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c62d2810 11. Compression algorithm (deflate)
2
3The deflation algorithm used by gzip (also zip and zlib) is a variation of
4LZ77 (Lempel-Ziv 1977, see reference below). It finds duplicated strings in
5the input data. The second occurrence of a string is replaced by a
6pointer to the previous string, in the form of a pair (distance,
7length). Distances are limited to 32K bytes, and lengths are limited
8to 258 bytes. When a string does not occur anywhere in the previous
932K bytes, it is emitted as a sequence of literal bytes. (In this
10description, `string' must be taken as an arbitrary sequence of bytes,
11and is not restricted to printable characters.)
12
13Literals or match lengths are compressed with one Huffman tree, and
14match distances are compressed with another tree. The trees are stored
15in a compact form at the start of each block. The blocks can have any
16size (except that the compressed data for one block must fit in
17available memory). A block is terminated when deflate() determines that
18it would be useful to start another block with fresh trees. (This is
19somewhat similar to the behavior of LZW-based _compress_.)
20
21Duplicated strings are found using a hash table. All input strings of
22length 3 are inserted in the hash table. A hash index is computed for
23the next 3 bytes. If the hash chain for this index is not empty, all
24strings in the chain are compared with the current input string, and
25the longest match is selected.
26
27The hash chains are searched starting with the most recent strings, to
28favor small distances and thus take advantage of the Huffman encoding.
29The hash chains are singly linked. There are no deletions from the
30hash chains, the algorithm simply discards matches that are too old.
31
32To avoid a worst-case situation, very long hash chains are arbitrarily
33truncated at a certain length, determined by a runtime option (level
34parameter of deflateInit). So deflate() does not always find the longest
35possible match but generally finds a match which is long enough.
36
37deflate() also defers the selection of matches with a lazy evaluation
38mechanism. After a match of length N has been found, deflate() searches for
39a longer match at the next input byte. If a longer match is found, the
40previous match is truncated to a length of one (thus producing a single
41literal byte) and the process of lazy evaluation begins again. Otherwise,
42the original match is kept, and the next match search is attempted only N
43steps later.
44
45The lazy match evaluation is also subject to a runtime parameter. If
46the current match is long enough, deflate() reduces the search for a longer
47match, thus speeding up the whole process. If compression ratio is more
48important than speed, deflate() attempts a complete second search even if
49the first match is already long enough.
50
51The lazy match evaluation is not performed for the fastest compression
52modes (level parameter 1 to 3). For these fast modes, new strings
53are inserted in the hash table only when no match was found, or
54when the match is not too long. This degrades the compression ratio
55but saves time since there are both fewer insertions and fewer searches.
56
57
582. Decompression algorithm (inflate)
59
602.1 Introduction
61
62The real question is, given a Huffman tree, how to decode fast. The most
63important realization is that shorter codes are much more common than
64longer codes, so pay attention to decoding the short codes fast, and let
65the long codes take longer to decode.
66
67inflate() sets up a first level table that covers some number of bits of
68input less than the length of longest code. It gets that many bits from the
69stream, and looks it up in the table. The table will tell if the next
70code is that many bits or less and how many, and if it is, it will tell
71the value, else it will point to the next level table for which inflate()
72grabs more bits and tries to decode a longer code.
73
74How many bits to make the first lookup is a tradeoff between the time it
75takes to decode and the time it takes to build the table. If building the
76table took no time (and if you had infinite memory), then there would only
77be a first level table to cover all the way to the longest code. However,
78building the table ends up taking a lot longer for more bits since short
79codes are replicated many times in such a table. What inflate() does is
80simply to make the number of bits in the first table a variable, and set it
81for the maximum speed.
82
83inflate() sends new trees relatively often, so it is possibly set for a
84smaller first level table than an application that has only one tree for
85all the data. For inflate, which has 286 possible codes for the
86literal/length tree, the size of the first table is nine bits. Also the
87distance trees have 30 possible values, and the size of the first table is
88six bits. Note that for each of those cases, the table ended up one bit
89longer than the ``average'' code length, i.e. the code length of an
90approximately flat code which would be a little more than eight bits for
91286 symbols and a little less than five bits for 30 symbols. It would be
92interesting to see if optimizing the first level table for other
93applications gave values within a bit or two of the flat code size.
