\subsubsection{Index Nested Loops Join (INLJ)}
\begin{algorithm}
    \caption{Nested Loops Join Algorithm}
    \begin{algorithmic}[1]
        \Procedure{IndexNestedLoopsJoin}{$R$, $S$}
            \For{each page $p_R \in R$} \Comment{Outer loop}
                \For{each tuple $t_r \in p_R$}
                    \State Probe the index on the join attribute of $S$ \Comment{Inner loop}
                    \State Add matching tuples to the join output
                \EndFor
            \EndFor
        \EndProcedure
    \end{algorithmic}
\end{algorithm}

The cost here is \cost{$P_R + T_R \cdot C(I_S)$ I/Os}, with $P_X$ the number of pages in the relation $X$, $T_X$ the number of tuples in said relation
and $C(I_X)$ is the cost to probe the index $I_X$ of relation $X$ and depends on the clustering of data and type of index.

This operation needs three buffer frames. $1$ for $R$, $1$ for output and $1$ to traverse $I_S$