LZ77 and LZ78

In the second of the two papers that introduced these algorithms they are analyzed as encoders defined by finite-state machines. A measure analogous to information entropy is developed for individual sequences (as opposed to probabilistic ensembles). This measure gives a bound on the data compression ratio that can be achieved. It is then shown that there exist finite lossless encoders for every sequence that achieve this bound as the length of the sequence grows to infinity. In this sense an algorithm based on this scheme produces asymptotically optimal encodings. This result can be proven more directly, as for example in notes by Peter Shor.[7]

LZ77 algorithms achieve compression by replacing repeated occurrences of data with references to a single copy of that data existing earlier in the uncompressed data stream. A match is encoded by a pair of numbers called a length-distance pair, which is equivalent to the statement "each of the next length characters is equal to the characters exactly distance characters behind it in the uncompressed stream". (The distance is sometimes called the offset instead.)

To spot matches, the encoder must keep track of some amount of the most recent data, such as the last 2 kB, 4 kB, or 32 kB. The structure in which this data is held is called a sliding window, which is why LZ77 is sometimes called sliding-window compression. The encoder needs to keep this data to look for matches, and the decoder needs to keep this data to interpret the matches the encoder refers to. The larger the sliding window is, the longer back the encoder may search for creating references.

It is not only acceptable but frequently useful to allow length-distance pairs to specify a length that actually exceeds the distance. As a copy command, this is puzzling: "Go back four characters and copy ten characters from that position into the current position". How can ten characters be copied over when only four of them are actually in the buffer? Tackling one byte at a time, there is no problem serving this request, because as a byte is copied over, it may be fed again as input to the copy command. When the copy-from position makes it to the initial destination position, it is consequently fed data that was pasted from the beginning of the copy-from position. The operation is thus equivalent to the statement "copy the data you were given and repetitively paste it until it fits". As this type of pair repeats a single copy of data multiple times, it can be used to incorporate a flexible and easy form of run-length encoding.

Another way to see things is as follows: While encoding, for the search pointer to continue finding matched pairs past the end of the search window, all characters from the first match at offset D and forward to the end of the search window must have matched input, and these are the (previously seen) characters that comprise a single run unit of length L_R, which must equal D. Then as the search pointer proceeds past the search window and forward, as far as the run pattern repeats in the input, the search and input pointers will be in sync and match characters until the run pattern is interrupted. Then L characters have been matched in total, L > D, and the code is [D, L, c].

Upon decoding [D, L, c], again, D = L_R. When the first L_R characters are read to the output, this corresponds to a single run unit appended to the output buffer. At this point, the read pointer could be thought of as only needing to return int(L/L_R) + (1 if L mod L_R ≠ 0) times to the start of that single buffered run unit, read L_R characters (or maybe fewer on the last return), and repeat until a total of L characters are read. But mirroring the encoding process, since the pattern is repetitive, the read pointer need only trail in sync with the write pointer by a fixed distance equal to the run length L_R until L characters have been copied to output in total.

Considering the above, especially if the compression of data runs is expected to predominate, the window search should begin at the end of the window and proceed backwards, since run patterns, if they exist, will be found first and allow the search to terminate, absolutely if the current maximal matching sequence length is met, or judiciously, if a sufficient length is met, and finally for the simple possibility that the data is more recent and may correlate better with the next input.

while input is not empty do
    prefix := longest prefix of input that begins in window
    
    if prefix exists then
        d := distance to start of prefix
        l := length of prefix
        c := char following prefix in input
    else
        d := 0
        l := 0
        c := first char of input
    end if
    
    output (d, l, c)
    
    discard l + 1 chars from front of window
    s := pop l + 1 chars from front of input
    append s to back of window
repeat

LZ78 algorithms achieve compression by replacing repeated occurrences of data with references to a dictionary that is built based on the input data stream. Each dictionary entry is of the form dictionary[...] = {index, character}, where index is the index to a previous dictionary entry, and character is appended to the string represented by dictionary[index]. For example, "abc" would be stored (in reverse order) as follows: dictionary[k] = {j, 'c'}, dictionary[j] = {i, 'b'}, dictionary[i] = {0, 'a'}, where an index of 0 specifies the first character of a string. The algorithm initializes last matching index = 0 and next available index = 1. For each character of the input stream, the dictionary is searched for a match: {last matching index, character}. If a match is found, then last matching index is set to the index of the matching entry, and nothing is output. If a match is not found, then a new dictionary entry is created: dictionary[next available index] = {last matching index, character}, and the algorithm outputs last matching index, followed by character, then resets last matching index = 0 and increments next available index. Once the dictionary is full, no more entries are added. When the end of the input stream is reached, the algorithm outputs last matching index. Note that strings are stored in the dictionary in reverse order, which an LZ78 decoder will have to deal with.

LZW is an LZ78-based algorithm that uses a dictionary pre-initialized with all possible characters (symbols) or emulation of a pre-initialized dictionary. The main improvement of LZW is that when a match is not found, the current input stream character is assumed to be the first character of an existing string in the dictionary (since the dictionary is initialized with all possible characters), so only the last matching index is output (which may be the pre-initialized dictionary index corresponding to the previous (or the initial) input character). Refer to the LZW article for implementation details.

BTLZ is an LZ78-based algorithm that was developed for use in real-time communications systems (originally modems) and standardized by CCITT/ITU as V.42bis. When the trie-structured dictionary is full, a simple re-use/recovery algorithm is used to ensure that the dictionary can keep adapting to changing data. A counter cycles through the dictionary. When a new entry is needed, the counter steps through the dictionary until a leaf node is found (a node with no dependents). This is deleted and the space re-used for the new entry. This is simpler to implement than LRU or LFU and achieves equivalent performance.

• Lempel–Ziv–Stac (LZS)
• Discrete cosine transform (DCT), a lossy compression algorithm used in JPEG and MPEG coding standards

• "The LZ77 algorithm". Data Compression Reference Center: RASIP working group. Faculty of Electrical Engineering and Computing, University of Zagreb. 1997. Archived from the original on 7 January 2013. Retrieved 22 June 2012.

• "The LZ78 algorithm". Data Compression Reference Center: RASIP working group. Faculty of Electrical Engineering and Computing, University of Zagreb. 1997. Archived from the original on 7 January 2013. Retrieved 22 June 2012.

• "The LZW algorithm". Data Compression Reference Center: RASIP working group. Faculty of Electrical Engineering and Computing, University of Zagreb. 1997. Archived from the original on 7 January 2013. Retrieved 22 June 2012.