src/zlib/algorithm.txt

   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 real question is, given a Huffman tree, how to decode fast.  The most
  63 important realization is that shorter codes are much more common than
  64 longer codes, so pay attention to decoding the short codes fast, and let
  65 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 set it
  81 for the maximum speed.
  82
  83 inflate() sends new trees relatively often, so it is possibly set for a
  84 smaller first level table than an application that has only one tree for
  85 all the data.  For inflate, which has 286 possible codes for the
  86 literal/length tree, the size of the first table is nine bits.  Also the
  87 distance trees have 30 possible values, and the size of the first table is
  88 six bits.  Note that for each of those cases, the table ended up one bit
  89 longer than the ``average'' code length, i.e. the code length of an
  90 approximately flat code which would be a little more than eight bits for
  91 286 symbols and a little less than five bits for 30 symbols.  It would be
  92 interesting to see if optimizing the first level table for other
  93 applications gave values within a bit or two of the flat code size.
  94
  95
  96 2.2 More details on the inflate table lookup
  97
  98 Ok, you want to know what this cleverly obfuscated inflate tree actually
  99 looks like.  You are correct that it's not a Huffman tree.  It is simply a
 100 lookup table for the first, let's say, nine bits of a Huffman symbol.  The
 101 symbol could be as short as one bit or as long as 15 bits.  If a particular
 102 symbol is shorter than nine bits, then that symbol's translation is duplicated
 103 in all those entries that start with that symbol's bits.  For example, if the
 104 symbol is four bits, then it's duplicated 32 times in a nine-bit table.  If a
 105 symbol is nine bits long, it appears in the table once.
 106
 107 If the symbol is longer than nine bits, then that entry in the table points
 108 to another similar table for the remaining bits.  Again, there are duplicated
 109 entries as needed.  The idea is that most of the time the symbol will be short
 110 and there will only be one table look up.  (That's whole idea behind data
 111 compression in the first place.)  For the less frequent long symbols, there
 112 will be two lookups.  If you had a compression method with really long
 113 symbols, you could have as many levels of lookups as is efficient.  For
 114 inflate, two is enough.
 115
 116 So a table entry either points to another table (in which case nine bits in
 117 the above example are gobbled), or it contains the translation for the symbol
 118 and the number of bits to gobble.  Then you start again with the next
 119 ungobbled bit.
 120
 121 You may wonder: why not just have one lookup table for how ever many bits the
 122 longest symbol is?  The reason is that if you do that, you end up spending
 123 more time filling in duplicate symbol entries than you do actually decoding.
 124 At least for deflate's output that generates new trees every several 10's of
 125 kbytes.  You can imagine that filling in a 2^15 entry table for a 15-bit code
 126 would take too long if you're only decoding several thousand symbols.  At the
 127 other extreme, you could make a new table for every bit in the code.  In fact,
 128 that's essentially a Huffman tree.  But then you spend two much time
 129 traversing the tree while decoding, even for short symbols.
 130
 131 So the number of bits for the first lookup table is a trade of the time to
 132 fill out the table vs. the time spent looking at the second level and above of
 133 the table.
 134
 135 Here is an example, scaled down:
 136
 137 The code being decoded, with 10 symbols, from 1 to 6 bits long:
 138
 139 A: 0
 140 B: 10
 141 C: 1100
 142 D: 11010
 143 E: 11011
 144 F: 11100
 145 G: 11101
 146 H: 11110
 147 I: 111110
 148 J: 111111
 149
 150 Let's make the first table three bits long (eight entries):
 151
 152 000: A,1
 153 001: A,1
 154 010: A,1
 155 011: A,1
 156 100: B,2
 157 101: B,2
 158 110: -> table X (gobble 3 bits)
 159 111: -> table Y (gobble 3 bits)
 160
 161 Each entry is what the bits decode to and how many bits that is, i.e. how
 162 many bits to gobble.  Or the entry points to another table, with the number of
 163 bits to gobble implicit in the size of the table.
 164
 165 Table X is two bits long since the longest code starting with 110 is five bits
 166 long:
 167
 168 00: C,1
 169 01: C,1
 170 10: D,2
 171 11: E,2
 172
 173 Table Y is three bits long since the longest code starting with 111 is six
 174 bits long:
 175
 176 000: F,2
 177 001: F,2
 178 010: G,2
 179 011: G,2
 180 100: H,2
 181 101: H,2
 182 110: I,3
 183 111: J,3
 184
 185 So what we have here are three tables with a total of 20 entries that had to
 186 be constructed.  That's compared to 64 entries for a single table.  Or
 187 compared to 16 entries for a Huffman tree (six two entry tables and one four
 188 entry table).  Assuming that the code ideally represents the probability of
 189 the symbols, it takes on the average 1.25 lookups per symbol.  That's compared
 190 to one lookup for the single table, or 1.66 lookups per symbol for the
 191 Huffman tree.
 192
 193 There, I think that gives you a picture of what's going on.  For inflate, the
 194 meaning of a particular symbol is often more than just a letter.  It can be a
 195 byte (a "literal"), or it can be either a length or a distance which
 196 indicates a base value and a number of bits to fetch after the code that is
 197 added to the base value.  Or it might be the special end-of-block code.  The
 198 data structures created in inftrees.c try to encode all that information
 199 compactly in the tables.
 200
 201
 202 Jean-loup Gailly        Mark Adler
 203 jloup@gzip.org          madler@alumni.caltech.edu
 204
 205
 206 References:
 207
 208 [LZ77] Ziv J., Lempel A., ``A Universal Algorithm for Sequential Data
 209 Compression,'' IEEE Transactions on Information Theory, Vol. 23, No. 3,
 210 pp. 337-343.
 211
 212 ``DEFLATE Compressed Data Format Specification'' available in
 213 ftp://ds.internic.net/rfc/rfc1951.txt