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2021-03-08
Many binary file formats need to store integer values. And often, those integer values are typically “small”. That is, while the field might technically be a 32-bit integer, that field will rarely hold values from that entire 32-bit range. Instead, values will usually be small in magnitude, and a large number of their higher-order bits will be 0. In those cases, it can be useful to try to save space by encoding the integer value in a smaller number of bytes. Doing so can substantially reduce the size of your file, especially when there are lots of these small-magnitude integers to store.
(Note that even though the goal, and end result, is that the file is smaller, this is not a _compression_ scheme, since it only works when your values are distributed in one particular way. Proper compression schemes are more sophisticated, so that they can find redundancies in many other patterns of data.)
With this goal, our job is to figure out a coding scheme that lets us:
This is not a new problem, and many solutions have existed for quite some time. The most popular these days is usually called “varint”, after its name in Google’s Protocol Buffers spec. This same encoding is sometimes also called LEB128, and its big-endian equivalent is sometimes called VLQ. For the rest of this post, I’m going to call this encoding scheme “metric varint”.
In short, metric varint uses the most-significant bit in each byte as a “continuation marker”. The lower 7 bits in each byte encode some of the bits of the integer value being encoded. A value of ‘1’ for the continuation bit means “more bytes to follow”; a value of ‘0’ means “this is the last byte for this value”.
To encode, you first encode the integer into binary, making sure to use a multiple of 7 bits. You then attach a ‘0’ bit to the beginning of the first (most-significant) 7-bit chunk, and a ‘1’ bit to the beginning of every other chunk. You then reverse the chunks (because the chunks are written in little-endian order), and write the sequence of chunks out to your file. Some examples:
0 ⇒ 0000000 (encode in binary)
⇒ 0_0000000 (prepend continuation markers)
⇒ 0_0000000 (reverse chunks)
⇒ 0x00
127 ⇒ 1111111
⇒ 0_1111111
⇒ 0_1111111
⇒ 0x7f
128 ⇒ 0000001 0000000
⇒ 0_0000001 1_0000000
⇒ 1_0000000 0_0000001
⇒ 0x80 0x01
50000 ⇒ 0000011 0000110 1010000
⇒ 0_0000011 1_0000110 1_1010000
⇒ 1_1010000 1_0000110 0_0000011
⇒ 0xd0 0x86 0x03
This encoding has a neat property that numbers less than 128 are encoded as themselves.
To decode, you keep reading in bytes until you see one with a ‘0’ for its MSB. For each of those bytes, you mask off the MSB, and ‘OR’ it into its place in the final result:
········ ········ ········ (start with empty result)
0xd0 ⇒ 1_1010000 ········ ········ ·1010000 (mask and OR; MSB 1 == continue)
0x86 ⇒ 1_0000110 ········ ··000011 01010000 (mask and OR; MSB 1 == continue)
0x03 ⇒ 0_0000011 ···00000 11000011 01010000 (mask and OR; MSB 0 == done)
00000000 11000011 01010000 => 50000
········
0x00 ⇒ 0_0000000 ·0000000
00000000 => 0
········
0xff ⇒ 0_1111111 ·1111111
01111111 => 127
········ ········
0x80 ⇒ 1_0000000 ········ ·0000000
0x01 ⇒ 0_0000001 ··000000 10000000
00000000 10000000 => 128
Note that the varint encoding described above is defined on _unsigned_ integers. If you want to encode a _signed_ integer, which might be negative, then you first need to figure out how to translate each signed integer into an unsigned integer. This is a separate step, independent of how you decide to then encode the translated unsigned integers.
Modern computers typically use _two’s complement_ to encode signed integers. Two’s complement is fantastic if you’re primarily concerned with performing arithmetic, since most arithmetic operations are _exactly the same_ for unsigned values and for two’s complement signed values. (This is a fascinating result! If you’re not familiar with it, I encourage you to check out the explanation in the Wikipedia article.)
However, for our purposes, two’s complement is one of the worst possible translations! We’re choosing a varint encoding scheme because our values are typically small. In two’s complement, the smallest negative values (-1, -2, etc.) get translated into the largest possible unsigned values (‘UINT_MAX’, ‘UINT_MAX - 1’, etc.), meaning that they take up the most amount of space when encoded.
To get around this, Protocol Buffers (and most other file formats that use varint) use the “zig-zag” method to translate signed integers into unsigned. The zig-zag translation has the beneficial property that small values — regardless of whether they’re negative or positive — end up taking the least amount of space when varint-encoded.
Having just described a perfectly fine (and widespread!) varint encoding, I’d like to propose a slight variation that I like better. (I’m calling this variant “imperial” varint to distinguish it from the “metric” varint described above.)
The two encodings are very similar. They both use continuation bits to describe how many bytes there are in the encoded value, and in particular, use one continuation bit per encoded byte.
There are three important differences that are worth noting. The first is that the bytes will be written in big-endian order instead of little-endian order. The second (and biggest) difference is that instead of placing one continuation bit into each encoded byte, imperial varint places _all_ of the continuation bits at the beginning of the encoded value, at the MSB end of the first encoded byte. And, for reasons we’ll see in a bit, we flip the meaning of the continuation bits: we’ll use ‘0’ to mean “another byte follows”, and ‘1’ to mean “this is the last byte”.
Repeating our examples from above:
0 ⇒ 0000000 (encode in binary)
⇒ 1 0000000 (prepend continuation markers)
⇒ 10000000 (group into bytes)
⇒ 0x80
127 ⇒ 1111111
⇒ 1 1111111
⇒ 11111111
⇒ 0xff
128 ⇒ 0000001 0000000
⇒ 01 0000001 0000000
⇒ 01000000 10000000
⇒ 0x40 0x80
50000 ⇒ 0000011 0000110 1010000
⇒ 001 0000011 0000110 1010000
⇒ 00100000 11000011 01010000
⇒ 0x20 0xc3 0x50
By placing all of the continuation bits at the beginning of the encoded value, we avoid loops in both the encoding and decoding processes. When we encode, we just have to figure out how many bytes we need. Each byte length has a corresponding continuation bit pattern (1 byte ⇒ ‘0x80’, 2 bytes ⇒ ‘0x40’, 3 bytes ⇒ ‘0x20’, etc). After ‘OR’ing that bit pattern into place, we can write out the correct number of bytes as a single operation.
On the decoding side, we take advantage of the fact that each continuation bit pattern consists of zero or more ‘0’ bits, followed by exactly one ‘1’ bit. That means that we can read in the first byte, and then use the “count leading zeros” operation to easily determine how many _additional_ bytes we have to read. “Count leading zeros” is available in most programming languages as an instrinsic, and also has a dedicated instruction on most modern CPUs.
Processor support for “count leading zeros”
All together, this makes encoding and decoding of imperial varints much simpler and faster! To see this concretely, you can check out the implementations in my Swanson programming language project:
So there you go! If you find yourself designing a binary file format in the future (admittedly a pretty niche concern), consider using imperial varints instead of metric varints.
vu128: Efficient variable-length integers
Uses units of full octets, not of 7-bit quantities like in imperial or metric varint.