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43-397: 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

86-451: A substitution coder , is a class of lossless data compression algorithms which operate by searching for matches between the text to be compressed and a set of strings contained in a data structure (called the 'dictionary') maintained by the encoder. When the encoder finds such a match, it substitutes a reference to the string's position in the data structure. Some dictionary coders use a 'static dictionary', one whose full set of strings

129-503: A book. More common are methods where the dictionary starts in some predetermined state but the contents change during the encoding process, based on the data that has already been encoded. Both the LZ77 and LZ78 algorithms work on this principle. In LZ77, a circular buffer called the "sliding window" holds the last N bytes of data processed. This window serves as the dictionary, effectively storing every substring that has appeared in

172-404: A generalization of run-length encoding that can take advantage of runs of strings of characters (such as BWWBWWBWWBWW ). Run-length encoding can be expressed in multiple ways to accommodate data properties as well as additional compression algorithms. For instance, one popular method encodes run lengths for runs of two or more characters only, using an "escape" symbol to identify runs, or using

215-433: A length–distance pair, and distinguish their length–distance pairs from literals (raw data encoded as itself, rather than as part of a length–distance pair). A few examples: The LZ78 algorithms compress sequential data by building a dictionary of token sequences from the input, and then replacing the second and subsequent occurrence of the sequence in the data stream with a reference to the dictionary entry. The observation

258-417: A potential issue since the decoder output is one step behind the dictionary. Refer to LZW for how this is handled. Enhancements to LZW include handing symbol sizes other than 8 bits and having reserved codes to reset the dictionary and to indicate end of data. Run-length encoding Run-length encoding ( RLE ) is a form of lossless data compression in which runs of data (consecutive occurrences of

301-467: A screen containing plain black text on a solid white background. There will be many long runs of white pixels in the blank space, and many short runs of black pixels within the text. A hypothetical scan line , with B representing a black pixel and W representing white, might read as follows: With a run-length encoding (RLE) data compression algorithm applied to the above hypothetical scan line, it can be rendered as follows: This can be interpreted as

344-494: A sequence of twelve Ws, one B, twelve Ws, three Bs, etc., and represents the original 67 characters in only 18. While the actual format used for the storage of images is generally binary rather than ASCII characters like this, the principle remains the same. Even binary data files can be compressed with this method; file format specifications often dictate repeated bytes in files as padding space. However, newer compression methods such as DEFLATE often use LZ77 -based algorithms,

387-433: A standard to encode run-length colour for fax machines, known as T.45. That fax colour coding standard, which along with other techniques is incorporated into Modified Huffman coding , is relatively efficient because most faxed documents are primarily white space, with occasional interruptions of black. RLE has a space complexity of ⁠ O ( n ) {\displaystyle O(n)} ⁠ , where n

430-402: Is 1B and results in entry number 2 in the dictionary, {1,B} . The token "B" is output, preceded by the sequence represented by dictionary entry 1. Entry 1 is an 'A' (followed by "entry 0" – nothing) so AB is added to the output. Next 0B is added to the dictionary as the next entry, 3 {0,B} , and B (preceded by nothing) is added to the output. Finally a dictionary entry for 1$

473-428: Is created and A$ is output resulting in A AB B A$ or AABBA removing the spaces and EOF marker. 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

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516-402: Is determined before coding begins and does not change during the coding process. This approach is most often used when the message or set of messages to be encoded is fixed and large; for instance, an application that stores the contents of a book in the limited storage space of a PDA generally builds a static dictionary from a concordance of the text and then uses that dictionary to compress

559-497: 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

602-413: Is output. LZW is similar to LZ78, but, the dictionary is initialized to all possible symbols. The typical implementation works with 8 bit symbols, so the dictionary "codes" for hex 00 to hex FF (decimal 255) are pre-defined. Dictionary entries would be added starting with code value hex 100. Unlike LZ78, if a match is not found (or if the end of data), then only the dictionary code is output. This creates

645-445: Is set to the index of the matching entry, nothing is output, and last matching index is left representing the input so far. Input is processed until a match is not found. Then a new dictionary entry is created, dictionary[next available index] = {last matching index, token} , and the algorithm outputs last matching index, followed by token, then resets last matching index = 0 and increments next available index. As an example consider

688-415: Is that the number of repeated sequences is a good measure of the non random nature of a sequence. The algorithms represent the dictionary as an n-ary tree where n is the number of tokens used to form token sequences. Each dictionary entry is of the form dictionary[...] = {index, token} , where index is the index to a dictionary entry representing a previously seen sequence, and token is the next token from

731-418: Is the entropy rate of the source. Similar theorems apply to other versions of LZ algorithm. 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

774-490: Is the size of the input data. Run-length encoding compresses data by reducing the physical size of a repeating string of characters. This process involves converting the input data into a compressed format by identifying and counting consecutive occurrences of each character. The steps are as follows: The decoding process involves reconstructing the original data from the encoded format by repeating characters according to their counts. The steps are as follows: Consider

