%root:main.tex
\begin{algorithm}[t]
	\caption{$\approxq(\circuit, \vct{p}, \conf, \error)$}
	\label{alg:mon-sam}
	\begin{algorithmic}[1]
		\Require \circuit: Circuit
		\Require $\vct{p} = (\prob_1,\ldots, \prob_\numvar)$ $\in [0, 1]^N$
		\Require $\conf$ $\in [0, 1]$
		\Require $\error$ $\in [0, 1]$
		\Ensure \vari{acc} $\in \mathbb{R}$

		\State $\accum \gets 0$\label{alg:mon-sam-global1}
		\State $\numsamp \gets \ceil{\frac{2 \log{\frac{2}{\conf}}}{\error^2}}$\label{alg:mon-sam-global2}
		\State $(\circuit_\vari{mod}, \vari{size}) \gets $ \onepass($\circuit$)\label{alg:mon-sam-onepass}\Comment{$\onepass$ is \Cref{alg:one-pass-iter}}
	
		\For{$\vari{i} \in 1 \text{ to }\numsamp$}\label{alg:sampling-loop}\Comment{Perform the required number of samples}
			\State $(\vari{M}, \vari{sgn}_\vari{i}) \gets  $ \sampmon($\circuit_\vari{mod}$)\label{alg:mon-sam-sample}\Comment{\sampmon is \Cref{alg:sample}. Note that $\vari{sgn}_\vari{i}$ is the \emph{sign} of the monomial's coefficient and \emph{not} the coefficient itself}
			\If{$\vari{M}$ has at most one variable from each block}\label{alg:check-duplicate-block}
				\State $\vari{Y}_\vari{i} \gets \prod_{X_j\in\vari{M}}p_j$\label{alg:mon-sam-assign1}\Comment{\vari{M} is the sampled monomial's set of variables (cref. \cref{subsec:sampmon-remarks})}
				\State $\vari{Y}_\vari{i} \gets \vari{Y}_\vari{i} \times\; \vari{sgn}_\vari{i}$\label{alg:mon-sam-product}
			\State $\accum \gets \accum + \vari{Y}_\vari{i}$\Comment{Store the sum over all samples}\label{alg:mon-sam-add}
			\EndIf
		\EndFor

		\State  $\vari{acc} \gets \vari{acc} \times \frac{\vari{size}}{\numsamp}$\label{alg:mon-sam-global3}
		\State \Return \vari{acc}
	\end{algorithmic}
\end{algorithm}