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11. Compression algorithm (deflate) 11. Compression algorithm (deflate)
2 2
3The deflation algorithm used by gzip (also zip and zlib) is a variation of 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 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 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, 6pointer to the previous string, in the form of a pair (distance,
7length). Distances are limited to 32K bytes, and lengths are limited 7length). Distances are limited to 32K bytes, and lengths are limited
8to 258 bytes. When a string does not occur anywhere in the previous 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 932K bytes, it is emitted as a sequence of literal bytes. (In this
10description, `string' must be taken as an arbitrary sequence of bytes, 10description, `string' must be taken as an arbitrary sequence of bytes,
11and is not restricted to printable characters.) 11and is not restricted to printable characters.)
12 12
13Literals or match lengths are compressed with one Huffman tree, and 13Literals or match lengths are compressed with one Huffman tree, and
14match distances are compressed with another tree. The trees are stored 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 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 16size (except that the compressed data for one block must fit in
17available memory). A block is terminated when deflate() determines that 17available memory). A block is terminated when deflate() determines that
18it would be useful to start another block with fresh trees. (This is 18it would be useful to start another block with fresh trees. (This is
19somewhat similar to the behavior of LZW-based _compress_.) 19somewhat similar to the behavior of LZW-based _compress_.)
20 20
21Duplicated strings are found using a hash table. All input strings of 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 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 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 24strings in the chain are compared with the current input string, and
25the longest match is selected. 25the longest match is selected.
26 26
27The hash chains are searched starting with the most recent strings, to 27The hash chains are searched starting with the most recent strings, to
28favor small distances and thus take advantage of the Huffman encoding. 28favor small distances and thus take advantage of the Huffman encoding.
29The hash chains are singly linked. There are no deletions from the 29The hash chains are singly linked. There are no deletions from the
30hash chains, the algorithm simply discards matches that are too old. 30hash chains, the algorithm simply discards matches that are too old.
31 31
32To avoid a worst-case situation, very long hash chains are arbitrarily 32To avoid a worst-case situation, very long hash chains are arbitrarily
33truncated at a certain length, determined by a runtime option (level 33truncated at a certain length, determined by a runtime option (level
34parameter of deflateInit). So deflate() does not always find the longest 34parameter of deflateInit). So deflate() does not always find the longest
35possible match but generally finds a match which is long enough. 35possible match but generally finds a match which is long enough.
36 36
37deflate() also defers the selection of matches with a lazy evaluation 37deflate() also defers the selection of matches with a lazy evaluation
38mechanism. After a match of length N has been found, deflate() searches for 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 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 40previous match is truncated to a length of one (thus producing a single
41literal byte) and the process of lazy evaluation begins again. Otherwise, 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 42the original match is kept, and the next match search is attempted only N
43steps later. 43steps later.
44 44
45The lazy match evaluation is also subject to a runtime parameter. If 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 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 47match, thus speeding up the whole process. If compression ratio is more
48important than speed, deflate() attempts a complete second search even if 48important than speed, deflate() attempts a complete second search even if
49the first match is already long enough. 49the first match is already long enough.
50 50
51The lazy match evaluation is not performed for the fastest compression 51The lazy match evaluation is not performed for the fastest compression
52modes (level parameter 1 to 3). For these fast modes, new strings 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 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 54when the match is not too long. This degrades the compression ratio
55but saves time since there are both fewer insertions and fewer searches. 55but saves time since there are both fewer insertions and fewer searches.
56 56
57 57
582. Decompression algorithm (inflate) 582. Decompression algorithm (inflate)
59 59
602.1 Introduction 602.1 Introduction
61 61
62The key question is how to represent a Huffman code (or any prefix code) so 62The key question is how to represent a Huffman code (or any prefix code) so
63that you can decode fast. The most important characteristic is that shorter 63that you can decode fast. The most important characteristic is that shorter
64codes are much more common than longer codes, so pay attention to decoding the 64codes are much more common than longer codes, so pay attention to decoding the
65short codes fast, and let the long codes take longer to decode. 65short codes fast, and let the long codes take longer to decode.
66 66
67inflate() sets up a first level table that covers some number of bits of 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 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 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 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() 71the value, else it will point to the next level table for which inflate()
72grabs more bits and tries to decode a longer code. 72grabs more bits and tries to decode a longer code.
73 73
74How many bits to make the first lookup is a tradeoff between the time it 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 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 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, 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 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 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 then 80simply to make the number of bits in the first table a variable, and then
81to set that variable for the maximum speed. 81to set that variable for the maximum speed.
