Data Compression Algorithms of LARC and LHarc
                      Haruhiko Okumura*
*The author is the sysop of the Science SIG of PV-VAN.  His
 address is: 12-2-404 Green Heights, 580 Nagasawa, Yokosuka 239
 Japan
---------------------------------------------------------------

1. Introduction

In the spring of 1988, I wrote a very simple data compression program
named LZSS in C language, and uploaded it to the Science SIG (forum) of
PC-VAN, Japan's biggest personal computer network. 

That program was based on Storer and Szymanski's slightly modified
version of one of Lempel and Ziv's algorithms.  Despite its simplicity,
for most files its compression outperformed the archivers then widely
used. 

Kazuhiko Miki rewrote my LZSS in Turbo Pascal and assembly language, and
soon made it evolve into a complete archiver, which he named LARC. 

The first versions of LZSS and LARC were rather slow.  So I rewrote my
LZSS using a binary tree, and so did Miki.  Although LARC's encoding was
slower than the fastest archiver available, its decoding was quite fast,
and its algorithm was so simple that even self-extracting files
(compressed files plus decoder) it created were usually smaller than
non-self-extracting files from other archivers. 

Soon many hobby programmers joined the archiver project at the forum. 
Very many suggestions were made, and LARC was revised again and again. 
By the summer of 1988, LARC's speed and compression have improved so
much that LARC-compressed programs were beginning to be uploaded in many
forums of PC-VAN and other networks. 

In that summer I wrote another program, LZARI, which combined the LZSS
algorithm with adaptive arithmetic compression.  Although it was slower
than LZSS, its compression performance was amazing. 

Miki, the author of LARC, uploaded LZARI to NIFTY-Serve, another big
information network in Japan.  In NIFTY-Serve, Haruyasu Yoshizaki
replaced LZARI's adaptive arithmetic coding with a version of adaptive
Huffman coding to increase speed.  Based on this algorithm, which he
called LZHUF, he developed yet another archiver, LHarc. 

In what follows, I will review several of these algorithms and supply
simplified codes in C language. 

2. Simple coding methods

Replacing several (usually 8 or 4) "space" characters by one "tab"
character is a very primitive method for data compression.  Another
simple method is run-length coding, which encodes the message
"AAABBBBAACCCC" into "3A4B2A4C", for example. 

3. LZSS coding

This scheme is initiated by Ziv and Lempel [1].  A slightly modified
version is described by Storer and Szymanski [2].  An implementation
using a binary tree is proposed by Bell [3].  The algorithm is quite
simple: Keep a ring buffer, which initially contains "space" characters
only.  Read several letters from the file to the buffer.  Then search
the buffer for the longest string that matches the letters just read,
and send its length and position in the buffer. 

If the buffer size is 4096 bytes, the position can be encoded in 12
bits.  If we represent the match length in four bits, the <position,
length> pair is two bytes long.  If the longest match is no more than
two characters, then we send just one character without encoding, and
restart the process with the next letter.  We must send one extra bit
each time to tell the decoder whether we are sending a <position,
length> pair or an unencoded character. 

The accompanying file LZSS.C is a version of this algorithm.  This
implementation uses multiple binary trees to speed up the search for the
longest match.  All the programs in this article are written in
draft-proposed ANSI C.  I tested them with Turbo C 2.0. 

4. LZW coding

This scheme was devised by Ziv and Lempel [4], and modified by Welch
[5]. 

The LZW coding has been adopted by most of the existing archivers, such
as ARC and PKZIP.  The algorithm can be made relatively fast, and is
suitable for hardware implementation as well. 

The algorithm can be outlined as follows: Prepare a table that can
contain several thousand items.  Initially register in its 0th through
255th positions the usual 256 characters.  Read several letters from the
file to be encoded, and search the table for the longest match.  Suppose
the longest match is given by the string "ABC".  Send the position of
"ABC" in the table.  Read the next character from the file.  If it is
"D", then register a new string "ABCD" in the table, and restart the
process with the letter "D".  If the table becomes full, discard the
oldest item or, preferably, the least used. 

A Pascal program for this algorithm is given in Storer's book [6]. 

5. Huffman coding

Classical Huffman coding is invented by Huffman [7].  A fairly readable
accound is given in Sedgewick [8]. 

Suppose the text to be encoded is "ABABACA", with four A's, two B's, and
a C.  We represent this situation as follows:

        4    2    1
        |    |    |
        A    B    C

Combine the least frequent two characters into one, resulting in the new
frequency 2 + 1 = 3:

        4      3
        |     /  \
        A    B    C

Repeat the above step until the whole characters combine into a tree:

            7
          /  \
         /     3
        /    /  \
       A    B    C

Start at the top ("root") of this encoding tree, and travel to the
character you want to encode.  If you go left, send a "0"; otherwise
send a "1".  Thus, "A" is encoded by "0", "B" by "10", "C" by "11". 
Algotether, "ABABACA" will be encoded into ten bits, "0100100110". 

To decode this code, the decoder must know the encoding tree, which must
be sent separately. 

A modification to this classical Huffman coding is the adaptive, or
dynamic, Huffman coding.  See, e.g., Gallager [9].  In this method, the
encoder and the decoder processes the first letter of the text as if the
frequency of each character in the file were one, say.  After the first
letter has been processed, both parties increment the frequency of that
character by one.  For example, if the first letter is 'C', then
freq['C'] becomes two, whereas every other frequencies are still one. 
Then the both parties modify the encoding tree accordingly.  Then the
second letter will be encoded and decoded, and so on. 

