# Integer Compression

## It never occurred to me that there would be a way to compress an integer...

I recently learned about three methods of encoding integer values:

This sort of blew my mind because I never thought about trying to take a basic, primitive type (such as an integer) and store it using *less* disk space.

In a simple example, let’s assume a basic signed integer in C which takes 16 bits. If we were to save the number 3, we would be storing the following bit sequence:

`00000000 00000011`

BUT, we can apply one of the methods above and actually use less than 16 bits. Let’s take a look at each method.

*(It’s important to note that in all scenarios in this post we will be using log base 2)*

## Unary encoding

Unary encoding is by far the simplest. For an integer \(x\) where \(x \geq 1\), the unary encoding is \(x-1\) 1s, followed by a 0.

For example, the number 3 will be two 1s, followed by a 0:

`110`

And the number 9 will be eight 1s, followed by a 0:

`111111110`

For small numbers this is great - we’ve saved 13 bits and 7 bits respectively in these examples. However, you’ll likely notice that as the number grows larger, we end up taking *more* space than we would have by just storing the integer itself.

## γ-encoding

For γ-encoding we need to start doing some math.

We start by finding the unary code for \(1+\lfloor log x \rfloor\). So, for the number 3 we end up with:

\(1+\lfloor log 3 \rfloor = 2\), which in unary code becomes:

`10`

Next, we find the uniform (binary) code for \(x-2^{\lfloor log x \rfloor}\), but represented in \(\lfloor log x \rfloor\) bits:

\(3-2^{\lfloor log 3 \rfloor} = 3-2^1 = 1\), which when represented in \(\lfloor log 3 \rfloor = 1\) bit is:

`1`

Piecing the two parts together (unary part first, then the uniform), we get:

`101`

Definitely a savings compared to storing the raw value of 3 in 16 bits, but there is no gain over the unary encoding. What about the number 9?

\(1+\lfloor log 9 \rfloor = 4\), which in unary code becomes:

`1110`

\(9-2^{\lfloor log 9 \rfloor} = 9-2^3 = 1\), represented in \(\lfloor log 9 \rfloor = 3\) bits is:

`001`

Piecing the two parts together (unary part first, then the uniform), we get:

`1110001`

A 2-bit savings over the unary encoding! As the numbers grow, so do the savings.

For example, 15 in unary-encoding:

`111111111111110`

…and in γ-encoding:

`1110111`

## δ-encoding

The δ encoding is more complicated. It is nearly identical to γ-encoding, however instead of starting with the unary encoding of \(1+\lfloor log x \rfloor\), we start with the *γ-encoding* of \(1+\lfloor log x \rfloor\). This can get tricky to keep track of.

Let’s use the number 3 again for our first example. We start with the γ-encoding of \(1+\lfloor log x \rfloor\):

1+⌊log 3⌋ = 1+1 = 2, so we find the γ-encoding of the number 2. Following the steps in the section above, you will see that the γ-encoding is:

`100`

The rest is identical to the γ-encoding. We find the uniform (binary) code for \(x-2^{\lfloor log x \rfloor}\), but represented in \(\lfloor log x \rfloor\) bits:

\(3-2^{\lfloor log 3 \rfloor} = 1\), represented in \(\lfloor log 3\rfloor = 1\) bit is simply:

`1`

Piecing the two parts together (γ-encoding first, then the uniform):

`1001`

Yikes! This took 4 bits instead of the 3 bits used by the γ-encoding! What about a larger number like 125?

We start with the γ-encoding of \(1 + \lfloor log x \rfloor\):

\(1 + \lfloor log 125 \rfloor = 7\), represented in γ-encoding is:

`11011`

Next we find the uniform (binary) code for \(x - 2^{\lfloor log x \rfloor}\), but represented in \(\lfloor log x \rfloor\) bits:

\(125-2^{\lfloor log 125 \rfloor} = 125 - 2^6 = 61\), represented in binary in \(\lfloor log 125 \rfloor = 6\) bits is:

`111101`

Piecing the two parts together (γ-encoding first, then the uniform):

`11011111101`

On the other hand, the γ-encoding for 125 is:

`1111110111101`

A 2-bit savings!

## Summary

Long story short, I though these were some pretty cool methods for encoding integers. I won’t write about it here, but you can just as simply decode these bit strings. Actually, I won’t lie, it’s not quite as simple.

Here is a link to a GitHub repo where I’ve put together a couple Python scripts to perform both the compression and decompression techniques: