\begin{algorithm}
\caption{List Viterbi Algorithm \cite{seshadri1994list}}\label{alg:cap_lva}
\hspace*{\algorithmicindent} \textbf{Input: $L, \tau, \left\{\mathbf{\pi}_i\right\}_{i=1}^N, \{\mathbf{\Phi}_i \}_{i=1}^N$} \\
\hspace*{\algorithmicindent} \textbf{Output: ${\mathbf{Z}}$} 
\begin{algorithmic}[1]

    \Statex \textbf{\underline{Initialization}}:
    \For{$s \gets 1$ to $2^{\tau}$}
        \State $\Psi(1,s,1) \gets \left\{\mathbf{\pi}_i\right\}_{i=1}^N\left(s\right)$ \Comment{initial probability for each \newline \hspace*{12em} state} \label{alg:lva:line:ini1}
        \State $\xi\left(n,s,l\right) \gets s$ \Comment{previous state of each state}\label{alg:lva:line:ini2}
        \For{$l \gets 2$ to $L$}
            \State $\Psi(1,s,l) \gets 0$
            \State $\xi\left(n,s,l\right) \gets s$ 
        \EndFor
    \EndFor
    
    \Statex \textbf{\underline{Recursion}}:
    \For{$n \gets 2$ to $N$}
    \For{$s_2 \gets 1$ to $2^{\tau}$}
    \For{$l\gets 1$ to $L$}
    \State \Comment{find the probability of the $l$-most likely  \newline \hspace*{6em}  previous state and its rank}

        \State $\Psi(n,s_2,l) = \max^{(l)}_{{\substack{s_1\in\{1,...,2^{\tau}\} \\ k\in\{1,..,L\}}}}
        \newline \hspace*{11.5em} 
                                \left\{ \Psi(n-1,s_1,k) \Phi_n \left( s_1,s_2\right)\right\}$ \label{alg:lva:line:rec1}
        \State $\left({s_1}^*, k^*\right) = \arg\max^{(l)}_{{\substack{s_1\in\{1,...,2^{\tau}\} \\ k\in\{1,..,L\}}}} \newline \hspace*{11.5em} 
                                        \left\{ \Psi(n-1, s_1,k) \Phi_n \left( s_1,s_2\right)\right\}$ 
       \State $\xi\left(n,s_2,l\right) \gets {s_1}^*$ \label{alg:lva:line:rec2}
       \State $\chi\left(n,l\right) \gets k^*$ \label{alg:lva:line:rec3}
    \EndFor
    \EndFor
    \EndFor
    
   \Statex \textbf{\underline{Backtracking}}:
   \For{$l \gets 1$ to $L$}
        \State $\mathbf{Z}_{l,N} \gets \arg\max_{s\in\{1,...,2^{\tau}\}}\left\{ \Psi(N,s,l)\right\}$ \label{alg:lva:line:back1}
        \State $l_N \gets \chi\left(n,l\right)$
       \For{$n \gets N-1$ to $1$}
            \State $\mathbf{Z}_{l,n} \gets \xi\left(n+1,\mathbf{Z}_{l_{n+1},n+1},l_{n+1}\right)$ \label{alg:lva:line:back2}
            \State $l_{n} \gets \chi\left(n+1,l_{n+1}\right) $
       \EndFor
    \EndFor
    \State \Return $\mathbf{Z}$  
\end{algorithmic}
\end{algorithm}

\begin{algorithm}
\caption{List Viterbi Algorithm \cite{seshadri1994list}}\hspace*{\algorithmicindent} \textbf{Input: $L, \tau, \left\{\mathbf{\pi}_i\right\}_{i=1}^N, \{\mathbf{\Phi}_i \}_{i=1}^N$} \\
\hspace*{\algorithmicindent} \textbf{Output: ${\mathbf{Z}}$}
\begin{algorithmic}
[1]
\Statex \textbf{\underline{Initialization}}:
\For{$s \gets 1$ to $2^{\tau}$}
\State $\Psi(1,s,1) \gets \left\{\mathbf{\pi}_i\right\}_{i=1}^N\left(s\right)$ \Comment{initial probability for each \newline \hspace*{12em} state} \State $\xi\left(n,s,l\right) \gets s$ \Comment{previous state of each state} \For{$l \gets 2$ to $L$}
\State $\Psi(1,s,l) \gets 0$
\State $\xi\left(n,s,l\right) \gets s$
\EndFor
\EndFor
\Statex \textbf{\underline{Recursion}}:
\For{$n \gets 2$ to $N$}
\For{$s_2 \gets 1$ to $2^{\tau}$}
\For{$l\gets 1$ to $L$}
\State \Comment{find the probability of the $l$-most likely \newline \hspace*{6em} previous state and its rank}
\State $\Psi(n,s_2,l) = \max^{(l)}_{{\substack{s_1\in\{1,...,2^{\tau}\} \\ k\in\{1,..,L\}}}}
\newline \hspace*{11.5em}
\left\{ \Psi(n-1,s_1,k) \Phi_n \left( s_1,s_2\right)\right\}$ \State $\left({s_1}^*, k^*\right) = \arg\max^{(l)}_{{\substack{s_1\in\{1,...,2^{\tau}\} \\ k\in\{1,..,L\}}}} \newline \hspace*{11.5em}
\left\{ \Psi(n-1, s_1,k) \Phi_n \left( s_1,s_2\right)\right\}$
\State $\xi\left(n,s_2,l\right) \gets {s_1}^*$ \State $\chi\left(n,l\right) \gets k^*$ \EndFor
\EndFor
\EndFor
\Statex \textbf{\underline{Backtracking}}:
\For{$l \gets 1$ to $L$}
\State $\mathbf{Z}_{l,N} \gets \arg\max_{s\in\{1,...,2^{\tau}\}}\left\{ \Psi(N,s,l)\right\}$ \State $l_N \gets \chi\left(n,l\right)$
\For{$n \gets N-1$ to $1$}
\State $\mathbf{Z}_{l,n} \gets \xi\left(n+1,\mathbf{Z}_{l_{n+1},n+1},l_{n+1}\right)$ \State $l_{n} \gets \chi\left(n+1,l_{n+1}\right) $
\EndFor
\EndFor
\State \Return $\mathbf{Z}$
\end{algorithmic}
\end{algorithm}
{
    "title": "multi-stage algorithm for group testing with prior statistics",
    "id": "2402.10018",
    "abstract": "in this paper, we propose an efficient multi-stage algorithm for non-adaptive group testing (gt) with general correlated prior statistics. the proposed solution can be applied to any correlated statistical prior represented in trellis, e.g., finite state machines and markov processes. we introduce a variation of list viterbi algorithm (lva) to enable accurate recovery using much fewer tests than objectives, which efficiently gains from the correlated prior statistics structure. our numerical results demonstrate that the proposed multi-stage gt (msgt) algorithm can obtain the optimal maximum a posteriori (map) performance with feasible complexity in practical regimes, such as with covid-19 and sparse signal recovery applications, and reduce in the scenarios tested the number of pooled tests by at least $25\\%$ compared to existing classical low complexity gt algorithms. moreover, we analytically characterize the complexity of the proposed msgt algorithm that guarantees its efficiency.",
    "categories": "cs.it math.it q-bio.qm stat.ap",
    "doi": "",
    "created": "2024-02-15",
    "updated": "",
    "authors": [
        "ayelet c. portnoy",
        "alejandro cohen"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2402.10018"
}
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