Common join algorithms
Working in the Azure Disaster Recovery (ASR) team for over 6 months now, I have been using Kusto (a log analytics platform developed at Microsoft) extensively for interactive analysis and monitoring of internal service components and flows.
Kusto is modeled in a typical RDBMS fashion and it supports complex analytical queries over stored entities. The ingestion and querying performance is topnotch, which comes at the cost of sacrificing the ability to carry out inplace updates of individual rows. In the Kusto documentation the best practices have mentioned that when doing joins (synonymous to joins in SQL terminology), always keep the computationally heavy part on the right side of the join for better performance. I used to wonder why this is recommended until I read about the most common join algorithms. Perhaps reading about how joins are implemented will help me write more efficient queries? This is a post about some very primitive join algorithms.
Notation
 : Number of records in relation X
 : A block (partition) in relation X
 : The number of blocks (partitions) of X
 : Number of records in a block of relation X
Nested loop algorithm
This is the trivial join algorithm with two nested loops. A brute force on all rowrow combinations of both sides. For joining two relations X and Y, it runs in operations.
The following would be a crude pseudocode for this.
Hash join algorithm
In this algorithm one of the tables is loaded into memory and hashed on the joining key. Then while scanning the second table, the hashes are matched to verify the join condition. To judge if this is a better algorithm we need to consider all pros and cons of the algorithm. First let us look at the pseudocode. In the example below, an inner join is performed. The primary thing to consider is that the hash function has the join attributes as keys and the entire row as the value.
This algorithm hence consists of two “phases”
 Build phase  where we build the hash table from relation X
 Probe phase  where we scan (probe) the relation Y to match hashes
The build phase runs in and the probe phase runs in because hash table lookup is .
Overall complexity: which is linear and much better than the quadratic nested loop.
As you might have guessed, two limitations immediately pop up when considering the hash join
 What if during the build phase, the relation (table) does not fit into available memory?
 What about nonequality conditions? Comparing hashes would work only for equijoins and not for any generic join conditions.
Dealing with the limitations

Memory constraint
If the whole relation does not fit into memory, then one way is to partition the relation into blocks of size that fit in memory, hash each block and then probe the other relation for each block of the first relation.
For joining X and Y, if we partition X into blocks then the time taken for each block to be joined with relation Y is , similar to the classical hashjoin above. Overall for all blocks this will take which is still better than the nested loop.

Equijoin constraint
We cannot use hash join with a nonequality condition (because hashing). This remains a limitation of the algorithm.
Sortmerge algorithm
The hashjoin does not work for conditions other than equality, that’s where sortmerge algorithm hops in. This is the most commonly used algorithm in most RDBMS implementations. The special idea here is to first sort both the relations (tables) by the join attribute so that a linear scan with two probes (one for each relation) will be able to deal with both relations at the same time. Therefore, practically the costliest part of this algorithm is sorting the inputs. Sorting can be done in 2 ways
 Explicit external sort.
 Exploit a preexisting ordering in the join relations. For instance if the join input is produced by an index scan* then we already have that relation ordered.
Therefore, for two relations X and Y, if X fits in memory pages and Y fits in memory pages, then the worst case running time would be .
There are numerous other join algorithms that leverage the ideas in the above mentioned basic algorithms, for instance the hybrid hashjoin partitions each relation using a hash function for saving probe time on the second relation when performing the actual join. Knowing how joins are implemented in the DBMS being used, one might therefore write more efficient queries.
* Recommended reading: B+ tree, Bx tree