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4710c53d | 1 | 1. Compression algorithm (deflate)\r |
2 | \r | |
3 | The deflation algorithm used by gzip (also zip and zlib) is a variation of\r | |
4 | LZ77 (Lempel-Ziv 1977, see reference below). It finds duplicated strings in\r | |
5 | the input data. The second occurrence of a string is replaced by a\r | |
6 | pointer to the previous string, in the form of a pair (distance,\r | |
7 | length). Distances are limited to 32K bytes, and lengths are limited\r | |
8 | to 258 bytes. When a string does not occur anywhere in the previous\r | |
9 | 32K bytes, it is emitted as a sequence of literal bytes. (In this\r | |
10 | description, `string' must be taken as an arbitrary sequence of bytes,\r | |
11 | and is not restricted to printable characters.)\r | |
12 | \r | |
13 | Literals or match lengths are compressed with one Huffman tree, and\r | |
14 | match distances are compressed with another tree. The trees are stored\r | |
15 | in a compact form at the start of each block. The blocks can have any\r | |
16 | size (except that the compressed data for one block must fit in\r | |
17 | available memory). A block is terminated when deflate() determines that\r | |
18 | it would be useful to start another block with fresh trees. (This is\r | |
19 | somewhat similar to the behavior of LZW-based _compress_.)\r | |
20 | \r | |
21 | Duplicated strings are found using a hash table. All input strings of\r | |
22 | length 3 are inserted in the hash table. A hash index is computed for\r | |
23 | the next 3 bytes. If the hash chain for this index is not empty, all\r | |
24 | strings in the chain are compared with the current input string, and\r | |
25 | the longest match is selected.\r | |
26 | \r | |
27 | The hash chains are searched starting with the most recent strings, to\r | |
28 | favor small distances and thus take advantage of the Huffman encoding.\r | |
29 | The hash chains are singly linked. There are no deletions from the\r | |
30 | hash chains, the algorithm simply discards matches that are too old.\r | |
31 | \r | |
32 | To avoid a worst-case situation, very long hash chains are arbitrarily\r | |
33 | truncated at a certain length, determined by a runtime option (level\r | |
34 | parameter of deflateInit). So deflate() does not always find the longest\r | |
35 | possible match but generally finds a match which is long enough.\r | |
36 | \r | |
37 | deflate() also defers the selection of matches with a lazy evaluation\r | |
38 | mechanism. After a match of length N has been found, deflate() searches for\r | |
39 | a longer match at the next input byte. If a longer match is found, the\r | |
40 | previous match is truncated to a length of one (thus producing a single\r | |
41 | literal byte) and the process of lazy evaluation begins again. Otherwise,\r | |
42 | the original match is kept, and the next match search is attempted only N\r | |
43 | steps later.\r | |
44 | \r | |
45 | The lazy match evaluation is also subject to a runtime parameter. If\r | |
46 | the current match is long enough, deflate() reduces the search for a longer\r | |
47 | match, thus speeding up the whole process. If compression ratio is more\r | |
48 | important than speed, deflate() attempts a complete second search even if\r | |
49 | the first match is already long enough.\r | |
50 | \r | |
51 | The lazy match evaluation is not performed for the fastest compression\r | |
52 | modes (level parameter 1 to 3). For these fast modes, new strings\r | |
53 | are inserted in the hash table only when no match was found, or\r | |
54 | when the match is not too long. This degrades the compression ratio\r | |
55 | but saves time since there are both fewer insertions and fewer searches.\r | |
56 | \r | |
57 | \r | |
58 | 2. Decompression algorithm (inflate)\r | |
59 | \r | |
60 | 2.1 Introduction\r | |
61 | \r | |
