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1 | 1. Compression algorithm (deflate) |
2 | |
3 | The deflation algorithm used by gzip (also zip and zlib) is a variation of |
4 | LZ77 (Lempel-Ziv 1977, see reference below). It finds duplicated strings in |
5 | the input data. The second occurrence of a string is replaced by a |
6 | pointer to the previous string, in the form of a pair (distance, |
7 | length). Distances are limited to 32K bytes, and lengths are limited |
8 | to 258 bytes. When a string does not occur anywhere in the previous |
9 | 32K bytes, it is emitted as a sequence of literal bytes. (In this |
10 | description, `string' must be taken as an arbitrary sequence of bytes, |
11 | and is not restricted to printable characters.) |
12 | |
13 | Literals or match lengths are compressed with one Huffman tree, and |
14 | match distances are compressed with another tree. The trees are stored |
15 | in a compact form at the start of each block. The blocks can have any |
16 | size (except that the compressed data for one block must fit in |
17 | available memory). A block is terminated when deflate() determines that |
18 | it would be useful to start another block with fresh trees. (This is |
19 | somewhat similar to the behavior of LZW-based _compress_.) |
20 | |
21 | Duplicated strings are found using a hash table. All input strings of |
22 | length 3 are inserted in the hash table. A hash index is computed for |
23 | the next 3 bytes. If the hash chain for this index is not empty, all |
24 | strings in the chain are compared with the current input string, and |
25 | the longest match is selected. |
26 | |
27 | The hash chains are searched starting with the most recent strings, to |
28 | favor small distances and thus take advantage of the Huffman encoding. |
29 | The hash chains are singly linked. There are no deletions from the |
30 | hash chains, the algorithm simply discards matches that are too old. |
31 | |
32 | To avoid a worst-case situation, very long hash chains are arbitrarily |
33 | truncated at a certain length, determined by a runtime option (level |
34 | parameter of deflateInit). So deflate() does not always find the longest |
35 | possible match but generally finds a match which is long enough. |
36 | |
37 | deflate() also defers the selection of matches with a lazy evaluation |
38 | mechanism. After a match of length N has been found, deflate() searches for |
39 | a longer match at the next input byte. If a longer match is found, the |
40 | previous match is truncated to a length of one (thus producing a single |
41 | literal byte) and the process of lazy evaluation begins again. Otherwise, |
42 | the original match is kept, and the next match search is attempted only N |
43 | steps later. |
44 | |
45 | The lazy match evaluation is also subject to a runtime parameter. If |
46 | the current match is long enough, deflate() reduces the search for a longer |
47 | match, thus speeding up the whole process. If compression ratio is more |
48 | important than speed, deflate() attempts a complete second search even if |
49 | the first match is already long enough. |
50 | |
51 | The lazy match evaluation is not performed for the fastest compression |
52 | modes (level parameter 1 to 3). For these fast modes, new strings |
53 | are inserted in the hash table only when no match was found, or |
54 | when the match is not too long. This degrades the compression ratio |
55 | but saves time since there are both fewer insertions and fewer searches. |
56 | |
57 | |
58 | 2. Decompression algorithm (inflate) |
59 | |
60 | 2.1 Introduction |
61 | |
62 | The key question is how to represent a Huffman code (or any prefix code) so |
63 | that you can decode fast. The most important characteristic is that shorter |
64 | codes are much more common than longer codes, so pay attention to decoding the |
65 | short codes fast, and let the long codes take longer to decode. |
66 | |
67 | inflate() sets up a first level table that covers some number of bits of |
68 | input less than the length of longest code. It gets that many bits from the |
69 | stream, and looks it up in the table. The table will tell if the next |
70 | code is that many bits or less and how many, and if it is, it will tell |
71 | the value, else it will point to the next level table for which inflate() |
72 | grabs more bits and tries to decode a longer code. |
73 | |
74 | How many bits to make the first lookup is a tradeoff between the time it |
75 | takes to decode and the time it takes to build the table. If building the |
76 | table took no time (and if you had infinite memory), then there would only |
77 | be a first level table to cover all the way to the longest code. However, |
78 | building the table ends up taking a lot longer for more bits since short |
79 | codes are replicated many times in such a table. What inflate() does is |
80 | simply to make the number of bits in the first table a variable, and then |
81 | to set that variable for the maximum speed. |
82 | |
83 | For inflate, which has 286 possible codes for the literal/length tree, the size |
84 | of the first table is nine bits. Also the distance trees have 30 possible |
85 | values, and the size of the first table is six bits. Note that for each of |
86 | those cases, the table ended up one bit longer than the ``average'' code |
87 | length, i.e. the code length of an approximately flat code which would be a |
