Algorithm A1 An algorithm for carving a universe U into a set of non-overlapping complexes $\left\{S^{{\left(i\right)}^{*}}\right\}$

$\mathcal{U}=\mathbb{P}\left(U\right)$ $i=1$ while $\mathcal{U}\ne ⌀$do     $\mathbb{S}=\{S^{*}\in \mathcal{U}:\phi \left(S^{*}\right)=\underset{S\in \mathcal{U}}{\max }\phi \left(S\right)\}$     if $\left|\mathbb{S}\right|=1$ then         $S^{{\left(i\right)}^{*}}=\underset{S\in \mathcal{U}}{\mathrm{argmax}}\phi \left(S\right)$         $\mathcal{U}=\mathcal{U}\setminus \{S\in \mathcal{U}:S\cap S^{{\left(i\right)}^{*}}\ne ⌀\}$         $i=i+1$     else         $\mathbb{T}=\{S\in \mathbb{S}:\exists S^{\prime }\in \mathbb{S}\mathrm{with}S\ne S^{\prime }\mathrm{and}S\cap S^{\prime }\ne ⌀\}$         $\mathcal{U}=\mathcal{U}\setminus \mathbb{T}$         $\mathbb{S}=\mathbb{S}\setminus \mathbb{T}$         if $\left|\mathbb{S}\right|>0$ then            for $S\in \mathbb{S}$ do                $S^{{\left(i\right)}^{*}}=S$                $\mathcal{U}=\mathcal{U}\setminus \{S\in \mathcal{U}:S\cap S^{{\left(i\right)}^{*}}\ne ⌀\}$                $i=i+1$            end for         end if     end if end while