Block nested loop

A block-nested loop (BNL) is an algorithm used to join two relations in a relational database.[1]

This algorithm[2] is a variation of the simple nested loop join and joins two relations ${\displaystyle R}$ and ${\displaystyle S}$ (the "outer" and "inner" join operands, respectively). Suppose ${\displaystyle |R|<|S|}$. In a traditional nested loop join, ${\displaystyle S}$ will be scanned once for every tuple of ${\displaystyle R}$. If there are many qualifying ${\displaystyle R}$ tuples, and particularly if there is no applicable index for the join key on ${\displaystyle S}$, this operation will be very expensive.

The block nested loop join algorithm improves on the simple nested loop join by only scanning ${\displaystyle S}$ once for every group of ${\displaystyle R}$ tuples. Here groups are disjoint sets of tuples in ${\displaystyle R}$ and the union of all groups has the same tuples as ${\displaystyle R}$. For example, one variant of the block nested loop join reads an entire page of ${\displaystyle R}$ tuples into memory and loads them into a hash table. It then scans ${\displaystyle S}$, and probes the hash table to find ${\displaystyle S}$ tuples that match any of the tuples in the current page of ${\displaystyle R}$. This reduces the number of scans of ${\displaystyle S}$ that are necessary.

algorithm block_nested_loop_join is
    for each page pr in R do
        for each page ps in S do
            for each tuple r in pr do
                for each tuple s in ps do
                    if r and s satisfy the join condition then
                        yield tuple <r,s>

A more aggressive variant of this algorithm loads as many pages of ${\displaystyle R}$ as can be fit in the available memory, loading all such tuples into a hash table, and then repeatedly scans ${\displaystyle S}$. This further reduces the number of scans of ${\displaystyle S}$ that are necessary. In fact, this algorithm is essentially a special-case of the classic hash join algorithm.

The block nested loop runs in ${\displaystyle O(P_{r}P_{s}/M)}$ I/Os where ${\displaystyle M}$ is the number of available pages of internal memory and ${\displaystyle P_{r}}$ and ${\displaystyle P_{s}}$ is size of ${\displaystyle R}$ and ${\displaystyle S}$ respectively in pages. Note that block nested loop runs in ${\displaystyle O(P_{r}+P_{s})}$ I/Os if ${\displaystyle R}$ fits in the available internal memory.