11. Compression algorithm (deflate)
  2
  3The deflation algorithm used by gzip (also zip and zlib) is a variation of
  4LZ77 (Lempel-Ziv 1977, see reference below). It finds duplicated strings in
  5the input data.  The second occurrence of a string is replaced by a
  6pointer to the previous string, in the form of a pair (distance,
  7length).  Distances are limited to 32K bytes, and lengths are limited
  8to 258 bytes. When a string does not occur anywhere in the previous
  932K bytes, it is emitted as a sequence of literal bytes.  (In this
 10description, `string' must be taken as an arbitrary sequence of bytes,
 11and is not restricted to printable characters.)
 12
 13Literals or match lengths are compressed with one Huffman tree, and
 14match distances are compressed with another tree. The trees are stored
 15in a compact form at the start of each block. The blocks can have any
 16size (except that the compressed data for one block must fit in
 17available memory). A block is terminated when deflate() determines that
 18it would be useful to start another block with fresh trees. (This is
 19somewhat similar to the behavior of LZW-based _compress_.)
 20
 21Duplicated strings are found using a hash table. All input strings of
 22length 3 are inserted in the hash table. A hash index is computed for
 23the next 3 bytes. If the hash chain for this index is not empty, all
 24strings in the chain are compared with the current input string, and
 25the longest match is selected.
 26
 27The hash chains are searched starting with the most recent strings, to
 28favor small distances and thus take advantage of the Huffman encoding.
 29The hash chains are singly linked. There are no deletions from the
 30hash chains, the algorithm simply discards matches that are too old.
 31
 32To avoid a worst-case situation, very long hash chains are arbitrarily
 33truncated at a certain length, determined by a runtime option (level
 34parameter of deflateInit). So deflate() does not always find the longest
 35possible match but generally finds a match which is long enough.
 36
 37deflate() also defers the selection of matches with a lazy evaluation
 38mechanism. After a match of length N has been found, deflate() searches for
 39a longer match at the next input byte. If a longer match is found, the
 40previous match is truncated to a length of one (thus producing a single
 41literal byte) and the process of lazy evaluation begins again. Otherwise,
 42the original match is kept, and the next match search is attempted only N
 43steps later.
 44
 45The lazy match evaluation is also subject to a runtime parameter. If
 46the current match is long enough, deflate() reduces the search for a longer
 47match, thus speeding up the whole process. If compression ratio is more
 48important than speed, deflate() attempts a complete second search even if
 49the first match is already long enough.
 50
 51The lazy match evaluation is not performed for the fastest compression
 52modes (level parameter 1 to 3). For these fast modes, new strings
 53are inserted in the hash table only when no match was found, or
 54when the match is not too long. This degrades the compression ratio
 55but saves time since there are both fewer insertions and fewer searches.
 56
 57
 582. Decompression algorithm (inflate)
 59
 602.1 Introduction
 61
 62The key question is how to represent a Huffman code (or any prefix code) so
 63that you can decode fast.  The most important characteristic is that shorter
 64codes are much more common than longer codes, so pay attention to decoding the
 65short codes fast, and let the long codes take longer to decode.
 66
 67inflate() sets up a first level table that covers some number of bits of
 68input less than the length of longest code.  It gets that many bits from the
 69stream, and looks it up in the table.  The table will tell if the next
 70code is that many bits or less and how many, and if it is, it will tell
 71the value, else it will point to the next level table for which inflate()
 72grabs more bits and tries to decode a longer code.
 73
 74How many bits to make the first lookup is a tradeoff between the time it
 75takes to decode and the time it takes to build the table.  If building the
 76table took no time (and if you had infinite memory), then there would only
 77be a first level table to cover all the way to the longest code.  However,
 78building the table ends up taking a lot longer for more bits since short
 79codes are replicated many times in such a table.  What inflate() does is
 80simply to make the number of bits in the first table a variable, and  then
 81to set that variable for the maximum speed.
 82
 83For inflate, which has 286 possible codes for the literal/length tree, the size
 84of the first table is nine bits.  Also the distance trees have 30 possible
 85values, and the size of the first table is six bits.  Note that for each of
 86those cases, the table ended up one bit longer than the ``average'' code
 87length, i.e. the code length of an approximately flat code which would be a
 88little more than eight bits for 286 symbols and a little less than five bits
 89for 30 symbols.
