1. Compression algorithm (deflate) | |
The deflation algorithm used by gzip (also zip and zlib) is a variation of | |
LZ77 (Lempel-Ziv 1977, see reference below). It finds duplicated strings in | |
the input data. The second occurrence of a string is replaced by a | |
pointer to the previous string, in the form of a pair (distance, | |
length). Distances are limited to 32K bytes, and lengths are limited | |
to 258 bytes. When a string does not occur anywhere in the previous | |
32K bytes, it is emitted as a sequence of literal bytes. (In this | |
description, `string' must be taken as an arbitrary sequence of bytes, | |
and is not restricted to printable characters.) | |
Literals or match lengths are compressed with one Huffman tree, and | |
match distances are compressed with another tree. The trees are stored | |
in a compact form at the start of each block. The blocks can have any | |
size (except that the compressed data for one block must fit in | |
available memory). A block is terminated when deflate() determines that | |
it would be useful to start another block with fresh trees. (This is | |
somewhat similar to the behavior of LZW-based _compress_.) | |
Duplicated strings are found using a hash table. All input strings of | |
length 3 are inserted in the hash table. A hash index is computed for | |
the next 3 bytes. If the hash chain for this index is not empty, all | |
strings in the chain are compared with the current input string, and | |
the longest match is selected. | |
The hash chains are searched starting with the most recent strings, to | |
favor small distances and thus take advantage of the Huffman encoding. | |
The hash chains are singly linked. There are no deletions from the | |
hash chains, the algorithm simply discards matches that are too old. | |
To avoid a worst-case situation, very long hash chains are arbitrarily | |
truncated at a certain length, determined by a runtime option (level | |
parameter of deflateInit). So deflate() does not always find the longest | |
possible match but generally finds a match which is long enough. | |
deflate() also defers the selection of matches with a lazy evaluation | |
mechanism. After a match of length N has been found, deflate() searches for | |
a longer match at the next input byte. If a longer match is found, the | |
previous match is truncated to a length of one (thus producing a single | |
literal byte) and the process of lazy evaluation begins again. Otherwise, | |
the original match is kept, and the next match search is attempted only N | |
steps later. | |
The lazy match evaluation is also subject to a runtime parameter. If | |
the current match is long enough, deflate() reduces the search for a longer | |
match, thus speeding up the whole process. If compression ratio is more | |
important than speed, deflate() attempts a complete second search even if | |
the first match is already long enough. | |
The lazy match evaluation is not performed for the fastest compression | |
modes (level parameter 1 to 3). For these fast modes, new strings | |
are inserted in the hash table only when no match was found, or | |
when the match is not too long. This degrades the compression ratio | |
but saves time since there are both fewer insertions and fewer searches. | |
2. Decompression algorithm (inflate) | |
2.1 Introduction | |
The key question is how to represent a Huffman code (or any prefix code) so | |
that you can decode fast. The most important characteristic is that shorter | |
codes are much more common than longer codes, so pay attention to decoding the | |
short codes fast, and let the long codes take longer to decode. | |
inflate() sets up a first level table that covers some number of bits of | |
input less than the length of longest code. It gets that many bits from the | |
stream, and looks it up in the table. The table will tell if the next | |
code is that many bits or less and how many, and if it is, it will tell | |
the value, else it will point to the next level table for which inflate() | |
grabs more bits and tries to decode a longer code. | |
How many bits to make the first lookup is a tradeoff between the time it | |
takes to decode and the time it takes to build the table. If building the | |
table took no time (and if you had infinite memory), then there would only | |
be a first level table to cover all the way to the longest code. However, | |
building the table ends up taking a lot longer for more bits since short | |
codes are replicated many times in such a table. What inflate() does is | |
simply to make the number of bits in the first table a variable, and then | |
to set that variable for the maximum speed. | |
For inflate, which has 286 possible codes for the literal/length tree, the size | |
of the first table is nine bits. Also the distance trees have 30 possible | |
values, and the size of the first table is six bits. Note that for each of | |
