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annotate mupdf-source/thirdparty/zlib/doc/algorithm.txt @ 46:7ee69f120f19 default tip

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>>>>> tag v1.26.5+1 for changeset b74429b0f5c4
author Franz Glasner <fzglas.hg@dom66.de>
date Sat, 11 Oct 2025 17:17:30 +0200
parents b50eed0cc0ef
children
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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1 1. Compression algorithm (deflate)
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2
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3 The deflation algorithm used by gzip (also zip and zlib) is a variation of
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4 LZ77 (Lempel-Ziv 1977, see reference below). It finds duplicated strings in
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5 the input data. The second occurrence of a string is replaced by a
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6 pointer to the previous string, in the form of a pair (distance,
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7 length). Distances are limited to 32K bytes, and lengths are limited
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8 to 258 bytes. When a string does not occur anywhere in the previous
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9 32K bytes, it is emitted as a sequence of literal bytes. (In this
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10 description, `string' must be taken as an arbitrary sequence of bytes,
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11 and is not restricted to printable characters.)
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12
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13 Literals or match lengths are compressed with one Huffman tree, and
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14 match distances are compressed with another tree. The trees are stored
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15 in a compact form at the start of each block. The blocks can have any
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16 size (except that the compressed data for one block must fit in
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17 available memory). A block is terminated when deflate() determines that
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18 it would be useful to start another block with fresh trees. (This is
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19 somewhat similar to the behavior of LZW-based _compress_.)
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21 Duplicated strings are found using a hash table. All input strings of
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22 length 3 are inserted in the hash table. A hash index is computed for
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23 the next 3 bytes. If the hash chain for this index is not empty, all
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24 strings in the chain are compared with the current input string, and
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25 the longest match is selected.
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27 The hash chains are searched starting with the most recent strings, to
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28 favor small distances and thus take advantage of the Huffman encoding.
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29 The hash chains are singly linked. There are no deletions from the
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30 hash chains, the algorithm simply discards matches that are too old.
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32 To avoid a worst-case situation, very long hash chains are arbitrarily
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33 truncated at a certain length, determined by a runtime option (level
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34 parameter of deflateInit). So deflate() does not always find the longest
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35 possible match but generally finds a match which is long enough.
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37 deflate() also defers the selection of matches with a lazy evaluation
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38 mechanism. After a match of length N has been found, deflate() searches for
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39 a longer match at the next input byte. If a longer match is found, the
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40 previous match is truncated to a length of one (thus producing a single
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41 literal byte) and the process of lazy evaluation begins again. Otherwise,
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42 the original match is kept, and the next match search is attempted only N
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43 steps later.
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45 The lazy match evaluation is also subject to a runtime parameter. If
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46 the current match is long enough, deflate() reduces the search for a longer
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47 match, thus speeding up the whole process. If compression ratio is more
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48 important than speed, deflate() attempts a complete second search even if
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49 the first match is already long enough.
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51 The lazy match evaluation is not performed for the fastest compression
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52 modes (level parameter 1 to 3). For these fast modes, new strings
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53 are inserted in the hash table only when no match was found, or
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54 when the match is not too long. This degrades the compression ratio
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55 but saves time since there are both fewer insertions and fewer searches.
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58 2. Decompression algorithm (inflate)
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60 2.1 Introduction
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62 The key question is how to represent a Huffman code (or any prefix code) so
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63 that you can decode fast. The most important characteristic is that shorter
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64 codes are much more common than longer codes, so pay attention to decoding the
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65 short codes fast, and let the long codes take longer to decode.
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67 inflate() sets up a first level table that covers some number of bits of
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68 input less than the length of longest code. It gets that many bits from the
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69 stream, and looks it up in the table. The table will tell if the next
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70 code is that many bits or less and how many, and if it is, it will tell
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71 the value, else it will point to the next level table for which inflate()
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72 grabs more bits and tries to decode a longer code.
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74 How many bits to make the first lookup is a tradeoff between the time it
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75 takes to decode and the time it takes to build the table. If building the
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76 table took no time (and if you had infinite memory), then there would only
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77 be a first level table to cover all the way to the longest code. However,
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78 building the table ends up taking a lot longer for more bits since short
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79 codes are replicated many times in such a table. What inflate() does is
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80 simply to make the number of bits in the first table a variable, and then
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81 to set that variable for the maximum speed.
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83 For inflate, which has 286 possible codes for the literal/length tree, the size
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84 of the first table is nine bits. Also the distance trees have 30 possible
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85 values, and the size of the first table is six bits. Note that for each of
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86 those cases, the table ended up one bit longer than the ``average'' code
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87 length, i.e. the code length of an approximately flat code which would be a
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88 little more than eight bits for 286 symbols and a little less than five bits
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89 for 30 symbols.
