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    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 two 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@gzip.org          madler@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://www.ietf.org/rfc/rfc1951.txt