817-532: 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. Dictionary coder A dictionary coder , also sometimes known as

860-492: The Windows 3.x startup screen. Run-length encoding (RLE) schemes were employed in the transmission of analog television signals as far back as 1967. In 1983, run-length encoding was patented by Hitachi . RLE is particularly well suited to palette -based bitmap images (which use relatively few colours) such as computer icons , and was a popular image compression method on early online services such as CompuServe before

903-408: 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

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946-509: The advent of more sophisticated formats such as GIF . It does not work well on continuous-tone images (which use very many colours) such as photographs, although JPEG uses it on the coefficients that remain after transforming and quantizing image blocks. Common formats for run-length encoded data include Truevision TGA , PackBits (by Apple, used in MacPaint ), PCX and ILBM . The International Telecommunication Union also describes

989-496: 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. In

1032-432: The character itself as the escape, so that any time a character appears twice it denotes a run. On the previous example, this would give the following: This would be interpreted as a run of twelve Ws, a B, a run of twelve Ws, a run of three Bs, etc. In data where runs are less frequent, this can significantly improve the compression rate. One other matter is the application of additional compression algorithms. Even with

1075-414: 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

1118-408: 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

1161-453: The data is more recent and may correlate better with the next input. The following pseudocode is a reproduction of the LZ77 compression algorithm sliding window. Even though all LZ77 algorithms work by definition on the same basic principle, they can vary widely in how they encode their compressed data to vary the numerical ranges of a length–distance pair, alter the number of bits consumed for

1204-432: 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

1247-513: 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

1290-407: The dictionary grows, and in binary the indexes need not be represented by any more than the minimum number of bits. Decompression consists of rebuilding the dictionary from the compressed sequence. From the sequence 0A1B0B1$ the first entry is always the terminator 0 {...} , and the first from the sequence would be 1 {0,A} . The A is added to the output. The second pair from the input

1333-429: The encoding process, the dictionary is empty. An index value of zero is used to represent the end of a string, so the first index of the dictionary is one. At each step of the encoding process, if there is no match, then the last matching index (or zero) and character are both added to the dictionary and output to the compressed stream. If there is a match, then the working index is updated to the matching index, and nothing

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1376-462: 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 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

1419-456: The file size. RLE may also refer in particular to an early graphics file format supported by CompuServe for compressing black and white images, that was widely supplanted by their later Graphics Interchange Format (GIF). RLE also refers to a little-used image format in Windows 3.x that is saved with the file extension rle ; it is a run-length encoded bitmap, and the format was used for

1462-415: The input that makes this entry unique in the dictionary. Note how the algorithm is greedy, and so nothing is added to the table until a unique making token is found. The algorithm is to initialize last matching index = 0 and next available index = 1 and then, for each token of the input stream, the dictionary searched for a match: {last matching index, token} . If a match is found, then last matching index

1505-728: 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 ). LZ78 is universal and entropic  —  If X {\textstyle X} is a binary source that is stationary and ergodic, then lim sup n 1 n l L Z 78 ( X 1 : n ) ≤ h ( X ) {\displaystyle \limsup _{n}{\frac {1}{n}}l_{LZ78}(X_{1:n})\leq h(X)} with probability 1. Here h ( X ) {\textstyle h(X)}

1548-402: The past N bytes as dictionary entries. Instead of a single index identifying a dictionary entry, two values are needed: the length , indicating the length of the matched text, and the offset (also called the distance ), indicating that the match is found in the sliding window starting offset bytes before the current text. LZ78 uses a more explicit dictionary structure; at the beginning of

1591-448: 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

1634-402: The runs extracted, the frequencies of different characters may be large, allowing for further compression; however, if the run lengths are written in the file in the locations where the runs occurred, the presence of these numbers interrupts the normal flow and makes it harder to compress. To overcome this, some run-length encoders separate the data and escape symbols from the run lengths, so that

1677-564: The same data value) are stored as a single occurrence of that data value and a count of its consecutive occurrences, rather than as the original run. As an imaginary example of the concept, when encoding an image built up from colored dots, the sequence "green green green green green green green green green" is shortened to "green x 9". This is most efficient on data that contains many such runs, for example, simple graphic images such as icons, line drawings, games, and animations. For files that do not have many runs, encoding them with RLE could increase

1720-436: 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

1763-438: The sequence of tokens AABBA which would assemble the dictionary; and the output sequence of the compressed data would be 0A1B0B . Note that the last A is not represented yet as the algorithm cannot know what comes next. In practice an EOF marker is added to the input – AABBA$ for example. Note also that in this case the output 0A1B0B1$ is longer than the original input but compression ratio improves considerably as

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1806-416: 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

1849-414: The verses. This scheme of using Huffman coding to represent indices into a concordance has been called "Huffword". In a related and more general method, a dictionary is built from redundancy extracted from a data environment (various input streams) which dictionary is then used statically to compress a further input stream. For example, a dictionary is built from old English texts then is used to compress

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