62 | The key question is how to represent a Huffman code (or any prefix code) so\r | |
63 | that you can decode fast. The most important characteristic is that shorter\r | |
64 | codes are much more common than longer codes, so pay attention to decoding the\r | |
65 | short codes fast, and let the long codes take longer to decode.\r | |
66 | \r | |
67 | inflate() sets up a first level table that covers some number of bits of\r | |
68 | input less than the length of longest code. It gets that many bits from the\r | |
69 | stream, and looks it up in the table. The table will tell if the next\r | |
70 | code is that many bits or less and how many, and if it is, it will tell\r | |
71 | the value, else it will point to the next level table for which inflate()\r | |
72 | grabs more bits and tries to decode a longer code.\r | |
73 | \r | |
74 | How many bits to make the first lookup is a tradeoff between the time it\r | |
75 | takes to decode and the time it takes to build the table. If building the\r | |
76 | table took no time (and if you had infinite memory), then there would only\r | |
77 | be a first level table to cover all the way to the longest code. However,\r | |
78 | building the table ends up taking a lot longer for more bits since short\r | |
79 | codes are replicated many times in such a table. What inflate() does is\r | |
80 | simply to make the number of bits in the first table a variable, and then\r | |
81 | to set that variable for the maximum speed.\r | |
82 | \r | |
83 | For inflate, which has 286 possible codes for the literal/length tree, the size\r | |
84 | of the first table is nine bits. Also the distance trees have 30 possible\r | |
85 | values, and the size of the first table is six bits. Note that for each of\r | |
86 | those cases, the table ended up one bit longer than the ``average'' code\r | |
87 | length, i.e. the code length of an approximately flat code which would be a\r | |
88 | little more than eight bits for 286 symbols and a little less than five bits\r | |
89 | for 30 symbols.\r | |
90 | \r | |
91 | \r | |
92 | 2.2 More details on the inflate table lookup\r | |
93 | \r | |
94 | Ok, you want to know what this cleverly obfuscated inflate tree actually\r | |
95 | looks like. You are correct that it's not a Huffman tree. It is simply a\r | |
96 | lookup table for the first, let's say, nine bits of a Huffman symbol. The\r | |
97 | symbol could be as short as one bit or as long as 15 bits. If a particular\r | |
98 | symbol is shorter than nine bits, then that symbol's translation is duplicated\r | |
99 | in all those entries that start with that symbol's bits. For example, if the\r | |
100 | symbol is four bits, then it's duplicated 32 times in a nine-bit table. If a\r | |
101 | symbol is nine bits long, it appears in the table once.\r | |
102 | \r | |
103 | If the symbol is longer than nine bits, then that entry in the table points\r | |
104 | to another similar table for the remaining bits. Again, there are duplicated\r | |
105 | entries as needed. The idea is that most of the time the symbol will be short\r | |
106 | and there will only be one table look up. (That's whole idea behind data\r | |
107 | compression in the first place.) For the less frequent long symbols, there\r | |
108 | will be two lookups. If you had a compression method with really long\r | |
109 | symbols, you could have as many levels of lookups as is efficient. For\r | |
110 | inflate, two is enough.\r | |
111 | \r | |
112 | So a table entry either points to another table (in which case nine bits in\r | |
113 | the above example are gobbled), or it contains the translation for the symbol\r | |
114 | and the number of bits to gobble. Then you start again with the next\r | |
115 | ungobbled bit.\r | |
116 | \r | |
117 | You may wonder: why not just have one lookup table for how ever many bits the\r | |
118 | longest symbol is? The reason is that if you do that, you end up spending\r | |
