\newpage
\subsection{Concurrency Control}
The goal of concurrency control is to ensure the correct executions of concurrent transactions.
The baseline for correctness is, as previously mentioned, serial execution.

\inlinedefinition[Serial History] A history $H$ is serial if, for every pair of two transactions $T_i$ and $T_j$ that appear in $H$,
either all operations from $T_i$ appear before all all operations of $T_j$, or vice versa.
A serial history with only committed transactions is correct by definition, because transactions are isolated and each transaction starts in a consistent state
and leaves the DB in a consistent state.

\inlinedefinition[Equivalent History] Two histories are equivalent if and only if
\begin{enumerate}
    \item They are over the same transactions and contain the same operations
    \item Conflicting operations of non-aborted transactions are ordered in the same way in both histories
\end{enumerate}

\inlinedefinition[Serializable History] A history is serializable if and only it is equivalent to a serial history.

\inlinetheorem[Serializability] A history is serializable if and only if its serializability graph is acyclic.
The seerializabililty graph is constructed from the history graph as follows:
\begin{enumerate}
    \item The start node is the transaction the first operation.
    \item Then, follow the operations, adding any new transaction to the tree and connect any further dependencies.
\end{enumerate}
More visually, see the below graphics

\begin{figure}[h!]
    \begin{center}
        \begin{subfigure}{0.49\textwidth}
            \includegraphics[width=0.9\textwidth]{assets/serializability-graph-history.png}
            \caption{The history}
        \end{subfigure}
        \begin{subfigure}{0.49\textwidth}
            \includegraphics[width=0.7\textwidth]{assets/serializability-graph-built.png}
            \caption{The resulting serializability graph}
        \end{subfigure}
    \end{center}
    \caption{Building a serializability graph (Figure from lecture slides 19, slide 37 and 38)}
\end{figure}