TripleDNN

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1. Compression algorithm (deflate)

The deflation algorithm used by gzip (also zip and zlib) is a variation of
LZ77 (Lempel-Ziv 1977, see reference below). It finds duplicated strings in
the input data.  The second occurrence of a string is replaced by a
pointer to the previous string, in the form of a pair (distance,
length).  Distances are limited to 32K bytes, and lengths are limited
to 258 bytes. When a string does not occur anywhere in the previous
32K bytes, it is emitted as a sequence of literal bytes.  (In this
description, `string' must be taken as an arbitrary sequence of bytes,
and is not restricted to printable characters.)

Literals or match lengths are compressed with one Huffman tree, and
match distances are compressed with another tree. The trees are stored
in a compact form at the start of each block. The blocks can have any
size (except that the compressed data for one block must fit in
available memory). A block is terminated when deflate() determines that
it would be useful to start another block with fresh trees. (This is
somewhat similar to the behavior of LZW-based _compress_.)

Duplicated strings are found using a hash table. All input strings of
length 3 are inserted in the hash table. A hash index is computed for
the next 3 bytes. If the hash chain for this index is not empty, all
strings in the chain are compared with the current input string, and
the longest match is selected.

The hash chains are searched starting with the most recent strings, to
favor small distances and thus take advantage of the Huffman encoding.
The hash chains are singly linked. There are no deletions from the
hash chains, the algorithm simply discards matches that are too old.

To avoid a worst-case situation, very long hash chains are arbitrarily
truncated at a certain length, determined by a runtime option (level
parameter of deflateInit). So deflate() does not always find the longest
possible match but generally finds a match which is long enough.

deflate() also defers the selection of matches with a lazy evaluation
mechanism. After a match of length N has been found, deflate() searches for
a longer match at the next input byte. If a longer match is found, the
previous match is truncated to a length of one (thus producing a single
literal byte) and the process of lazy evaluation begins again. Otherwise,
the original match is kept, and the next match search is attempted only N
steps later.

The lazy match evaluation is also subject to a runtime parameter. If
the current match is long enough, deflate() reduces the search for a longer
match, thus speeding up the whole process. If compression ratio is more
important than speed, deflate() attempts a complete second search even if
the first match is already long enough.

The lazy match evaluation is not performed for the fastest compression
modes (level parameter 1 to 3). For these fast modes, new strings
are inserted in the hash table only when no match was found, or
when the match is not too long. This degrades the compression ratio
but saves time since there are both fewer insertions and fewer searches.

2. Decompression algorithm (inflate)

2.1 Introduction

The key question is how to represent a Huffman code (or any prefix code) so
that you can decode fast.  The most important characteristic is that shorter
codes are much more common than longer codes, so pay attention to decoding the
short codes fast, and let the long codes take longer to decode.

inflate() sets up a first level table that covers some number of bits of
input less than the length of longest code.  It gets that many bits from the
stream, and looks it up in the table.  The table will tell if the next
code is that many bits or less and how many, and if it is, it will tell
the value, else it will point to the next level table for which inflate()
grabs more bits and tries to decode a longer code.

How many bits to make the first lookup is a tradeoff between the time it
takes to decode and the time it takes to build the table.  If building the
table took no time (and if you had infinite memory), then there would only
be a first level table to cover all the way to the longest code.  However,
building the table ends up taking a lot longer for more bits since short
codes are replicated many times in such a table.  What inflate() does is
simply to make the number of bits in the first table a variable, and  then
to set that variable for the maximum speed.

For inflate, which has 286 possible codes for the literal/length tree, the size
of the first table is nine bits.  Also the distance trees have 30 possible
values, and the size of the first table is six bits.  Note that for each of
those cases, the table ended up one bit longer than the ``average'' code
length, i.e. the code length of an approximately flat code which would be a
little more than eight bits for 286 symbols and a little less than five bits
for 30 symbols.

