LZ77 and LZ78 are the two lossless data compression algorithms published in papers by Abraham Lempel and Jacob Ziv in 1977 and 1978. They are also known as Lempel-Ziv 1 (LZ1) and Lempel-Ziv 2 (LZ2) respectively. These two algorithms form the basis for many variations including LZW, LZSS, LZMA and others. Besides their academic influence, these algorithms formed the basis of several ubiquitous compression schemes, including the one used in GIF and the DEFLATE algorithm used in PNG and ZIP. They are both theoretically dictionary coders. LZ77 maintains a sliding window during compression. This was later shown to be equivalent to the explicit dictionary constructed by LZ78—however, they are only equivalent when the entire data is intended to be decompressed. Since LZ77 encodes and decodes from a sliding window over previously seen characters, decompression must always start at the beginning of the input. Conceptually, LZ78 decompression could allow random access to the input if the entire dictionary were known in advance. However, in practice the dictionary is created during encoding and decoding by creating a new phrase whenever a token is output. The algorithms were named an IEEE Milestone in 2004. In 2021 Jacob Ziv was awarded the IEEE Medal of Honor for his involvement in their development.
Theoretical efficiency 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 exists 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. Formally, (Theorem 13.5.2 ).
Similar theorems apply to other versions of LZ algorithm.
LZ77 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 compose a single run unit of length LR, 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 = LR. When the first LR 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/LR) + (1 if L mod LR ≠ 0) times to the start of that single buffered run unit, read LR 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 LR 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.
Pseudocode The following pseudocode is a reproduction of the LZ77 compression algorithm sliding window.
while input is not empty do match := longest repeated occurrence of input that begins in window if match exists then d := distance to start of match l := length of match c := char following match 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
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