Algorithm 2 rpTree(D)

$$\begin{array}{cc}\hfill & 1:\text{Let}D\text{be the root node of tree}t;\hfill \\ \hfill & 2:\text{Initialize the set of working nodes}\mathcal{W}\leftarrow \{D\};\hfill \\ \hfill & 3:\mathbf{while}\mathcal{W}\text{is not empty}\mathbf{do}\hfill \\ \hfill & 4:\text{Randomly pick}W\in \mathcal{W}\text{and set}\mathcal{W}\leftarrow \mathcal{W}-\{W\};\hfill \\ \hfill & 5:\mathbf{if}\mid W\mid <n_s\mathbf{then}\hfill \\ \hfill & 6:\text{Skip to the next round of the while loop};\hfill \\ \hfill & 7:\mathbf{end}\mathbf{if}\hfill \\ \hfill & 8:\text{Generate a random line}\overrightarrow{r};\hfill \\ \hfill & 9:\text{Project points in}W\text{onto}\overrightarrow{r},\text{let}M=\{P_{\overrightarrow{r}}(w):w\in W\};\hfill \\ \hfill & 10:\text{Let}a=\text{min}(M)\text{and}b=\max (M);\hfill \\ \hfill & 11:\text{Generate a splitting point}s\text{uniformly over interval}[a,b];\hfill \\ \hfill & 12:\text{Split node}W\text{by}{W}_L=\{a:P_{\overrightarrow{r}}(a)<s\}\text{and}{W}_R=\{a:\hfill \\ \hfill & P_{\overrightarrow{r}}(a)\ge s\};\hfill \\ \hfill & 13:W.left\leftarrow W_L\text{and}W.right\leftarrow W_R;\hfill \\ \hfill & 14:\text{Update the working set by}\mathcal{W}\leftarrow \mathcal{W}\cup \{W_L,W_R\};\hfill \\ \hfill & 15:\mathbf{end}\mathbf{while}\hfill \\ \hfill & 16:\text{return}(t);\hfill \end{array}$$