82 82
83For inflate, which has 286 possible codes for the literal/length tree, the size 83For inflate, which has 286 possible codes for the literal/length tree, the size
84of the first table is nine bits. Also the distance trees have 30 possible 84of the first table is nine bits. Also the distance trees have 30 possible
85values, and the size of the first table is six bits. Note that for each of 85values, and the size of the first table is six bits. Note that for each of
86those cases, the table ended up one bit longer than the ``average'' code 86those cases, the table ended up one bit longer than the ``average'' code
87length, i.e. the code length of an approximately flat code which would be a 87length, i.e. the code length of an approximately flat code which would be a
88little more than eight bits for 286 symbols and a little less than five bits 88little more than eight bits for 286 symbols and a little less than five bits
89for 30 symbols. 89for 30 symbols.
90 90
91 91
922.2 More details on the inflate table lookup 922.2 More details on the inflate table lookup
93 93
94Ok, you want to know what this cleverly obfuscated inflate tree actually 94Ok, you want to know what this cleverly obfuscated inflate tree actually
95looks like. You are correct that it's not a Huffman tree. It is simply a 95looks like. You are correct that it's not a Huffman tree. It is simply a
96lookup table for the first, let's say, nine bits of a Huffman symbol. The 96lookup table for the first, let's say, nine bits of a Huffman symbol. The
97symbol could be as short as one bit or as long as 15 bits. If a particular 97symbol could be as short as one bit or as long as 15 bits. If a particular
98symbol is shorter than nine bits, then that symbol's translation is duplicated 98symbol is shorter than nine bits, then that symbol's translation is duplicated
99in all those entries that start with that symbol's bits. For example, if the 99in all those entries that start with that symbol's bits. For example, if the
100symbol is four bits, then it's duplicated 32 times in a nine-bit table. If a 100symbol is four bits, then it's duplicated 32 times in a nine-bit table. If a
101symbol is nine bits long, it appears in the table once. 101symbol is nine bits long, it appears in the table once.
102 102
103If the symbol is longer than nine bits, then that entry in the table points 103If the symbol is longer than nine bits, then that entry in the table points
104to another similar table for the remaining bits. Again, there are duplicated 104to another similar table for the remaining bits. Again, there are duplicated
105entries as needed. The idea is that most of the time the symbol will be short 105entries as needed. The idea is that most of the time the symbol will be short
106and there will only be one table look up. (That's whole idea behind data 106and there will only be one table look up. (That's whole idea behind data
107compression in the first place.) For the less frequent long symbols, there 107compression in the first place.) For the less frequent long symbols, there
108will be two lookups. If you had a compression method with really long 108will be two lookups. If you had a compression method with really long
109symbols, you could have as many levels of lookups as is efficient. For 109symbols, you could have as many levels of lookups as is efficient. For
110inflate, two is enough. 110inflate, two is enough.
111 111
112So a table entry either points to another table (in which case nine bits in 112So a table entry either points to another table (in which case nine bits in
113the above example are gobbled), or it contains the translation for the symbol 113the above example are gobbled), or it contains the translation for the symbol
114and the number of bits to gobble. Then you start again with the next 114and the number of bits to gobble. Then you start again with the next
115ungobbled bit. 115ungobbled bit.
116 116
117You may wonder: why not just have one lookup table for how ever many bits the 117You may wonder: why not just have one lookup table for how ever many bits the
118longest symbol is? The reason is that if you do that, you end up spending 118longest symbol is? The reason is that if you do that, you end up spending
119more time filling in duplicate symbol entries than you do actually decoding. 119more time filling in duplicate symbol entries than you do actually decoding.
120At least for deflate's output that generates new trees every several 10's of 120At least for deflate's output that generates new trees every several 10's of
121kbytes. You can imagine that filling in a 2^15 entry table for a 15-bit code 121kbytes. You can imagine that filling in a 2^15 entry table for a 15-bit code
122would take too long if you're only decoding several thousand symbols. At the 122would take too long if you're only decoding several thousand symbols. At the
123other extreme, you could make a new table for every bit in the code. In fact, 123other extreme, you could make a new table for every bit in the code. In fact,
124that's essentially a Huffman tree. But then you spend two much time 124that's essentially a Huffman tree. But then you spend two much time
125traversing the tree while decoding, even for short symbols. 125traversing the tree while decoding, even for short symbols.
126 126
127So the number of bits for the first lookup table is a trade of the time to 127So the number of bits for the first lookup table is a trade of the time to
128fill out the table vs. the time spent looking at the second level and above of 128fill out the table vs. the time spent looking at the second level and above of
129the table. 129the table.
