Algorithm 1 Solve globally optimal ZOSC with channel gains from finite alphabet $\mathcal{M}$

0. Initialize the marginal distributions ${\mathit{P}}_{X_1}\in {\mathbb{R}}_{+}^{\left|\mathcal{M}\right|\times 1},{\mathit{P}}_{X_2}\in {\mathbb{R}}_{+}^{\left|\mathcal{M}\right|\times 1},{\mathit{P}}_Y\in {\mathbb{R}}_{+}^{\left|\mathcal{M}\right|\times 1}$ and define $\mathit{P}:={[{\mathit{P}}_{X_1}^T,{\mathit{P}}_{X_2}^T,{\mathit{P}}_Y^T]}^T$.

1. Construct the $\left|\mathcal{M}\right|$-ary expansion matrix $\mathit{E}$, where each row of $\mathit{E}$ is a tuple $(x_1,x_2,y)\in {\mathcal{M}}^3$ (the 1st row is (0,0,0) and the proceedings follow an increasing order with respect to the $\left|\mathcal{M}\right|$-ary expansion).

2. Construct $\mathit{A}\in {[0,1]}^{3\left|\mathcal{M}\right|\times {\left|\mathcal{M}\right|}^3}$, where $A_{i,j}=0$, if by the j-th row of $\mathit{E}$, $x_1+x_2\le y$ and $p_j$ is not used in calculating the marginal probability $P_i$, where $P_i$ is the i-row of $\mathit{P}$, $i=1,\cdots ,3\left|\mathcal{M}\right|$, $j={1,\cdots ,\left|\mathcal{M}\right|}^3$. Define $L=3\left|\mathcal{M}\right|$.

3. Reorder $\mathit{A}$ and $\mathit{E}$ as $\tilde{\mathit{A}}$ and $\tilde{\mathit{E}}$, respectively, such that ${\{\frac{1}{2}lng\left(\widetilde{{\mathit{E}}_i}\right)\}}_{i\in \mathcal{L}}$ are in an increasing manner.

repeat

4. Update $\tilde{\mathit{A}}$: set $\tilde{L}-L$ columns of $\tilde{\mathit{A}}$ as a zero vectors, where the indices of those columns correspond to the rows of $\tilde{\mathit{E}}$ having the smallest rates.

5. Solve $\tilde{\mathit{A}}\tilde{\widetilde{\tilde{\mathit{p}}}}=\tilde{\mathit{P}}$, $0<{\widetilde{\tilde{\tilde{p}}}}_i\le 1$, $\sum p_i=1$, $i\in \tilde{\mathcal{L}}$, by CVX.

6. Set L = L + 1.

until$\tilde{\mathit{A}}\tilde{\widetilde{\tilde{\mathit{p}}}}=\tilde{\mathit{P}}$ is feasible,

7. ZOSC = ${\displaystyle \underset{(x_1,x_2,y)\in \mathcal{F}}{min}}\frac{1}{2}[{log}_2(1+x_1+x_2)-{log}_2(1+y)]$, where

$\mathcal{F}:=\left\{{\mathit{E}}_j\right|\mathrm{if}{p}_j>0,j=1\cdots ,\left|\mathcal{M}{|}^3\right\}$.