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.

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.

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.

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].

Huffman coding

Classical Huffman coding is invented by Huffman [7]. A fairly readable account 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.

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 < 1Since the first character is "A" whose probability is 3/4, we shrink the interval to the lower 3/4:

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

0 <= x < 9/16Next comes "B" whose probability is 1/4, so we take the upper 1/4:

27/64 <= x < 9/16because "B" is the second element in our alphabet, {A, B}. The last character is "A" and the interval is

27/64 <= x < 135/256which can be written in binary notation

0.011011 <= x < 0.10000111Choose 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.

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.

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).