6. Arithmetic coding

The original concept of arithmetic coding is proposed by P.  Elias.  An
implementation in C language is described by Witten and others [10]. 

Although the Huffman coding is optimal if each character must be encoded
into a fixed (integer) number of bits, arithmetic coding wins if no such
restriction is made. 

As an example we shall encode "AABA" using arithmetic coding.  For
simplicity suppose we know beforehand that the probabilities for "A" and
"B" to appear in the text are 3/4 and 1/4, respectively. 

Initially, consider an interval:

              0 <= x < 1.

Since the first character is "A" whose probability is 3/4, we shrink the
interval to the lower 3/4:

              0 <= x < 3/4.

The next character is "A" again, so we take the lower 3/4:

              0 <= x < 9/16.

Next comes "B" whose probability is 1/4, so we take the upper 1/4:

              27/64 <= x < 9/16,

because "B" is the second element in our alphabet, {A, B}.  The last
character is "A" and the interval is

              27/64 <= x < 135/256,

which can be written in binary notation

              0.011011 <= x < 0.10000111.

Choose from this interval any number that can be represented in fewest
bits, say 0.1, and send the bits to the right of "0."; in this case we
send only one bit, "1".  Thus we have encoded four letters into one bit!
With the Huffman coding, four letters could not be encoded into less
than four bits. 

To decode the code "1", we just reverse the process: First, we supply
the "0." to the right of the received code "1", resulting in "0.1" in
binary notation, or 1/2.  Since this number is in the first 3/4 of the
initial interval 0 <= x < 1, the first character must be "A".  Shrink
the interval into the lower 3/4.  In this new interval, the number 1/2
lies in the lower 3/4 part, so the second character is again "A", and so
on.  The number of letters in the original file must be sent separately
(or a special 'EOF' character must be appended at the end of the file). 

The algorithm described above requires that both the sender and receiver
know the probability distribution for the characters.  The adaptive
version of the algorithm removes this restriction by first supposing
uniform or any agreed-upon distribution of characters that approximates
the true distribution, and then updating the distribution after each
character is sent and received. 

7. LZARI

In each step the LZSS algorithm sends either a character or a <position,
length> pair.  Among these, perhaps character "e" appears more
frequently than "x", and a <position, length> pair of length 3 might be
commoner than one of length 18, say.  Thus, if we encode the more
frequent in fewer bits and the less frequent in more bits, the total
length of the encoded text will be diminished.  This consideration
suggests that we use Huffman or arithmetic coding, preferably of
adaptive kind, along with LZSS. 

This is easier said than done, because there are many possible
<position, length> combinations.  Adaptive compression must keep running
statistics of frequency distribution.  Too many items make statistics
unreliable. 

What follows is not even an approximate solution to the problem posed
above, but anyway this was what I did in the summer of 1988. 

I extended the character set from 256 to three-hundred or so in size,
and let characters 0 through 255 be the usual 8-bit characters, whereas
characters 253 + n represent that what follows is a position of string
of length n, where n = 3, 4 , ....  These extended set of characters
will be encoded with adaptive arithmetic compression. 

I also observed that longest-match strings tend to be the ones that were
read relatively recently.  Therefore, recent positions should be encoded
into fewer bits.  Since 4096 positions are too many to encode
adaptively, I fixed the probability distribution of the positions "by
hand." The distribution function given in the accompanying LZARI.C is
rather tentative; it is not based on thorough experimentation.  In
retrospect, I could encode adaptively the most significant 6 bits, say,
or perhaps by some more ingenious method adapt the parameters of the
distribution function to the running statistics. 

At any rate, the present version of LZARI treats the positions rather
separately, so that the overall compression is by no means optimal. 
Furthermore, the string length threshold above which strings are coded
into <position, length> pairs is fixed, but logically its value must
change according to the length of the <position, length> pair we would
get. 

8. LZHUF

LZHUF, the algorithm of Haruyasu Yoshizaki's archiver LHarc, replaces
LZARI's adaptive arithmetic coding with adaptive Huffman.  LZHUF encodes
the most significant 6 bits of the position in its 4096-byte buffer by
table lookup.  More recent, and hence more probable, positions are coded
in less bits.  On the other hand, the remaining 6 bits are sent
verbatim.  Because Huffman coding encodes each letter into a fixed
number of bits, table lookup can be easily implemented. 

Though theoretically Huffman cannot exceed arithmetic compression, the
difference is very slight, and LZHUF is fairly fast. 

The accompanying file LZHUF.C was written by Yoshizaki.  I translated
the comments into English and made a few trivial changes to make it
conform to the ANSI C standard. 

References
  [1] J. Ziv and A. Lempel, IEEE Trans. IT-23, 337-343 (1977).
  [2] J. A. Storer and T. G. Szymanski, J. ACM, 29, 928-951
      (1982).
  [3] T. C. Bell, IEEE Trans. COM-34, 1176-1182 (1986).
  [4] J. Ziv and A. Lempel, IEEE Trans. IT-24, 530-536 (1978).
  [5] T. A. Welch, Computer, 17, No.6, 8-19 (1984).
  [6] J. A. Storer, Data Compression: Methods and Theory
      (Computer Science Press, 1988).
  [7] D. A. Huffman, Proc IRE 40, 1098-1101 (1952).
  [8] R. Sedgewick, Algorithms, 2nd ed. (Addison-Wesley, 1988).
  [9] R. G. Gallager, IEEE Trans. IT-24, 668-674 (1978).
 [10] I. E. Witten, R. M. Neal, and J. G. Cleary, Commun. ACM
      30, 520-540 (1987).