Algorithm 2 Greedy refining algorithm

1:$\mathcal{A}\leftarrow \left\{\{1,\dots ,|\mathcal{Y}\left|\right\}\right\}$       // $\mathcal{A}$ starts by being the coarsest partition of $\mathcal{Y}$, i.e., $g_{\mathcal{A}}=Id$. 2:Front $\leftarrow [\mathcal{A},\mathrm{ndef},\dots ,\mathrm{ndef}]$       // Will contain the list of the $\left|\mathcal{Y}\right|$ partitions chosen during the execution 3:  4:for $i\in \{1,\dots ,|\mathcal{Y}|-1\}$ do 5:    // Minimize over $\mathcal{B}$ splitting of $\mathcal{A}$ the slope between $\left(H(g_{\mathcal{A}}(Y)),R^{*}(g_{\mathcal{A}})\right)$ and $\left(H(g_{\mathcal{B}}(Y)),R^{*}(g_{\mathcal{B}})\right)$. 6:    $\mathcal{A}\leftarrow {argmin}_{\mathcal{B}\in Split(\mathcal{A})}\frac{R^{*}(g_{\mathcal{B}})-R^{*}(g_{\mathcal{A}})}{H(g_{\mathcal{B}}(Y))-H(g_{\mathcal{A}}(Y))}$ 7:    Front$\left[i\right]\leftarrow \mathcal{A}$ 8:  9:return Front      // $\mathcal{A}=\left\{\left\{1\right\},\dots \left\{\right|\mathcal{Y}\left|\right\}\right\}$ at this point