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