2.2 More details on the inflate table lookup

Ok, you want to know what this cleverly obfuscated inflate tree actually
looks like.  You are correct that it's not a Huffman tree.  It is simply a
lookup table for the first, let's say, nine bits of a Huffman symbol.  The
symbol could be as short as one bit or as long as 15 bits.  If a particular
symbol is shorter than nine bits, then that symbol's translation is duplicated
in all those entries that start with that symbol's bits.  For example, if the
symbol is four bits, then it's duplicated 32 times in a nine-bit table.  If a
symbol is nine bits long, it appears in the table once.

If the symbol is longer than nine bits, then that entry in the table points
to another similar table for the remaining bits.  Again, there are duplicated
entries as needed.  The idea is that most of the time the symbol will be short
and there will only be one table look up.  (That's whole idea behind data
compression in the first place.)  For the less frequent long symbols, there
will be two lookups.  If you had a compression method with really long
symbols, you could have as many levels of lookups as is efficient.  For
inflate, two is enough.

So a table entry either points to another table (in which case nine bits in
the above example are gobbled), or it contains the translation for the symbol
and the number of bits to gobble.  Then you start again with the next
ungobbled bit.

You may wonder: why not just have one lookup table for how ever many bits the
longest symbol is?  The reason is that if you do that, you end up spending
more time filling in duplicate symbol entries than you do actually decoding.
At least for deflate's output that generates new trees every several 10's of
kbytes.  You can imagine that filling in a 2^15 entry table for a 15-bit code
would take too long if you're only decoding several thousand symbols.  At the
other extreme, you could make a new table for every bit in the code.  In fact,
that's essentially a Huffman tree.  But then you spend too much time
traversing the tree while decoding, even for short symbols.

So the number of bits for the first lookup table is a trade of the time to
fill out the table vs. the time spent looking at the second level and above of
the table.

Here is an example, scaled down:

The code being decoded, with 10 symbols, from 1 to 6 bits long:

A: 0
B: 10
C: 1100
D: 11010
E: 11011
F: 11100
G: 11101
H: 11110
I: 111110
J: 111111

Let's make the first table three bits long (eight entries):

000: A,1
001: A,1
010: A,1
011: A,1
100: B,2
101: B,2
110: -> table X (gobble 3 bits)
111: -> table Y (gobble 3 bits)

Each entry is what the bits decode as and how many bits that is, i.e. how
many bits to gobble.  Or the entry points to another table, with the number of
bits to gobble implicit in the size of the table.

Table X is two bits long since the longest code starting with 110 is five bits
long:

00: C,1
01: C,1
10: D,2
11: E,2

Table Y is three bits long since the longest code starting with 111 is six
bits long:

000: F,2
001: F,2
010: G,2
011: G,2
100: H,2
101: H,2
110: I,3
111: J,3

So what we have here are three tables with a total of 20 entries that had to
be constructed.  That's compared to 64 entries for a single table.  Or
compared to 16 entries for a Huffman tree (six two entry tables and one four
entry table).  Assuming that the code ideally represents the probability of
the symbols, it takes on the average 1.25 lookups per symbol.  That's compared
to one lookup for the single table, or 1.66 lookups per symbol for the
Huffman tree.

There, I think that gives you a picture of what's going on.  For inflate, the
meaning of a particular symbol is often more than just a letter.  It can be a
byte (a "literal"), or it can be either a length or a distance which
indicates a base value and a number of bits to fetch after the code that is
added to the base value.  Or it might be the special end-of-block code.  The
data structures created in inftrees.c try to encode all that information
compactly in the tables.

Jean-loup Gailly        Mark Adler
jloup@gzip.org          madler@alumni.caltech.edu

References:

[LZ77] Ziv J., Lempel A., ``A Universal Algorithm for Sequential Data
Compression,'' IEEE Transactions on Information Theory, Vol. 23, No. 3,
pp. 337-343.

``DEFLATE Compressed Data Format Specification'' available in
http://tools.ietf.org/html/rfc1951