Dlugosz' Variable-Length Integer Encoding — Revision 2


Since the whole idea of the ZIP2 file format is to save room, it is wasteful to store values with a larger number of bytes when a small number is generally enough. Also, since it is designed with the future in mind, what is the upper limit of the value's size? For example, 32 bits might be a good choice for a length value now, but insufficient in the near future. So it is natural that many integer values will be stored as variable-sized integers. It will be short when possible, keeping the file efficient. However, it may be long when necessary.

My first idea was to simply use UTF-8 encoding. However, this is less efficient for small integers than it could be (two byte forms only goes up to 2K) and is limited to handling numbers up to 31 bits. A design goal was to handle 64 bits, so this would not do.

Since I was looking for something different anyway, it became clear that the design considerations for UTF-8 were simply not applicable for my needs. A simpler format would work quite well, and offer other benefits in exchange.

History and Previous Versions

Version 1 of this idea used a very simple and elegant approach, where an n- byte field contained n-1 1-bits, a 0-bit, and finally 7*n bits making up the number. This proved more difficult than anticipated to implement efficiently in Perl, and has the disadvantage of not knowing how many bytes you need to read from a stream before you know the length.

Version 2 is totally revised, but has some features in common. It further optimizes some common number ranges, and makes reading from a stream easier.

It can't be explained in a sentence anymore, but requires some tables. But, real implementations used tables anyway, so it's not actually more difficult to program.


The first byte of the variable-length field will indicate how long the field is, and may contain some bits of the value. Subsequent bytes contain the rest of the number's value. In addition, there is a super-extended form that encodes the length in the second (or more) bytes, but this is not needed except for very large multi-precision integers.

In all cases, masking off the highest bits will leave the value readable in big-endian format in the field, once you know the field's length (which is encoded in those high bits).

prefix bitsbytesdata bitsunsigned range
0 1 7 127
10 2 14 16,383
110 3 21 2,097,151
111 00 4 27 134,217,727 (128K)
111 01 5 35 34,359,738,368 (32G)
111 10 8 59 holds the significant part of a Win32 FILETIME
111 11 000 6 40 1,099,511,627,776 (1T)
111 11 001 9 64 A full 64-bit value with one byte overhead
111 11 010 17 128 A GUID/UUID
111 11 111 nany Any multi-precision integer

A byte with its high bit cleared is a one-byte value, holding a number from 0- 127. If the high bit is set, the next highest bit indicates that the value is two bytes long when clear, longer when set.

So a two-byte value can hold up to 16K, which is sufficient for ordinals most of the time. A 4-byte value can hold up to 128 Meg, which is sufficient for most length values. If you're storing files longer than that, an extra byte for a longer length field is not significant.

Longer fields still have less than 1 byte of overhead to encode the length. A 64-bit Windows FILETIME really has 57 bits for current time values, and that will still encode to a 8-byte field (5 bits overhead for the length).

There are 4 unused encodings with 1 byte of overhead, and they can be assigned to optimize cases for common needs, as they are discovered. For more arbitrary storage, a final form gives the length as a separate value, and can encode any length.

Benifits and Features


There are four general forms. The shortest preserve minimum size and maximum range for ordinal values. They have n= 0, 1, or 2 leading 1 bits followed by a 0 bit, followed by n additional bytes.

The second format has 3 leading 1 bits, followed by a 2-bit selector. The selector encodes for 3, 4 or 7 additional bytes.

When the selector is binary 11, the rest of the byte encodes a 3-bit selector that encodes more lengths.

When that selector is binary 111, the length follows the first byte. The length is itself encoded as a VLI. That is followed by that many value bytes.

[0 xxxxxxx] holds 0—127 in one byte.

The first bit is zero, indicating this form. The remaining 7 bits hold the value.

[10 xxxxxx] [yyyyyyyy] holds 0—16,383 in two bytes.

The first two bits are 10, indicating this form. The remaining 6 x bits are the most-significant bits of the value, and the second byte is the least-significant 8 bits of the value.

If you mask off the first two bits (set them both to zero for an unsigned vli), the two bytes will hold the value in big-endian.

[110 xxxxx] [yyyyyyyy] [zzzzzzzz] holds 0—2,097,151 in three bytes.

The first three bits are 110, indicating this form. The remaining 5 x bits are the most-significant bits of the value, followed by two more bytes of value.

[111 ff xxx] [[ n bytes ]]

The first three bits are 110, indicating this form. The following two f bits encode the length. The remaining 3 x bits are the most-significant bits of the value, followed by the specified number of additional value bytes.

Values of ff encode: 00 = 3 more bytes (27 bits of value total), 01 = 4 more bytes (35 bits total), 10 =7 more bytes (59 bits total), and 11 means to use the even longer g format.

These are listed individually below.

[111 00 xxx] [[ 3 bytes ]] holds 0—128K in 4 bytes.
[111 01 xxx] [[ 4 bytes ]] holds a 35-bit value.
[111 10 xxx] [[ 7 bytes ]] holds a 59-bit value.

[111 11 ggg] [[ n bytes ]]

The first five bits are 11111, indicating this form. The following three g bits encode the length. This is followed by the indicated number of value bytes.

Values of ggg encode: 000 = 5 more bytes, 001 = 8 more bytes, 010 = 16 more bytes, and 111 means to use the even longer v format. The other 4 values of ggg are reserved for future definition.

[111 11 000] [[ 5 bytes ]]
[111 11 001] [[ 8 bytes ]]
[111 11 010] [[ 16 bytes ]]
[111 11 011] [[ ??? ]] is reserved
[111 11 100] [[ ??? ]] is reserved
[111 11 101] [[ ??? ]] is reserved
[111 11 110] [[ ??? ]] is reserved
[111 11 111] [[ v-field ]] [[ lots more bytes]]


Example numbers are written with a leading +/- to indicate signed values, and with no prefix to indicate unsigned values. The prefix (+) means that the number has the same encoding whether treated as signed or unsigned.

number encoding form notes
(+) 1 01 0 {7 bits}
(+) 5 05 0 {7 bits}
(+) 20 14 0 {7 bits}
(+) 200 80 c8 10 {14 bits}
(+) 400 81 90 10 {14 bits}
(+) 10,000 a7 10 10 {14 bits}
(+) 16,384 c0 40 00 110 {21 bits} Smallest unsigned number that needs 3 bytes
(+) 2,000,000 de 84 80 110 {21 bits}

Page content copyright 2003 by John M. Dlugosz. Home:http://www.dlugosz.com, email:mailto:john@dlugosz.com