Algorithm 6 SWFDM

Input:Stream $X={\bigcup }_{i=1}^m{X}_i$, distance metric $d(·,·)$, parameter $\epsilon \in (0,1)$, window size $w\in {\mathbb{Z}}^{+}$, size constraints $k_1,\dots ,k_m$ ($k={\sum }_{i=1}^m{k}_i$) Output: A set $S\subseteq W$ s.t. $|S\cap X_i|=k_i$ for $i\in \left[m\right]$ ▹Stream processing 1:$\Lambda ,\mathcal{U}=\{\frac{d_{min}}{{(1-\epsilon )}^j}:j\in {\mathbb{Z}}_0^{+}\wedge {(1-\epsilon )}^j\ge \frac{d_{min}}{d_{max}}\}$ 2:for all $\lambda \in \Lambda ,\mu \in \mathcal{U}$ do 3:    Initialize $A_{\lambda ,\mu },A_{\lambda ,\mu }^{\prime },B_{\lambda ,\mu },B_{\lambda ,\mu }^{\prime }=\varnothing$ 4:    for all $i\in \left[m\right]$ do 5:        Initialize $A_{\lambda ,\mu }^{\left(i\right)},A_{\lambda ,\mu }^{\prime \left(i\right)},B_{\lambda ,\mu }^{\left(i\right)},B_{\lambda ,\mu }^{\prime \left(i\right)}=\varnothing$ 6:for all $x\in X$ do 7:    Run Lines 5–16 of Algorithm 5 to update $A_{\lambda ,\mu }$, $A_{\lambda ,\mu }^{\prime }$, $B_{\lambda ,\mu }$, and $B_{\lambda ,\mu }^{\prime }$ w.r.t. x 8:    if $m=2\wedge c\left(x\right)=i$ and ‘SWFDM1’ is used then 9:        Run Lines 5–16 of Algorithm 5 to update to update $A_{\lambda ,\mu }^{\left(i\right)}$, $A_{\lambda ,\mu }^{\prime \left(i\right)}$, $B_{\lambda ,\mu }^{\left(i\right)}$, and $B_{\lambda ,\mu }^{\prime \left(i\right)}$ w.r.t. x under size constraint $k_i$ 10:    else if $c\left(x\right)=i$ and ‘SWFDM2’ is used then 11:        Run Lines 5–16 of Algorithm 5 to update to update $A_{\lambda ,\mu }^{\left(i\right)}$, $A_{\lambda ,\mu }^{\prime \left(i\right)}$, $B_{\lambda ,\mu }^{\left(i\right)}$, and $B_{\lambda ,\mu }^{\prime \left(i\right)}$ w.r.t. x under size constraint k ▹Post-processing 12:$W\leftarrow \{x\in X:max\{1,\left|X\right|-w+1\}\le t(x)\le |X\left|\right\}$ 13:for all$\lambda \in \Lambda$ and $\mu \in \mathcal{U}$ do 14:    if $A_{\lambda ,\mu }\subseteq W$ then 15:        $S_{\lambda ,\mu }\leftarrow \mathsf{ALG}(k,A_{\lambda ,\mu }\cup B_{\lambda ,\mu })$ 16:    else if $B_{\lambda ,\mu }\subseteq W$ then 17:        $S_{\lambda ,\mu }\leftarrow \mathsf{ALG}(k,(W\cap A_{\lambda ,\mu }^{\prime })\cup B_{\lambda ,\mu })$ 18:    if $m=2$ and ‘SWFDM1’ is used then 19:        if $\left|S_{\lambda ,\mu }\right|=k\wedge |S_{\lambda ,\mu }\cap X_i|<k_i$ then 20:           if $A_{\lambda ,\mu }^{\left(i\right)}\subseteq W$ then 21:               $S_{\lambda ,\mu }^{\left(i\right)}\leftarrow \mathsf{ALG}(k_i,A_{\lambda ,\mu }^{\left(i\right)}\cup B_{\lambda ,\mu }^{\left(i\right)})$ 22:           else if $B_{\lambda ,\mu }^{\left(i\right)}\subseteq W$ then 23:               $S_{\lambda ,\mu }^{\left(i\right)}\leftarrow \mathsf{ALG}(k_i,(W\cap A_{\lambda ,\mu }^{\prime \left(i\right)})\cup B_{\lambda ,\mu }^{\left(i\right)})$ 24:           Run Lines 10–15 of Algorithm 2 using $S_{\lambda ,\mu }$ and $S_{\lambda ,\mu }^{\left(i\right)}$ as input to find a fair solution $S_{\lambda ,\mu }$ 25:    else if ‘SWFDM2’ is used then 26:        for $i\in \left[m\right]$ do 17:           if $A_{\lambda ,\mu }^{\left(i\right)}\subseteq W$ then 28:               $S_{\lambda ,\mu }^{\left(i\right)}\leftarrow \mathsf{ALG}(k,A_{\lambda ,\mu }^{\left(i\right)}\cup B_{\lambda ,\mu }^{\left(i\right)})$ 29:           else if $B_{\lambda ,\mu }^{\left(i\right)}\subseteq W$ then 30:               $S_{\lambda ,\mu }^{\left(i\right)}\leftarrow \mathsf{ALG}(k,(W\cap A_{\lambda ,\mu }^{\prime \left(i\right)})\cup B_{\lambda ,\mu }^{\left(i\right)})$ 31:        Run Lines 10–17 (with $d(x,y)<\frac{\xi \mu }{m+1}$ in Line 13) of Algorithm 3 using $S_{\lambda ,\mu }$ and $S_{all}=$${\bigcup }_{i=1}^m{S}_{\lambda ,\mu }^{\left(i\right)}\cup S_{\lambda ,\mu }$ as input to find a fair solution $S_{\lambda ,\mu }$ 32:return $S\leftarrow {arg\; max}_{\lambda \in \Lambda ,\mu \in \mathcal{U}:\left|S_{\lambda ,\mu }\right|=k}div\left(S_{\lambda ,\mu }\right)$