those cases, the table ended up one bit longer than the ``average'' code | |
length, i.e. the code length of an approximately flat code which would be a | |
little more than eight bits for 286 symbols and a little less than five bits | |
for 30 symbols. | |
2.2 More details on the inflate table lookup | |
Ok, you want to know what this cleverly obfuscated inflate tree actually | |
looks like. You are correct that it's not a Huffman tree. It is simply a | |
lookup table for the first, let's say, nine bits of a Huffman symbol. The | |
symbol could be as short as one bit or as long as 15 bits. If a particular | |
symbol is shorter than nine bits, then that symbol's translation is duplicated | |
in all those entries that start with that symbol's bits. For example, if the | |
symbol is four bits, then it's duplicated 32 times in a nine-bit table. If a | |
symbol is nine bits long, it appears in the table once. | |
If the symbol is longer than nine bits, then that entry in the table points | |
to another similar table for the remaining bits. Again, there are duplicated | |
entries as needed. The idea is that most of the time the symbol will be short | |
and there will only be one table look up. (That's whole idea behind data | |
compression in the first place.) For the less frequent long symbols, there | |
will be two lookups. If you had a compression method with really long | |
symbols, you could have as many levels of lookups as is efficient. For | |
inflate, two is enough. | |
So a table entry either points to another table (in which case nine bits in | |
the above example are gobbled), or it contains the translation for the symbol | |
and the number of bits to gobble. Then you start again with the next | |
ungobbled bit. | |
You may wonder: why not just have one lookup table for how ever many bits the | |
longest symbol is? The reason is that if you do that, you end up spending | |
more time filling in duplicate symbol entries than you do actually decoding. | |
At least for deflate's output that generates new trees every several 10's of | |
kbytes. You can imagine that filling in a 2^15 entry table for a 15-bit code | |
would take too long if you're only decoding several thousand symbols. At the | |
other extreme, you could make a new table for every bit in the code. In fact, | |
that's essentially a Huffman tree. But then you spend two much time | |
traversing the tree while decoding, even for short symbols. | |
So the number of bits for the first lookup table is a trade of the time to | |
fill out the table vs. the time spent looking at the second level and above of | |
the table. | |
Here is an example, scaled down: | |
The code being decoded, with 10 symbols, from 1 to 6 bits long: | |
A: 0 | |
B: 10 | |
C: 1100 | |
D: 11010 | |
E: 11011 | |
F: 11100 | |
G: 11101 | |
H: 11110 | |
I: 111110 | |
J: 111111 | |
Let's make the first table three bits long (eight entries): | |
000: A,1 | |
001: A,1 | |
010: A,1 | |
011: A,1 | |
100: B,2 | |
101: B,2 | |
110: -> table X (gobble 3 bits) | |
111: -> table Y (gobble 3 bits) | |
Each entry is what the bits decode as and how many bits that is, i.e. how | |
many bits to gobble. Or the entry points to another table, with the number of | |
bits to gobble implicit in the size of the table. | |
Table X is two bits long since the longest code starting with 110 is five bits | |
long: | |
00: C,1 | |
01: C,1 | |
10: D,2 | |
11: E,2 | |
Table Y is three bits long since the longest code starting with 111 is six | |
bits long: | |
000: F,2 | |
001: F,2 | |
010: G,2 | |
011: G,2 | |
100: H,2 | |
101: H,2 | |
110: I,3 | |
111: J,3 | |
So what we have here are three tables with a total of 20 entries that had to | |
be constructed. That's compared to 64 entries for a single table. Or | |
compared to 16 entries for a Huffman tree (six two entry tables and one four | |
entry table). Assuming that the code ideally represents the probability of | |
the symbols, it takes on the average 1.25 lookups per symbol. That's compared | |
to one lookup for the single table, or 1.66 lookups per symbol for the | |
Huffman tree. | |
There, I think that gives you a picture of what's going on. For inflate, the | |
meaning of a particular symbol is often more than just a letter. It can be a | |
byte (a "literal"), or it can be either a length or a distance which | |
indicates a base value and a number of bits to fetch after the code that is | |
added to the base value. Or it might be the special end-of-block code. The | |
data structures created in inftrees.c try to encode all that information | |
compactly in the tables. | |
Jean-loup Gailly Mark Adler | |
jloup@gzip.org madler@alumni.caltech.edu | |
References: | |
[LZ77] Ziv J., Lempel A., ``A Universal Algorithm for Sequential Data | |
Compression,'' IEEE Transactions on Information Theory, Vol. 23, No. 3, | |
pp. 337-343. | |
``DEFLATE Compressed Data Format Specification'' available in | |
http://www.ietf.org/rfc/rfc1951.txt |