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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 real question is, given a Huffman tree, how to decode fast.  The most

important realization 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 set it

for the maximum speed.

inflate() sends new trees relatively often, so it is possibly set for a

smaller first level table than an application that has only one tree for

all the data.  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.  It would be

interesting to see if optimizing the first level table for other

applications gave values within a bit or two of the flat code size.

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 to 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

ftp://ds.internic.net/rfc/rfc1951.txt