119 | more time filling in duplicate symbol entries than you do actually decoding.\r | |
120 | At least for deflate's output that generates new trees every several 10's of\r | |
121 | kbytes. You can imagine that filling in a 2^15 entry table for a 15-bit code\r | |
122 | would take too long if you're only decoding several thousand symbols. At the\r | |
123 | other extreme, you could make a new table for every bit in the code. In fact,\r | |
124 | that's essentially a Huffman tree. But then you spend two much time\r | |
125 | traversing the tree while decoding, even for short symbols.\r | |
126 | \r | |
127 | So the number of bits for the first lookup table is a trade of the time to\r | |
128 | fill out the table vs. the time spent looking at the second level and above of\r | |
129 | the table.\r | |
130 | \r | |
131 | Here is an example, scaled down:\r | |
132 | \r | |
133 | The code being decoded, with 10 symbols, from 1 to 6 bits long:\r | |
134 | \r | |
135 | A: 0\r | |
136 | B: 10\r | |
137 | C: 1100\r | |
138 | D: 11010\r | |
139 | E: 11011\r | |
140 | F: 11100\r | |
141 | G: 11101\r | |
142 | H: 11110\r | |
143 | I: 111110\r | |
144 | J: 111111\r | |
145 | \r | |
146 | Let's make the first table three bits long (eight entries):\r | |
147 | \r | |
148 | 000: A,1\r | |
149 | 001: A,1\r | |
150 | 010: A,1\r | |
151 | 011: A,1\r | |
152 | 100: B,2\r | |
153 | 101: B,2\r | |
154 | 110: -> table X (gobble 3 bits)\r | |
155 | 111: -> table Y (gobble 3 bits)\r | |
156 | \r | |
157 | Each entry is what the bits decode as and how many bits that is, i.e. how\r | |
158 | many bits to gobble. Or the entry points to another table, with the number of\r | |
159 | bits to gobble implicit in the size of the table.\r | |
160 | \r | |
161 | Table X is two bits long since the longest code starting with 110 is five bits\r | |
162 | long:\r | |
163 | \r | |
164 | 00: C,1\r | |
165 | 01: C,1\r | |
166 | 10: D,2\r | |
167 | 11: E,2\r | |
168 | \r | |
169 | Table Y is three bits long since the longest code starting with 111 is six\r | |
170 | bits long:\r | |
171 | \r | |
172 | 000: F,2\r | |
173 | 001: F,2\r | |
174 | 010: G,2\r | |
175 | 011: G,2\r | |
176 | 100: H,2\r | |
177 | 101: H,2\r | |
178 | 110: I,3\r | |
179 | 111: J,3\r | |
180 | \r | |
181 | So what we have here are three tables with a total of 20 entries that had to\r | |
182 | be constructed. That's compared to 64 entries for a single table. Or\r | |
183 | compared to 16 entries for a Huffman tree (six two entry tables and one four\r | |
184 | entry table). Assuming that the code ideally represents the probability of\r | |
185 | the symbols, it takes on the average 1.25 lookups per symbol. That's compared\r | |
186 | to one lookup for the single table, or 1.66 lookups per symbol for the\r | |
187 | Huffman tree.\r | |
188 | \r | |
189 | There, I think that gives you a picture of what's going on. For inflate, the\r | |
190 | meaning of a particular symbol is often more than just a letter. It can be a\r | |
191 | byte (a "literal"), or it can be either a length or a distance which\r | |
192 | indicates a base value and a number of bits to fetch after the code that is\r | |
193 | added to the base value. Or it might be the special end-of-block code. The\r | |
194 | data structures created in inftrees.c try to encode all that information\r | |
195 | compactly in the tables.\r | |
196 | \r | |
197 | \r | |
198 | Jean-loup Gailly Mark Adler\r | |
199 | jloup@gzip.org madler@alumni.caltech.edu\r | |
200 | \r | |
201 | \r | |
202 | References:\r | |
203 | \r | |
204 | [LZ77] Ziv J., Lempel A., ``A Universal Algorithm for Sequential Data\r | |
205 | Compression,'' IEEE Transactions on Information Theory, Vol. 23, No. 3,\r | |
206 | pp. 337-343.\r | |
207 | \r | |
208 | ``DEFLATE Compressed Data Format Specification'' available in\r | |
209 | http://www.ietf.org/rfc/rfc1951.txt\r |