implementation-transitive-closure.tex

\begin{figure*}
    \centering
    \begin{subfigure}[b]{0.4\textwidth}
        \centering
        \includegraphics[width=\textwidth]{figs/tree-simple.pdf}
        \caption{We can think of this tree as visualizing a relation~$R$ where
          $(X, Y)~\in~R$ iff there is an edge from~$X$ to~$Y$.}
    \end{subfigure}
    \hspace{5mm}
    \begin{subfigure}[b]{0.4\textwidth}
        \centering
        \includegraphics[width=\textwidth]{figs/tree-transitive.pdf}
        \caption{Transitive closure~$S$ of relation~$R$. Additionally to
          each tuple from~$R$ (solid edges), $S$~also contains additional
          transitive edges (dotted edges).}
    \end{subfigure}
    \caption{Illustrating the idea behind transitive closures. A
      transitive closure~$S$ of relation~$R$ is defined as the
      ``minimal transitive relation that contains~$R$''~\cite{tc}.}\label{fig:tc}
\end{figure*}