right)+{\overrightarrow{M}}^{*}\left(k\right)$ 31.  (where ${\overrightarrow{W}}_{dist}=|{\overrightarrow{M}}_p^{*}\left(k\right)-\overrightarrow{M}\left(k\right)|$ It means the distance between $p^{th}$ whale and prey (i.e., the best solution attained till now); $s\in \left[-1,1\right]$ and $lo{g}_{spiral}$ is the constant for defining the shape of the logarithmic spiral) 32.  end if1 33. end for 34. Return ${\mathit{M}}^{\mathbf{*}}$  //GWO-based update 35. For each search solution 36.  Update the current search agent 37.  $M\left(k+1\right)=\frac{\left(M_1+M_2+M_3\right)}{3}$ 38.  where $M_1=M_{\alpha }\left(k\right)-C_1\cdot P_{\alpha };M_2=M_{\beta }\left(k\right)-C_2\cdot P_{\beta };M_3=M_{\delta }\left(k\right)-C_3\cdot P_{\delta };$            $P_{\alpha }=|V_1\cdot M_{\alpha }-M|;P_{\beta }=|V_2\cdot M_{\beta }-M|;P_{\delta }=|V_3\cdot M_{\delta }-M|;$ 39. End for 40. Check if any search agent goes beyond the search space and adjust it 41. Calculate the fitness of each search agent (i.e., relay set) obtained through both search algorithms 42. Compare and update $M\left(final\_best\right)$ if there is a better solution 43. Store the best solution attained so far 44. $k=k+1$ 45. end while 46. return $M\left(final\_best\right)$ 47. Stop