94
95
962.2 More details on the inflate table lookup
97
98Ok, you want to know what this cleverly obfuscated inflate tree actually
99looks like. You are correct that it's not a Huffman tree. It is simply a
100lookup table for the first, let's say, nine bits of a Huffman symbol. The
101symbol could be as short as one bit or as long as 15 bits. If a particular
102symbol is shorter than nine bits, then that symbol's translation is duplicated
103in all those entries that start with that symbol's bits. For example, if the
104symbol is four bits, then it's duplicated 32 times in a nine-bit table. If a
105symbol is nine bits long, it appears in the table once.
106
107If the symbol is longer than nine bits, then that entry in the table points
108to another similar table for the remaining bits. Again, there are duplicated
109entries as needed. The idea is that most of the time the symbol will be short
110and there will only be one table look up. (That's whole idea behind data
111compression in the first place.) For the less frequent long symbols, there
112will be two lookups. If you had a compression method with really long
113symbols, you could have as many levels of lookups as is efficient. For
114inflate, two is enough.
115
116So a table entry either points to another table (in which case nine bits in
117the above example are gobbled), or it contains the translation for the symbol
118and the number of bits to gobble. Then you start again with the next
119ungobbled bit.
120
121You may wonder: why not just have one lookup table for how ever many bits the
122longest symbol is? The reason is that if you do that, you end up spending
123more time filling in duplicate symbol entries than you do actually decoding.
124At least for deflate's output that generates new trees every several 10's of
125kbytes. You can imagine that filling in a 2^15 entry table for a 15-bit code
126would take too long if you're only decoding several thousand symbols. At the
127other extreme, you could make a new table for every bit in the code. In fact,
128that's essentially a Huffman tree. But then you spend two much time
129traversing the tree while decoding, even for short symbols.
130
131So the number of bits for the first lookup table is a trade of the time to
132fill out the table vs. the time spent looking at the second level and above of
133the table.
134
135Here is an example, scaled down:
136
137The code being decoded, with 10 symbols, from 1 to 6 bits long:
138
139A: 0
140B: 10
141C: 1100
142D: 11010
143E: 11011
144F: 11100
145G: 11101
146H: 11110
147I: 111110
148J: 111111
149
150Let's make the first table three bits long (eight entries):
151
152000: A,1
153001: A,1
154010: A,1
155011: A,1
156100: B,2
157101: B,2
158110: -> table X (gobble 3 bits)
159111: -> table Y (gobble 3 bits)
160
161Each entry is what the bits decode to and how many bits that is, i.e. how
162many bits to gobble. Or the entry points to another table, with the number of
163bits to gobble implicit in the size of the table.
164
165Table X is two bits long since the longest code starting with 110 is five bits
166long:
167
16800: C,1
16901: C,1
17010: D,2
17111: E,2
172
173Table Y is three bits long since the longest code starting with 111 is six
174bits long:
175
176000: F,2
177001: F,2
178010: G,2
179011: G,2
180100: H,2
181101: H,2
182110: I,3
183111: J,3
184
185So what we have here are three tables with a total of 20 entries that had to
186be constructed. That's compared to 64 entries for a single table. Or
187compared to 16 entries for a Huffman tree (six two entry tables and one four
188entry table). Assuming that the code ideally represents the probability of
189the symbols, it takes on the average 1.25 lookups per symbol. That's compared
190to one lookup for the single table, or 1.66 lookups per symbol for the
191Huffman tree.
192
193There, I think that gives you a picture of what's going on. For inflate, the
194meaning of a particular symbol is often more than just a letter. It can be a
195byte (a "literal"), or it can be either a length or a distance which
196indicates a base value and a number of bits to fetch after the code that is
197added to the base value. Or it might be the special end-of-block code. The
198data structures created in inftrees.c try to encode all that information
199compactly in the tables.
200
201
202Jean-loup Gailly Mark Adler
203jloup@gzip.org madler@alumni.caltech.edu
204
205
206References:
207
208[LZ77] Ziv J., Lempel A., ``A Universal Algorithm for Sequential Data
209Compression,'' IEEE Transactions on Information Theory, Vol. 23, No. 3,
210pp. 337-343.
211
212``DEFLATE Compressed Data Format Specification'' available in
213ftp://ds.internic.net/rfc/rfc1951.txt