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where n(j) is the size of set s(j). n(j) can be 0, in which
case the follower set for s(j) is empty. each n(j) value is
encoded in 6 bits, followed by n(j) eight bit character values
corresponding to s(j)[0] to s(j)[n(j)-1] respectively. if
n(j) is 0, then no values for s(j) are stored, and the value
for n(j-1) immediately follows.</pre>
<pre>immediately after the follower sets, is the compressed data
stream. the compressed data stream can be interpreted for the
probabilistic decompression as follows:
</pre>
<pre>let last-character <- 0.
loop until done
if the follower set s(last-character) is empty then
read 8 bits from the input stream, and copy this
value to the output stream.
otherwise if the follower set s(last-character) is non-empty then
read 1 bit from the input stream.
if this bit is not zero then
read 8 bits from the input stream, and copy this
value to the output stream.
otherwise if this bit is zero then
read b(n(last-character)) bits from the input
stream, and assign this value to i.
copy the value of s(last-character)[i] to the
output stream.</pre>
<pre> assign the last value placed on the output stream to
last-character.
end loop
</pre>
<pre>b(n(j)) is defined as the minimal number of bits required to
encode the value n(j)-1.
</pre>
<pre>the decompressed stream from above can then be expanded to
re-create the original file as follows:
</pre>
<pre>let state <- 0.</pre>
<pre>loop until done
read 8 bits from the input stream into c.
case state of
0: if c is not equal to dle (144 decimal) then
copy c to the output stream.
otherwise if c is equal to dle then
let state <- 1.</pre>
<pre> 1: if c is non-zero then
let v <- c.
let len <- l(v)
let state <- f(len).
otherwise if c is zero then
copy the value 144 (decimal) to the output stream.
let state <- 0</pre>
<pre> 2: let len <- len + c
let state <- 3.</pre>
<pre> 3: move backwards d(v,c) bytes in the output stream
(if this position is before the start of the output
stream, then assume that all the data before the
start of the output stream is filled with zeros).
copy len+3 bytes from this position to the output stream.
let state <- 0.
end case
end loop
</pre>
<pre>the functions f,l, and d are dependent on the 'compression
factor', 1 through 4, and are defined as follows:</pre>
<pre>for compression factor 1:
l(x) equals the lower 7 bits of x.
f(x) equals 2 if x equals 127 otherwise f(x) equals 3.
d(x,y) equals the (upper 1 bit of x) * 256 + y + 1.
for compression factor 2:
l(x) equals the lower 6 bits of x.
f(x) equals 2 if x equals 63 otherwise f(x) equals 3.
d(x,y) equals the (upper 2 bits of x) * 256 + y + 1.
for compression factor 3:
l(x) equals the lower 5 bits of x.
f(x) equals 2 if x equals 31 otherwise f(x) equals 3.
d(x,y) equals the (upper 3 bits of x) * 256 + y + 1.
for compression factor 4:
l(x) equals the lower 4 bits of x.
f(x) equals 2 if x equals 15 otherwise f(x) equals 3.
d(x,y) equals the (upper 4 bits of x) * 256 + y + 1.
</pre>
<pre>imploding - method 6
--------------------</pre>
<pre>the imploding algorithm is actually a combination of two distinct
algorithms. the first algorithm compresses repeated byte
sequences using a sliding dictionary. the second algorithm is
used to compress the encoding of the sliding dictionary output,
using multiple shannon-fano trees.</pre>
<pre>the imploding algorithm can use a 4k or 8k sliding dictionary
size. the dictionary size used can be determined by bit 1 in the
general purpose flag word; a 0 bit indicates a 4k dictionary
while a 1 bit indicates an 8k dictionary.</pre>
<pre>the shannon-fano trees are stored at the start of the compressed
file. the number of trees stored is defined by bit 2 in the
general purpose flag word; a 0 bit indicates two trees stored, a
1 bit indicates three trees are stored. if 3 trees are stored,
the first shannon-fano tree represents the encoding of the
literal characters, the second tree represents the encoding of
the length information, the third represents the encoding of the
distance information. when 2 shannon-fano trees are stored, the
length tree is stored first, followed by the distance tree.</pre>
<pre>the literal shannon-fano tree, if present is used to represent
the entire ascii character set, and contains 256 values. this
tree is used to compress any data not compressed by the sliding
dictionary algorithm. when this tree is present, the minimum
match length for the sliding dictionary is 3. if this tree is
not present, the minimum match length is 2.</pre>
<pre>the length shannon-fano tree is used to compress the length part
of the (length,distance) pairs from the sliding dictionary
output. the length tree contains 64 values, ranging from the
minimum match length, to 63 plus the minimum match length.</pre>
<pre>the distance shannon-fano tree is used to compress the distance
part of the (length,distance) pairs from the sliding dictionary
output. the distance tree contains 64 values, ranging from 0 to
