People is often said that indexing is a go-to technique to process efficiently queries in case your database is large enough. This post is for summarizing what database index is and revisiting hash and B+Tree.

Index is a data structure that organizes records to optimize certain kinds of retrieval operations. We may create index on a field of the table then retrieve all records that satisfy search conditions on `search-key`

field. Without index, our query would end up scanning linearly the entire content of the table to fetch only one or a few records.

In this post, I'd like to summarize the performance and use cases of two common indexing techniques: **Hash index** and **B+tree**

## Hash index

This technique is widely used for creating indices in *main memory* because its fast retrieval by nature. It has average O(1) operation complexity and O(n) storage complexity.

In many books, people use the term `bucket`

to denote a unit of storage that stores one or more records

There are two things to discuss when it comes to hashing:

- Hash function: maps search keys (as its input) to an integer representing that key in the bucket.
- Hashing scheme: how to deal with key collision after hashing.

Some people ask: why collision ? Does a perfect hash function ever exist ? In fact, let's say your keys is an infinite set, it's impossible to map them into a set of 32-bit integers without having no collision. There should be a trade-off between computation and collision rate.

There are a few hashing scheme worth mentioning: linear probing, chained hashing and extendible hashing. Lookup/insert/delete algorithms vary by hashing scheme, for example, chained hashing deal with key collisions by placing elements have the same hash value in the same bucket.

### Pros

- Hash index is suitable for equality or primary key lookup. Queries can benefit from hash index to get amortized O(1) lookup cost. For example:
`SELECT name, id FROM student WHERE id = '1315';`

### Cons

Hash table has some limitations:

- Range queries are not efficient. Hash table is based on uniform distribution. In other words, you have no control of where an index entry is going to be placed.
- Low scalability: performance of lookup operation can degrade when there a lot of collisions and it requires to resize the hash table then rehash existing index entries.

## B+Tree

This is a self-balancing tree data structure that keeps data in sorted order and allows fast search within each node, typically using binary search.

B+Tree is a standard index implementation in almost all relational database system.

B+Tree is basically a M-way search tree that have the following structure:

- perfectly balance: leaf nodes always have the same height.
- every inner node other than the root is at least half full (M/2 β 1 <= num of keys <= M β 1).
- every inner node with k keys has k+1 non-null children.

Every node of the tree has an array of sorted key-value pairs. The key-value pair is constructed from (search-key value, pointer) for root and inner nodes. Leaf node values can be 2 possibilities:

- the actual record
- the pointer to actual record

###
Lookup a value *v*

- Start with root node
- While node is not a leaf node, we do:
- Find the smallest Ki where Ki >= v
- If Ki == v: set current node to the node pointed by Pi+1
- Otherwise, set current node to node pointed by Pi

### Duplicate keys

In general, search-key can be duplicate, to solve this, most database implementations come up with composite search key. For example, we want to create an index on `student_name`

then our composite search key should be (student_name, Ap) where Ap is the primary key of the table.

### Pros

There're two major features that B+tree offers:

- Minimizing I/O operations
- Reduced height: B+Tree has quite large branching factor (value between 50 and 2000 often used) which makes the tree fat and short. The figure below illustrates a B+Tree with height of 2. As we can see nodes are spread out, it takes fewer nodes to traverse down to a leaf. The cost of looking up a single value is the height of the tree + 1 for the random access to the table.

- Scalability:
- You have predictable performance for all cases, O(log(n)) in particular. For databases, it is usually more important than having better best or average case performance.
- The tree always remain balanced by its implementation. A B+Tree with n keys always has a depth of O(log(n)). Thus, the performance will not degrade if the database grows bigger. A four-level tree with a branching factor of 500 can store up to 256 TB provided that a page is size of 4KB.

- B+Tree is most suited for range queries, for example
`"SELECT * FROM student WHERE age > 20 AND age < 22"`

## Conclusion

Although hash index performs better in terms of exact match queries, B+Tree is arguably the most widely used index structure in RDBMS thanks to its consistent performance in overall and high scalability.

B+Tree | Hash | |
---|---|---|

Lookup Time | O(log(n)) | O(log(1)) |

Insertion Time | O(log(n)) | O(log(1)) |

Deletion Time | O(log(n)) | O(log(1)) |

Recently, the log-structured merge tree (LSM-tree) has attracted significant interest as a contender to B+-tree, because its data structure could enable better storage space usage efficiency. I'll investigate it further and make a post about it in the near future.

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