1 1. Compression algorithm (deflate)

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.)

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_.)

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.

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.

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.

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.

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.

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.

58 2. Decompression algorithm (inflate)

60 2.1 Introduction

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.

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.

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.

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.

92 2.2 More details on the inflate table lookup

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.

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.

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.

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.

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.

131 Here is an example, scaled down:

133 The code being decoded, with 10 symbols, from 1 to 6 bits long:

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

146 Let's make the first table three bits long (eight entries):

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)

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.

161 Table X is two bits long since the longest code starting with 110 is five bits

162 long:

164 00: C,1

165 01: C,1

166 10: D,2

167 11: E,2

169 Table Y is three bits long since the longest code starting with 111 is six

170 bits long:

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

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.

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.

198 Jean-loup Gailly Mark Adler

199 jloup@gzip.org madler@alumni.caltech.edu

202 References:

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.

208 ``DEFLATE Compressed Data Format Specification'' available in

209 http://www.ietf.org/rfc/rfc1951.txt