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90
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92 2.2 More details on the inflate table lookup
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93
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94 Ok, you want to know what this cleverly obfuscated inflate tree actually
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95 looks like. You are correct that it's not a Huffman tree. It is simply a
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96 lookup table for the first, let's say, nine bits of a Huffman symbol. The
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97 symbol could be as short as one bit or as long as 15 bits. If a particular
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98 symbol is shorter than nine bits, then that symbol's translation is duplicated
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99 in all those entries that start with that symbol's bits. For example, if the
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100 symbol is four bits, then it's duplicated 32 times in a nine-bit table. If a
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101 symbol is nine bits long, it appears in the table once.
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102
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103 If the symbol is longer than nine bits, then that entry in the table points
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104 to another similar table for the remaining bits. Again, there are duplicated
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105 entries as needed. The idea is that most of the time the symbol will be short
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106 and there will only be one table look up. (That's whole idea behind data
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107 compression in the first place.) For the less frequent long symbols, there
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108 will be two lookups. If you had a compression method with really long
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109 symbols, you could have as many levels of lookups as is efficient. For
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110 inflate, two is enough.
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111
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112 So a table entry either points to another table (in which case nine bits in
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113 the above example are gobbled), or it contains the translation for the symbol
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114 and the number of bits to gobble. Then you start again with the next
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115 ungobbled bit.
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116
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117 You may wonder: why not just have one lookup table for how ever many bits the
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118 longest symbol is? The reason is that if you do that, you end up spending
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119 more time filling in duplicate symbol entries than you do actually decoding.
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120 At least for deflate's output that generates new trees every several 10's of
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121 kbytes. You can imagine that filling in a 2^15 entry table for a 15-bit code
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122 would take too long if you're only decoding several thousand symbols. At the
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123 other extreme, you could make a new table for every bit in the code. In fact,
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124 that's essentially a Huffman tree. But then you spend too much time
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125 traversing the tree while decoding, even for short symbols.
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126
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127 So the number of bits for the first lookup table is a trade of the time to
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128 fill out the table vs. the time spent looking at the second level and above of
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129 the table.
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130
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131 Here is an example, scaled down:
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132
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133 The code being decoded, with 10 symbols, from 1 to 6 bits long:
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134
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135 A: 0
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136 B: 10
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137 C: 1100
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138 D: 11010
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139 E: 11011
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140 F: 11100
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141 G: 11101
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142 H: 11110
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143 I: 111110
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144 J: 111111
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145
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146 Let's make the first table three bits long (eight entries):
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147
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148 000: A,1
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149 001: A,1
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150 010: A,1
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151 011: A,1
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152 100: B,2
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153 101: B,2
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154 110: -> table X (gobble 3 bits)
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155 111: -> table Y (gobble 3 bits)
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156
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157 Each entry is what the bits decode as and how many bits that is, i.e. how
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Franz Glasner <fzglas.hg@dom66.de>
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158 many bits to gobble. Or the entry points to another table, with the number of
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159 bits to gobble implicit in the size of the table.
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160
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161 Table X is two bits long since the longest code starting with 110 is five bits
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162 long:
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163
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164 00: C,1
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165 01: C,1
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166 10: D,2
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167 11: E,2
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168
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169 Table Y is three bits long since the longest code starting with 111 is six
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170 bits long:
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171
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172 000: F,2
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173 001: F,2
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174 010: G,2
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175 011: G,2
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176 100: H,2
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177 101: H,2
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178 110: I,3
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179 111: J,3
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180
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181 So what we have here are three tables with a total of 20 entries that had to
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Franz Glasner <fzglas.hg@dom66.de>
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182 be constructed. That's compared to 64 entries for a single table. Or
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183 compared to 16 entries for a Huffman tree (six two entry tables and one four
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184 entry table). Assuming that the code ideally represents the probability of
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185 the symbols, it takes on the average 1.25 lookups per symbol. That's compared
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186 to one lookup for the single table, or 1.66 lookups per symbol for the
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Franz Glasner <fzglas.hg@dom66.de>
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187 Huffman tree.
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188
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189 There, I think that gives you a picture of what's going on. For inflate, the
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190 meaning of a particular symbol is often more than just a letter. It can be a
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191 byte (a "literal"), or it can be either a length or a distance which
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Franz Glasner <fzglas.hg@dom66.de>
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192 indicates a base value and a number of bits to fetch after the code that is
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Franz Glasner <fzglas.hg@dom66.de>
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193 added to the base value. Or it might be the special end-of-block code. The
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Franz Glasner <fzglas.hg@dom66.de>
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194 data structures created in inftrees.c try to encode all that information
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Franz Glasner <fzglas.hg@dom66.de>
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195 compactly in the tables.
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196
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198 Jean-loup Gailly Mark Adler
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199 jloup@gzip.org madler@alumni.caltech.edu
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200
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202 References:
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203
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204 [LZ77] Ziv J., Lempel A., ``A Universal Algorithm for Sequential Data
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205 Compression,'' IEEE Transactions on Information Theory, Vol. 23, No. 3,
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206 pp. 337-343.
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207
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208 ``DEFLATE Compressed Data Format Specification'' available in
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209 http://tools.ietf.org/html/rfc1951