63, representing the upper 6 bits of the distance value. the
distance values themselves will be between 0 and the sliding
dictionary size, either 4k or 8k.</pre>
<pre>the shannon-fano trees themselves are stored in a compressed
format. the first byte of the tree data represents the number of
bytes of data representing the (compressed) shannon-fano tree
minus 1. the remaining bytes represent the shannon-fano tree
data encoded as:</pre>
<pre> high 4 bits: number of values at this bit length + 1. (1 - 16)
low 4 bits: bit length needed to represent value + 1. (1 - 16)</pre>
<pre>the shannon-fano codes can be constructed from the bit lengths
using the following algorithm:</pre>
<pre>1) sort the bit lengths in ascending order, while retaining the
order of the original lengths stored in the file.</pre>
<pre>2) generate the shannon-fano trees:</pre>
<pre> code <- 0
codeincrement <- 0
lastbitlength <- 0
i <- number of shannon-fano codes - 1 (either 255 or 63)</pre>
<pre> loop while i >= 0
code = code + codeincrement
if bitlength(i) <> lastbitlength then
lastbitlength=bitlength(i)
codeincrement = 1 shifted left (16 - lastbitlength)
shannoncode(i) = code
i <- i - 1
end loop
</pre>
<pre>3) reverse the order of all the bits in the above shannoncode()
vector, so that the most significant bit becomes the least
significant bit. for example, the value 0x1234 (hex) would
become 0x2c48 (hex).</pre>
<pre>4) restore the order of shannon-fano codes as originally stored
within the file.</pre>
<pre>example:</pre>
<pre> this example will show the encoding of a shannon-fano tree
of size 8. notice that the actual shannon-fano trees used
for imploding are either 64 or 256 entries in size.</pre>
<pre>example: 0x02, 0x42, 0x01, 0x13</pre>
<pre> the first byte indicates 3 values in this table. decoding the
bytes:
0x42 = 5 codes of 3 bits long
0x01 = 1 code of 2 bits long
0x13 = 2 codes of 4 bits long</pre>
<pre> this would generate the original bit length array of:
(3, 3, 3, 3, 3, 2, 4, 4)</pre>
<pre> there are 8 codes in this table for the values 0 thru 7. using the
algorithm to obtain the shannon-fano codes produces:</pre>
<pre> reversed order original
val sorted constructed code value restored length
--- ------ ----------------- -------- -------- ------
0: 2 1100000000000000 11 101 3
1: 3 1010000000000000 101 001 3
2: 3 1000000000000000 001 110 3
3: 3 0110000000000000 110 010 3
4: 3 0100000000000000 010 100 3
5: 3 0010000000000000 100 11 2
6: 4 0001000000000000 1000 1000 4
7: 4 0000000000000000 0000 0000 4
</pre>
<pre>the values in the val, order restored and original length columns
now represent the shannon-fano encoding tree that can be used for
decoding the shannon-fano encoded data. how to parse the
variable length shannon-fano values from the data stream is beyond the
scope of this document. (see the references listed at the end of
this document for more information.) however, traditional decoding
schemes used for huffman variable length decoding, such as the
greenlaw algorithm, can be successfully applied.</pre>
<pre>the compressed data stream begins immediately after the
compressed shannon-fano data. the compressed data stream can be
interpreted as follows:</pre>
<pre>loop until done
read 1 bit from input stream.</pre>
<pre> if this bit is non-zero then (encoded data is literal data)
if literal shannon-fano tree is present
read and decode character using literal shannon-fano tree.
otherwise
read 8 bits from input stream.
copy character to the output stream.
otherwise (encoded data is sliding dictionary match)
if 8k dictionary size
read 7 bits for offset distance (lower 7 bits of offset).
otherwise
read 6 bits for offset distance (lower 6 bits of offset).</pre>
<pre> using the distance shannon-fano tree, read and decode the
upper 6 bits of the distance value.</pre>
<pre> using the length shannon-fano tree, read and decode
the length value.</pre>
<pre> length <- length + minimum match length</pre>
<pre> if length = 63 + minimum match length
read 8 bits from the input stream,
add this value to length.</pre>
<pre> move backwards distance+1 bytes in the output stream, and
copy length characters from this position to the output
stream. (if this position is before the start of the output
stream, then assume that all the data before the start of
the output stream is filled with zeros).
end loop</pre>
<pre>tokenizing - method 7
--------------------</pre>
<pre>this method is not used by pkzip.</pre>
<pre>deflating - method 8
-----------------</pre>
<pre>the deflate algorithm is similar to the implode algorithm using
a sliding dictionary of up to 32k with secondary compression
from huffman/shannon-fano codes.</pre>
<pre>the compressed data is stored in blocks with a header describing
the block and the huffman codes used in the data block. the header
format is as follows:</pre>
<pre> bit 0: last block bit this bit is set to 1 if this is the last
compressed block in the data.
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