130 130
131Here is an example, scaled down: 131Here is an example, scaled down:
132 132
133The code being decoded, with 10 symbols, from 1 to 6 bits long: 133The code being decoded, with 10 symbols, from 1 to 6 bits long:
134 134
135A: 0 135A: 0
136B: 10 136B: 10
137C: 1100 137C: 1100
138D: 11010 138D: 11010
139E: 11011 139E: 11011
140F: 11100 140F: 11100
141G: 11101 141G: 11101
142H: 11110 142H: 11110
143I: 111110 143I: 111110
144J: 111111 144J: 111111
145 145
146Let's make the first table three bits long (eight entries): 146Let's make the first table three bits long (eight entries):
147 147
148000: A,1 148000: A,1
149001: A,1 149001: A,1
150010: A,1 150010: A,1
151011: A,1 151011: A,1
152100: B,2 152100: B,2
153101: B,2 153101: B,2
154110: -> table X (gobble 3 bits) 154110: -> table X (gobble 3 bits)
155111: -> table Y (gobble 3 bits) 155111: -> table Y (gobble 3 bits)
156 156
157Each entry is what the bits decode as and how many bits that is, i.e. how 157Each entry is what the bits decode as and how many bits that is, i.e. how
158many bits to gobble. Or the entry points to another table, with the number of 158many bits to gobble. Or the entry points to another table, with the number of
159bits to gobble implicit in the size of the table. 159bits to gobble implicit in the size of the table.
160 160
161Table X is two bits long since the longest code starting with 110 is five bits 161Table X is two bits long since the longest code starting with 110 is five bits
162long: 162long:
163 163
16400: C,1 16400: C,1
16501: C,1 16501: C,1
16610: D,2 16610: D,2
16711: E,2 16711: E,2
168 168
169Table Y is three bits long since the longest code starting with 111 is six 169Table Y is three bits long since the longest code starting with 111 is six
170bits long: 170bits long:
171 171
172000: F,2 172000: F,2
173001: F,2 173001: F,2
174010: G,2 174010: G,2
175011: G,2 175011: G,2
176100: H,2 176100: H,2
177101: H,2 177101: H,2
178110: I,3 178110: I,3
179111: J,3 179111: J,3
180 180
181So what we have here are three tables with a total of 20 entries that had to 181So what we have here are three tables with a total of 20 entries that had to
182be constructed. That's compared to 64 entries for a single table. Or 182be constructed. That's compared to 64 entries for a single table. Or
183compared to 16 entries for a Huffman tree (six two entry tables and one four 183compared to 16 entries for a Huffman tree (six two entry tables and one four
184entry table). Assuming that the code ideally represents the probability of 184entry table). Assuming that the code ideally represents the probability of
185the symbols, it takes on the average 1.25 lookups per symbol. That's compared 185the symbols, it takes on the average 1.25 lookups per symbol. That's compared
186to one lookup for the single table, or 1.66 lookups per symbol for the 186to one lookup for the single table, or 1.66 lookups per symbol for the
187Huffman tree. 187Huffman tree.
188 188
189There, I think that gives you a picture of what's going on. For inflate, the 189There, I think that gives you a picture of what's going on. For inflate, the
190meaning of a particular symbol is often more than just a letter. It can be a 190meaning of a particular symbol is often more than just a letter. It can be a
191byte (a "literal"), or it can be either a length or a distance which 191byte (a "literal"), or it can be either a length or a distance which
192indicates a base value and a number of bits to fetch after the code that is 192indicates a base value and a number of bits to fetch after the code that is
193added to the base value. Or it might be the special end-of-block code. The 193added to the base value. Or it might be the special end-of-block code. The
194data structures created in inftrees.c try to encode all that information 194data structures created in inftrees.c try to encode all that information
195compactly in the tables. 195compactly in the tables.
196 196
197 197
198Jean-loup Gailly Mark Adler 198Jean-loup Gailly Mark Adler
199jloup@gzip.org madler@alumni.caltech.edu 199jloup@gzip.org madler@alumni.caltech.edu
200 200
201 201
202References: 202References:
203 203
204[LZ77] Ziv J., Lempel A., ``A Universal Algorithm for Sequential Data 204[LZ77] Ziv J., Lempel A., ``A Universal Algorithm for Sequential Data
205Compression,'' IEEE Transactions on Information Theory, Vol. 23, No. 3, 205Compression,'' IEEE Transactions on Information Theory, Vol. 23, No. 3,
206pp. 337-343. 206pp. 337-343.
207 207
208``DEFLATE Compressed Data Format Specification'' available in 208``DEFLATE Compressed Data Format Specification'' available in
209http://www.ietf.org/rfc/rfc1951.txt 209http://www.ietf.org/rfc/rfc1951.txt