# 15.1. Introduction¶

## 15.1.1. Introduction¶

Hashing is a method for storing and retrieving records from a database. It lets you insert, delete, and search for records based on a search key value. When properly implemented, these operations can be performed in constant time. In fact, a properly tuned hash system typically looks at only one or two records for each search, insert, or delete operation. This is far better than the \(O(\log n)\) average cost required to do a binary search on a sorted array of \(n\) records, or the \(O(\log n)\) average cost required to do an operation on a binary search tree. However, even though hashing is based on a very simple idea, it is surprisingly difficult to implement properly. Designers need to pay careful attention to all of the details involved with implementing a hash system.

A hash system stores records in an array called a *hash table*,
which we will call `HT`

.
Hashing works by performing a computation on a search key
`K`

in a way that is intended to identify the position in
`HT`

that contains the record with key `K`

.
The function that does this calculation is called the
*hash function*,
and will be denoted by the letter **h**.
Since hashing schemes place records in the table in whatever order
satisfies the needs of the address calculation, records are
not ordered by value.
A position in the hash table is also known as a *slot*.
The number of slots in hash table `HT`

will be denoted by the
variable \(M\) with slots numbered from 0 to \(M-1\).

The goal for a hashing system is to arrange things such that,
for any key value `K`

and some hash function \(h\),
\(i = \mathbf{h}(K)\) is a slot in the table such that
\(0 <= i < M\),
and we have the key of the record stored at
`HT[i]`

equal to `K`

.

Hashing is not good for applications where multiple
records with the same key value are permitted.
Hashing is not a good method for answering range searches.
In other words, we cannot easily find all records (if any) whose key
values fall within a certain range.
Nor can we easily find the record with the minimum or maximum key
value, or visit the records in key order.
Hashing is most appropriate for answering the question, 'What record,
if any, has key value `K`

?'
**For applications where all search is done by exact-match queries,
hashing is the search method of choice because it is extremely
efficient when implemented correctly.**
As this tutorial shows, however, there are many approaches
to hashing and it is easy to devise an inefficient implementation.
Hashing is suitable for both in-memory and disk-based searching and
is one of the two most widely used methods for organizing large
databases stored on disk (the other is the B-tree).

As a simple (though unrealistic) example of hashing,
consider storing \(n\) records, each with a unique key value in
the range 0 to \(n-1\).
A record with key `k`

can be stored in
`HT[k]`

, and so the hash function is
\(\mathbf{h}(k) = k\).
To find the record with key value `k`

, look in
`HT[k]`

.

In most applications, there are many more values in the key range
than there are slots in the hash table.
For a more realistic example, suppose the key can take any value in
the range 0 to 65,535 (i.e., the key is a two-byte unsigned integer),
and that we expect to store approximately 1000 records at any given time.
It is impractical in this situation to use a hash table with
65,536 slots, because then the vast majority of the slots would be
left empty.
Instead, we must devise a hash function that allows us to store the
records in a much smaller table.
Because the key range is larger than the size of the table,
at least some of the slots must be mapped to from multiple key values.
Given a hash function **h** and two keys \(k_1\) and
\(k_2\), if
\(\mathbf{h}(k_1) = \beta = \mathbf{h}(k_2)\)
where \(\beta\) is a slot in
the table, then we say that \(k_1\) and \(k_2\) have a
*collision* at slot \(\beta\) under hash function **h**.

Finding a record with key value `K`

in a database organized by hashing
follows a two-step procedure:

- Compute the table location \(\mathbf{h}(K)\).
- Starting with slot \(\mathbf{h}(K)\), locate the record
containing key
`K`

using (if necessary) a*collision resolution*policy .