\begin{frame}
  \frametitle{Finding all Minimal Keys}

  \begin{code}{\textwidth}{\small Finding \emph{all} Minimal Keys}
    \small
    $
    \begin{array}{ll}
      \emph{Input:}  & A_1,A_2,\ldots,A_n \text{ (all attributes of $R$)} \text{ and } \mathcal{F} \text{ (set of FDs)} \\
      \emph{Output:} & \textit{Results} \text{ (the set of all minimal keys of $R$)} \\
    \end{array}%
    $\vspace{-1ex}
    \begin{algorithmic}
      \STATE $\textit{Results} = \varnothing$;
      \STATE $\textit{Candidates} = \{\; \{\, A_i \mid \text{$A_i$ is not part of any right-hand side in $\mathcal{F}$} \,\} \;\}$;
      \WHILE{$\textit{Candidates} \ne \varnothing$}
        \STATE choose and remove a smallest $\kappa \in \textit{Candidates}$;
        \IF{$\kappa$ contains no key in \textit{Results}}
          \IF{$\kappa^+_{\mathcal{F}} = \{\, A_1,A_2,\ldots,A_n \,\}$}
            \STATE $\textit{Results} = \textit{Results} \cup \{\kappa\}$;
          \ELSE
            \FORALL{$A_i \not\in \kappa^+_{\mathcal{F}}$} 
              \STATE $\kappa_i = \kappa \cup \{\,A_i\,\}$;
              \STATE $\textit{Candidates} = \textit{Candidates} \cup \{\,\kappa_i\,\}$;
            \ENDFOR
          \ENDIF
        \ENDIF
      \ENDWHILE
      \STATE \textbf{return} $\textit{Results}$;
    \end{algorithmic}
  \end{code}
\end{frame}