Algorithm 1 Proposed ECTCO algorithm

Input:K,B,${\sigma }^2$,$T_s^{max}$,$G_s$. ${\omega }_k$,$d_k$,$H_k$,$p_k^{tra}$,$p_k^{cir}$,$f_k^l$,$f_k^e$,$f_k^c$,$R_k^{EC}$,$\forall k\in \mathcal{K}$. Output:${\mathit{\gamma }}^{*}$. 1:Initialize: compute $P(k)$ by $G_s$ and initialize all the matrices in Equation (22). 2:solve the SDP problem Equation (22) without the rank-1 constraint to get its optimal solution ${\mathbf{G}}^{*}$. 3:extract the first $3K$ elements from the last column of ${\mathbf{G}}^{*}$ as ${\mathit{\gamma }}^{\prime }$. 4:if$\mathrm{rank}({\mathbf{G}}^{*})=1$then 5:    construct ${\mathit{\gamma }}^{*}$ from ${\mathit{\gamma }}^{\prime }$. 6:else 7:    for $l=1$ to L do 8:        generate random column vector ${\mathit{\xi }}_l\sim U(0,1)$. 9:        obtain random sample ${\mathit{\xi }}_l^{\prime }$ by offloading probability based stochastic mapping method according to Equation (25). 10:        $k=1$; 11:        repeat 12:           if $P(k)=⌀$ then 13:               ${RT}_k=0$; 14:           else 15:               compute $R{T}_k=\underset{j\in P(k)}{\max }{FT}_j$; 16:           end if 17:           compute $F{T}_k$ by ${\mathit{\xi }}_l^{\prime }$ and Equation (8); 18:           $k=k+1$; 19:        until $k=K+1$ 20:        if $F{T}_K>T_s^{max}$ then 21:           discard the random sample ${\mathit{\xi }}_l^{\prime }$. 22:        else 23:           ${\mathit{\xi }}_l^{\prime }$ as feasible solution ${\widehat{\mathit{\xi }}}^{(l)}$,compute the energy cost by ${\widehat{\mathit{\xi }}}^{(l)}$ and Equation (14) as ${\mathbf{E}}^{(l)}$. 24:        end if 25:    end for 26:    choose the solution among ${\widehat{\mathit{\xi }}}^{(1)},\dots ,{\widehat{\mathit{\xi }}}^{(L)}$ that yields the minimum energy cost: ${\mathit{\xi }}^{*}=\underset{{\widehat{\mathit{\xi }}}^{(l)}}{\mathrm{argmin}}{\mathbf{E}}^{(l)}$. 27:    compare ${\mathit{\xi }}^{*}$ with local computing only and cloud computing only solutions, update the one that yields the minimum energy cost as ${\mathit{\gamma }}^{*}$. 28:end if