88 | little more than eight bits for 286 symbols and a little less than five bits |
89 | for 30 symbols. |
90 | |
91 | |
92 | 2.2 More details on the inflate table lookup |
93 | |
94 | Ok, you want to know what this cleverly obfuscated inflate tree actually |
95 | looks like. You are correct that it's not a Huffman tree. It is simply a |
96 | lookup table for the first, let's say, nine bits of a Huffman symbol. The |
97 | symbol could be as short as one bit or as long as 15 bits. If a particular |
98 | symbol is shorter than nine bits, then that symbol's translation is duplicated |
99 | in all those entries that start with that symbol's bits. For example, if the |
100 | symbol is four bits, then it's duplicated 32 times in a nine-bit table. If a |
101 | symbol is nine bits long, it appears in the table once. |
102 | |
103 | If the symbol is longer than nine bits, then that entry in the table points |
104 | to another similar table for the remaining bits. Again, there are duplicated |
105 | entries as needed. The idea is that most of the time the symbol will be short |
106 | and there will only be one table look up. (That's whole idea behind data |
107 | compression in the first place.) For the less frequent long symbols, there |
108 | will be two lookups. If you had a compression method with really long |
109 | symbols, you could have as many levels of lookups as is efficient. For |
110 | inflate, two is enough. |
111 | |
112 | So a table entry either points to another table (in which case nine bits in |
113 | the above example are gobbled), or it contains the translation for the symbol |
114 | and the number of bits to gobble. Then you start again with the next |
115 | ungobbled bit. |
116 | |
117 | You may wonder: why not just have one lookup table for how ever many bits the |
118 | longest symbol is? The reason is that if you do that, you end up spending |
119 | more time filling in duplicate symbol entries than you do actually decoding. |
120 | At least for deflate's output that generates new trees every several 10's of |
121 | kbytes. You can imagine that filling in a 2^15 entry table for a 15-bit code |
122 | would take too long if you're only decoding several thousand symbols. At the |
123 | other extreme, you could make a new table for every bit in the code. In fact, |
124 | that's essentially a Huffman tree. But then you spend too much time |
125 | traversing the tree while decoding, even for short symbols. |
126 | |
127 | So the number of bits for the first lookup table is a trade of the time to |
128 | fill out the table vs. the time spent looking at the second level and above of |
129 | the table. |
130 | |
131 | Here is an example, scaled down: |
132 | |
133 | The code being decoded, with 10 symbols, from 1 to 6 bits long: |
134 | |
135 | A: 0 |
136 | B: 10 |
137 | C: 1100 |
138 | D: 11010 |
139 | E: 11011 |
140 | F: 11100 |
141 | G: 11101 |
142 | H: 11110 |
143 | I: 111110 |
144 | J: 111111 |
145 | |
146 | Let's make the first table three bits long (eight entries): |
147 | |
148 | 000: A,1 |
149 | 001: A,1 |
150 | 010: A,1 |
151 | 011: A,1 |
152 | 100: B,2 |
153 | 101: B,2 |
154 | 110: -> table X (gobble 3 bits) |
155 | 111: -> table Y (gobble 3 bits) |
156 | |
157 | Each entry is what the bits decode as and how many bits that is, i.e. how |
158 | many bits to gobble. Or the entry points to another table, with the number of |
159 | bits to gobble implicit in the size of the table. |
160 | |
161 | Table X is two bits long since the longest code starting with 110 is five bits |
162 | long: |
163 | |
164 | 00: C,1 |
165 | 01: C,1 |
166 | 10: D,2 |
167 | 11: E,2 |
168 | |
169 | Table Y is three bits long since the longest code starting with 111 is six |
170 | bits long: |
171 | |
172 | 000: F,2 |
173 | 001: F,2 |
174 | 010: G,2 |
175 | 011: G,2 |
176 | 100: H,2 |
177 | 101: H,2 |
178 | 110: I,3 |
179 | 111: J,3 |
180 | |
181 | So what we have here are three tables with a total of 20 entries that had to |
182 | be constructed. That's compared to 64 entries for a single table. Or |
183 | compared to 16 entries for a Huffman tree (six two entry tables and one four |
184 | entry table). Assuming that the code ideally represents the probability of |
185 | the symbols, it takes on the average 1.25 lookups per symbol. That's compared |
186 | to one lookup for the single table, or 1.66 lookups per symbol for the |
187 | Huffman tree. |
188 | |
189 | There, I think that gives you a picture of what's going on. For inflate, the |
190 | meaning of a particular symbol is often more than just a letter. It can be a |
191 | byte (a "literal"), or it can be either a length or a distance which |
192 | indicates a base value and a number of bits to fetch after the code that is |
193 | added to the base value. Or it might be the special end-of-block code. The |
194 | data structures created in inftrees.c try to encode all that information |
195 | compactly in the tables. |
196 | |
197 | |
198 | Jean-loup Gailly Mark Adler |
199 | jloup@gzip.org madler@alumni.caltech.edu |
200 | |
201 | |
202 | References: |
203 | |
204 | [LZ77] Ziv J., Lempel A., ``A Universal Algorithm for Sequential Data |
205 | Compression,'' IEEE Transactions on Information Theory, Vol. 23, No. 3, |
206 | pp. 337-343. |
207 | |
208 | ``DEFLATE Compressed Data Format Specification'' available in |
209 | http://tools.ietf.org/html/rfc1951 |