Algorithm 1 Search for ${\lambda }^{\mathrm{*}}$ with subsequent conditional independence test
Require: Samples ($\mathbf{X},\mathbf{Y},\mathbf{Z}$), Range ${\left\{{\lambda }_l\right\}}_{l=1}^L$, Significance level $\alpha$, Permutation iterations P, Rejection iterations B
Initalise:  $\mathrm{evaluation}\mathrm{rejection}\mathrm{rate}\left[{\lambda }^{\mathrm{*}}\right]=0$
1:	for $1\le l\le L$ do                              ▹Start: Search for ${\lambda }^{\mathrm{*}}$
2:	for $1\le b\le B$ do
3:	Permute $\mathbf{X}$ by cpt, call them ${\mathbf{X}}_b$
4:	$\mathrm{evaluation}\mathrm{test}\mathrm{statistic}\left[b\right]\leftarrow {\sum }_{z\in \mathbf{Z}}\mathrm{H}\mathrm{S}\mathrm{C}\mathrm{I}\mathrm{C}({\mathbf{X}}_b,\mathbf{Y},\mathbf{Z},{\lambda }_l)$
5:	for $1\le \pi \le P$ do
6:	Permute ${\mathbf{X}}_b$ by cpt, call them ${\mathbf{X}}_{b,\pi }$
7:	$\mathrm{evaluation}\mathrm{null}\mathrm{statistics}[b,\pi ]\leftarrow {\sum }_{z\in \mathbf{Z}}\mathrm{H}\mathrm{S}\mathrm{C}\mathrm{I}\mathrm{C}({\mathbf{X}}_{b,\pi },\mathbf{Y},\mathbf{Z},{\lambda }_l)$
8:	$\mathrm{evaluation}p-\mathrm{value}\leftarrow 1-$
9:	$\mathrm{percentile}\left(\mathrm{evaluation}\mathrm{test}\mathrm{statistic}\right[b],\mathrm{evaluation}\mathrm{null}\mathrm{statistics}[b,:\left]\right)$
10:	if $\mathrm{evaluation}p-\mathrm{value}\ge \alpha$ then
11:	Fail to reject $H_0:{\mathbf{X}}_b⫫\mathbf{Y}|\mathbf{Z}$
12:	rejects$\left[b\right]=0$
13:	else
14:	Reject $H_0:{\mathbf{X}}_b$ $\mathbf{Y}|\mathbf{Z}$
15:	$\mathrm{rejects}\left[b\right]=1$
16:	$\mathrm{evaluation}\mathrm{rejection}\mathrm{rate}\left[{\lambda }_l\right]\leftarrow {\mathrm{mean}}_{1\le b\le B}\left(\mathrm{rejects}\right)$
17:	if $|\mathrm{evaluation}\mathrm{rejection}\mathrm{rate}\left[{\lambda }^{\mathrm{*}}\right]-\alpha |\ge |\mathrm{evaluation}\mathrm{rejection}\mathrm{rate}\left[{\lambda }_l\right]-\alpha |$ then
18:	${\lambda }^{\mathrm{*}}\leftarrow {\lambda }_l$
19:	return ${\lambda }^{\mathrm{*}}$                               ▹End: Search for ${\lambda }^{\mathrm{*}}$
20:	$\mathrm{test}\mathrm{statistic}\leftarrow {\sum }_{z\in \mathbf{Z}}\mathrm{H}\mathrm{S}\mathrm{C}\mathrm{I}\mathrm{C}(\mathbf{X},\mathbf{Y},\mathbf{Z},{\lambda }^{\mathrm{*}})$                ▹Start: Conditional independence test
21:	for  $1\le p^{\prime }\le P$  do
22:	Permute $\mathbf{X}$ by cpt, call them ${\mathbf{X}}_{p^{\prime }}$
23:	$\mathrm{null}\mathrm{statistics}\left[p^{\prime }\right]\leftarrow {\sum }_{z\in \mathbf{Z}}\mathrm{H}\mathrm{S}\mathrm{C}\mathrm{I}\mathrm{C}({\mathbf{X}}_{p^{\prime }},\mathbf{Y},\mathbf{Z},{\lambda }^{\mathrm{*}})$
24:	$p-\mathrm{value}\leftarrow 1-\mathrm{percentile}(\mathrm{test}\mathrm{statistic},\mathrm{null}\mathrm{statistics})$
25:	if  $p-\mathrm{value} \ge \alpha$  then
26:	Fail to reject $H_0:X⫫Y|Z$
27:	else
28:	Reject $H_0:X$ $Y|Z$
	▹End: Conditional independence test