Atari Forever 4/Atari Forever 4.zip/Atari Forever 4.iso/PD_THEMA/GESETZE/LHARC.221/DOC/ALGORITH.DOC

Text File | 1993-03-16 | 14KB | 295 lines

                                                           April 7, 1989

------------------------------------------------------------------------
             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 simplic-
   ity, 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 ap-
   pended 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 LZHUF.C file 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).

                                - end -