Algorithm 3 Function pb_ch_pref

Require:${\tilde{V}}_i^n$, $R_i$, ${\Lambda }_{i-1}^{(X)}$

1: ${\tilde{T}}_i\left[{\mathcal{H}}_{X|VZ}^{(n)}\right]\leftarrow {\Lambda }_{i-1}^{(X)}$  2: ${\tilde{T}}_i\left[{\mathcal{H}}_{X|V}^{(n)}\setminus {\mathcal{H}}_{X|VZ}^{(n)}\right]\leftarrow R_i$  3: for $j\in {\left({\mathcal{H}}_{X|V}^{(n)}\right)}^{\mathrm{C}}$ do  4:    if $j\in {\left({\mathcal{H}}_{X|V}^{(n)}\right)}^{\mathrm{C}}\setminus {\mathcal{L}}_{X|V}^{(n)}$ then  5:        ${\tilde{T}}_i(j)\leftarrow p_{T(j)|T^{1:j-1}{V}^n}\left({\tilde{T}}_i(j)|{\tilde{T}}_i^{1:j-1}{\tilde{V}}_i^n\right)$  6:    else if $j\in {\mathcal{L}}_{X|V}^{(n)}$ then  7:        ${\tilde{T}}_i(j)\leftarrow {arg\; max}_{t\in \mathcal{X}}{p}_{T(j)|T^{1:j-1}{V}^n}\left(t|{\tilde{T}}_i^{1:j-1}{\tilde{V}}_i^n\right)$  8:    end if  9: end for 10: ${\tilde{X}}_i^n\leftarrow {\tilde{T}}_i^n{G}_n$ 11: ${\Lambda }_i^{(X)}\leftarrow {\tilde{T}}_i\left[{\mathcal{H}}_{X|V}^{(n)}\setminus {\mathcal{H}}_{X|VZ}^{(n)}\right]$ 12:return${\tilde{X}}_i^n$ and ${\Lambda }_i^{(X)}$