Nested loop join

A nested loop join is a naive algorithm that joins two relations $R$ and $S$ by making two nested loops:

  For each tuple r in R do
     For each tuple s in S do
        If r and s satisfy the join condition
           Then output the tuple <r,s>

This algorithm will involve n_r*b_s+ b_r block transfers and n_r+b_r seeks. Where b_r and b_s are number of blocks in relations r and s respectively, and n_r is the number of tuples in relation r.

The algorithm runs in $O(|R||S|)$ I/Os, where $|R|$ and $|S|$ is the number of tuples contained in $R$ and $S$ respectively. Can easily be generalized to join any number of relations.

The block nested loop join algorithm is a generalization of the simple nested loops algorithm that takes advantage of additional memory to reduce the number of times that the $S$ relation is scanned.

Improved version

The algorithm can be improved without requesting additional memory blocks to involve only b_r*b_s+ b_r seeks. For each read block from $R$, the relation $S$ can be read only once.

  For each block block_r in R do
     For each tuple s in S do
        For each tuple r in block_r do
           If r and s satisfy the join condition
              Then output the tuple <r,s>>>

Variable block_r is stored in memory, thus it is not needed to read it from disk for each tuple $s$.