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.
VII. Imploding - Method 6
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.
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.
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.
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.
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.
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.
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:
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)
The Shannon-Fano codes can be constructed from the bit lengths
using the following algorithm:
1) Sort the Bit Lengths in ascending order, while retaining the
order of the original lengths stored in the file.
2) Generate the Shannon-Fano trees:
Code <- 0
CodeIncrement <- 0
LastBitLength <- 0
i <- number of Shannon-Fano codes - 1 (either 255 or 63)
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
