\subsubsection{Partitioned Hash Join (PHJ)}
If the table doesn't fit into memory, then we can't use a normal hash join because that would be terribly inefficient.

Instead, we partition both tables into buckets and join the corresponding partitions.

\begin{algorithm}
    \caption{Partitioned Hash Join}
    \begin{algorithmic}[1]
        \Procedure{PartitionedHashJoin}{$R$, $S$}
            \State Partition $R$ into $RP$ partitions, using hash function $h$ on the join key
            \State Partition $S$ into $SP$ partitions, using hash function $h$ on the join key
            \State Join each partition $RP$ with the corresponding $SP$ partition using BNJL or building a hash table $SP$ or $RP$ in memory.
            This hash function should be different from $h$.
        \EndProcedure
    \end{algorithmic}
\end{algorithm}

The benefit of this approach is that in the third step, we are only operating on small numbers of records each, which can be done efficiently in memory.

What we ideally do in the above algorithm is to do \textit{recursive partitioning}, i.e. as already \textit{somewhat} described, we partition each partition again,
if it has more than one page. This further reduces access times.

The cost for this is \cost{$3 \cdot (P_S + P_R)$}, thus both OSMJ and PHJ are very similar in performance.