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r 2 111 8 * 2 = 16 3 * 2 = 6
c 1 1100 8 * 1 = 8 4 * 1 = 4
d 1 1101 8 * 1 = 8 4 * 1 = 4
all others 0 ---------- ----------
totals: 88 23
We could represent this information as a binary tree:
c
/
a b /---- d
/ / /
root --- --- --- r
To get the Huffman code we code a 0 bit each time we
traverse a branch to the left, and a 1 bit each time we
traverse a branch to the right. Thus the codes are generated
as in the table above and every character gets a unique
code. The decompressor simply starts at the root, reads the
squeezed file one bit at a time, and moves through the tree
until it reaches a terminal node and then sends the
character in that position to the output file.
The most frequently occurring characters are kept
closest to the root and thus have shorter codes. Those with
lower frequencies of occurrence are kept further away and
get longer codes. The result is often a file that is
significantly shorter than the original.
When all bytes occur with about the same frequency,
as in machine language program files, then all the codes
are about the same length and not much is gained. In fact,
since the de-coding information (the tree) must be
included in the output file, the result can often be
longer, particularly on short files.
ARC VERSION 2.20 PAGE - 32
For those of you that are interested in statistics, we
have included a small utility program with ARC that analyzes
the frequency distribution of the bytes in a file and
graphically displays the results. On the top portion of the
screen you will see the frequency distribution of the bytes
in the file. On the bottom portion is a bar graph
representing the lengths of the Huffman codes generated by
the squeeze algorithm. A huffman code can be anywhere from 0
to 24 bits in length. Each bit in the Huffman code is
represented by two pixels on the graphics screen. To run the
utility you must have ARC in memory and type:
a:analyze [d:]filename
The program will then read through 'd:filename' and
display a frequency distribution for the file.
But how do we come up with the best tree to use to
generate the Huffman codes? If you sit down and think about
it you will realize that even if only a dozen or so bytes
are used, the number of possible trees is quite large.
Huffman squeezing gets it name from the man that came
up with a solution to this problem. We actually tried to
figure it out for ourselves and ended up utterly confused
until we came across Huffman's article(2). Huffman makes it
look simple.
Lets go back to our previous example of "abracadabra".
We start out with the following frequency distribution:
character frequency
--------- ---------
a 5
b 2
r 2
c 1
d 1
We start by picking off the two lowest frequencies and
forming a partial tree with them. In this case "c" and "d".
This gives us a new table:
____________________
2. Huffman, D.A., "A METHOD FOR THE CONSTRUCTION OF MINIMUM
REDUNDANCY CODES", Proc. IRE, 40(9), 1098-1101(1952)
ARC VERSION 2.20 PAGE - 33
character frequency