# Nested loop join

A nested loop join is a naive algorithm that joins two sets by using two nested loops. Join operations are important to database management.

## Algorithm

Two relations $R$ and $S$ are joined as follows:

  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 nr*bs+ br block transfers and nr+br seeks, where br and bs are number of blocks in relations R and S respectively, and nr 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 br*bs+ br block transfers. 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 $s$.