# Linear hashing

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Linear hashing is a dynamic hash table algorithm invented by Witold Litwin (1980),[1] and later popularized by Paul Larson. Linear hashing allows for the expansion of the hash table one slot at a time. The frequent single slot expansion can very effectively control the length of the collision chain. The cost of hash table expansion is spread out across each hash table insertion operation, as opposed to being incurred all at once.[2] Linear hashing is therefore well suited for interactive applications.

## Algorithm Details

First the initial hash table is set up with some arbitrary initial number of buckets. The following values need to be kept track of:

• $N$: The initial number of buckets.
• $L$: The current level which is an integer that indicates on a logarithmic scale approximately how much the table has grown. This is initially $0$.
• $S$: The step pointer which points to a bucket. It initially points to the first bucket in the table.

Bucket collisions can be handled in a variety of ways but it is typical to have space for two items in each bucket and to add more buckets whenever a bucket overflows. Addresses are calculated in the following way:

• Apply a hash function to the key and call the result $H$.
• If $H \bmod N \times 2^L$ is an address that comes before $S$, the address is $H \bmod N \times 2^{L+1}$.
• If $H \bmod N \times 2^L$ is $S$ or an address that comes after $S$, the address is $H \bmod N \times 2^L$.

• Allocate a new bucket at the end of the table.
• If $S$ points to the $N \times 2^L$th bucket in the table, reset $S$ and increment $L$.
• Otherwise increment $S$.

The effect of all of this is that the table is split into three sections; the section before $S$, the section from $S$ to $N \times 2^L$, and the section after $N \times 2^L$. The first and last sections are stored using $H \bmod N \times 2^{L+1}$ and the middle section is stored using $H \bmod N \times 2^L$. Each time $S$ reaches $N \times 2^L$ the table has doubled in size.