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 (a] gzip.org madler (a] 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 210