 90
 91
 922.2 More details on the inflate table lookup
 93
 94Ok, you want to know what this cleverly obfuscated inflate tree actually
 95looks like.  You are correct that it's not a Huffman tree.  It is simply a
 96lookup table for the first, let's say, nine bits of a Huffman symbol.  The
 97symbol could be as short as one bit or as long as 15 bits.  If a particular
 98symbol is shorter than nine bits, then that symbol's translation is duplicated
 99in all those entries that start with that symbol's bits.  For example, if the
100symbol is four bits, then it's duplicated 32 times in a nine-bit table.  If a
101symbol is nine bits long, it appears in the table once.
102
103If the symbol is longer than nine bits, then that entry in the table points
104to another similar table for the remaining bits.  Again, there are duplicated
105entries as needed.  The idea is that most of the time the symbol will be short
106and there will only be one table look up.  (That's whole idea behind data
107compression in the first place.)  For the less frequent long symbols, there
108will be two lookups.  If you had a compression method with really long
109symbols, you could have as many levels of lookups as is efficient.  For
110inflate, two is enough.
111
112So a table entry either points to another table (in which case nine bits in
113the above example are gobbled), or it contains the translation for the symbol
114and the number of bits to gobble.  Then you start again with the next
115ungobbled bit.
116
117You may wonder: why not just have one lookup table for how ever many bits the
118longest symbol is?  The reason is that if you do that, you end up spending
119more time filling in duplicate symbol entries than you do actually decoding.
120At least for deflate's output that generates new trees every several 10's of
121kbytes.  You can imagine that filling in a 2^15 entry table for a 15-bit code
122would take too long if you're only decoding several thousand symbols.  At the
123other extreme, you could make a new table for every bit in the code.  In fact,
124that's essentially a Huffman tree.  But then you spend too much time
125traversing the tree while decoding, even for short symbols.
126
127So the number of bits for the first lookup table is a trade of the time to
128fill out the table vs. the time spent looking at the second level and above of
129the table.
130
131Here is an example, scaled down:
132
133The code being decoded, with 10 symbols, from 1 to 6 bits long:
134
135A: 0
136B: 10
137C: 1100
138D: 11010
139E: 11011
140F: 11100
141G: 11101
142H: 11110
143I: 111110
144J: 111111
145
146Let's make the first table three bits long (eight entries):
147
148000: A,1
149001: A,1
150010: A,1
151011: A,1
152100: B,2
153101: B,2
154110: -> table X (gobble 3 bits)
155111: -> table Y (gobble 3 bits)
156
157Each entry is what the bits decode as and how many bits that is, i.e. how
158many bits to gobble.  Or the entry points to another table, with the number of
159bits to gobble implicit in the size of the table.
160
161Table X is two bits long since the longest code starting with 110 is five bits
162long:
163
16400: C,1
16501: C,1
16610: D,2
16711: E,2
168
169Table Y is three bits long since the longest code starting with 111 is six
170bits long:
171
172000: F,2
173001: F,2
174010: G,2
175011: G,2
176100: H,2
177101: H,2
178110: I,3
179111: J,3
180
181So what we have here are three tables with a total of 20 entries that had to
182be constructed.  That's compared to 64 entries for a single table.  Or
183compared to 16 entries for a Huffman tree (six two entry tables and one four
184entry table).  Assuming that the code ideally represents the probability of
185the symbols, it takes on the average 1.25 lookups per symbol.  That's compared
186to one lookup for the single table, or 1.66 lookups per symbol for the
187Huffman tree.
188
189There, I think that gives you a picture of what's going on.  For inflate, the
190meaning of a particular symbol is often more than just a letter.  It can be a
191byte (a "literal"), or it can be either a length or a distance which
192indicates a base value and a number of bits to fetch after the code that is
193added to the base value.  Or it might be the special end-of-block code.  The
194data structures created in inftrees.c try to encode all that information
195compactly in the tables.
196
197
198Jean-loup Gailly        Mark Adler
199jloup@gzip.org          madler@alumni.caltech.edu
200
201
202References:
203
204[LZ77] Ziv J., Lempel A., ``A Universal Algorithm for Sequential Data
205Compression,'' IEEE Transactions on Information Theory, Vol. 23, No. 3,
206pp. 337-343.
207
208``DEFLATE Compressed Data Format Specification'' available in
209http://tools.ietf.org/html/rfc1951