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Processing of large document collections. Fall 2002, Part 3. Text compression. Despite a continuous increase in storage and transmission capacities, more and more effort has been put into using compression to increase the amount of data that can be handled

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text compression
Text compression
  • Despite a continuous increase in storage and transmission capacities, more and more effort has been put into using compression to increase the amount of data that can be handled
  • no matter how much storage space or transmission bandwidth is available, someone always finds ways to fill it with
text compression3
Text compression
  • Efficient storage and representation of information is an old problem (before the computer era)
    • Morse code: uses shorter representations for common characters
    • Braille code for the blind: includes contractions, which represent common words with 2 or 3 characters
text compression4
Text compression
  • On a computer: changing the representation of a file so that it takes less space to store or less time to transmit
    • original file can be reconstructed exactly from the compressed representation
  • different than data compression in general
    • text compression has to be lossless
    • compare with sound and images: small changes and noise is tolerated
text compression methods
Text compression methods
  • Huffman coding (in the 50’s)
    • compressing English: 5 bits/character
  • Ziv-Lempel compression (in the 70’s)
    • 4 bits/character
  • arithmetic coding
    • 2 bits/char (more processing power needed)
  • prediction by partial matching (80’s)
text compression methods6
Text compression methods
  • Since 80’s compression rate has been about the same
  • improvements are made in processor and memory utilization during compression
  • also: amount of compression may increase when more memory (for compression and uncompression) is available
text compression methods7
Text compression methods
  • Most text compression methods can be placed in one of two classes:
    • symbolwise methods
    • dictionary methods
symbolwise methods
Symbolwise methods
  • Work by estimating the probabilities of symbols (often characters)
    • coding one symbol at a time
    • using shorter codewords for the most likely symbols (in the same way as Morse code does)
symbolwise methods9
Symbolwise methods
  • variations differ mainly in how they estimate probabilities for symbols
    • the more accurate these estimates are, the greater the compression that can be achieved
    • to obtain good compression, the probability estimate is usually based on the context in which a symbol occurs
dictionary methods
Dictionary methods
  • compress by replacing words and other fragments of text with an index to an entry in a ”dictionary”
  • compression is achieved if the index is stored in fewer bits than the string it replaces
symbolwise methods11
Symbolwise methods
  • Modeling
    • estimating probabilities
    • there does not appear to be any single ”best” method
  • Coding
    • converting the probabilities into a bitstream for transmission
    • well understood, can be performed effectively
  • Compression methods obtain high compression by forming good models of the data that is to be coded
  • the function of a model is to predict symbols
    • e.g. during the encoding of a text , the ”prediction” for the next symbol might include a probability of 2% for the letter ’u’, based on its relative frequency in a sample of text
  • The set of all possible symbols is called the alphabet
  • the probability distribution provides an estimated probability for each symbol in the alphabet
encoding decoding
Encoding, decoding
  • the model provides the probability distribution to the encoder, which uses it to encode the symbol that actually occurs
  • the decoder uses an identical model together with the output of the encoder to find out what the encoded symbol was
information content of a symbol
Information content of a symbol
  • The number of bits in which a symbol s should be coded is called the information content I(s) of the symbol
  • the information content I(s) is directly related to the symbol’s predicted probability P(s), by the function
    • I(s) = -log P(s) bits
information content of a symbol16
Information content of a symbol
  • The average amount of information per symbol over the whole alphabet is known as the entropy of the probability distribution, denoted by H:
information content of a symbol17
Information content of a symbol
  • Provided that the symbols appear independently and with the assumed probabilities, H is a lower bound on compression, measured in bits per symbol, that can be achieved by any coding method
information content of a symbol18
Information content of a symbol
  • If the probability of symbol ’u’ is estimated to be 2%, the corresponding information content is 5.6 bits
  • if ’u’ happens to be the next symbol that is to be coded, it should be transmitted in 5.6 bits
information content of a symbol19
Information content of a symbol
  • predictions can usually be improved by taking account of the previous symbol
  • if a ’q’ has just occurred, the probability of ’u’ may jump to 95 %, based on how often ’q’ is followed by ’u’ in a sample of text
  • information content of ’u’ in this case is 0.074 bits
information content of a symbol20
Information content of a symbol
  • Models that take a few immediately preceding symbols into account to make a prediction are called finite-context models of order m
    • m is the number of previous symbols used to make a prediction
static models
Static models
  • There are many ways to estimate the probabilities in a model
  • we could use static modelling:
    • always use the same probabilities for symbols, regardless of what text is being coded
    • compressing system may not perform well, if different text is received
      • e.g. a model for English with a file of numbers
semi static models
Semi-static models
  • One solution is to generate a model specifically for each file that is to be compressed
  • an initial pass is made through the file to estimate symbol probabilities, and these are transmitted to the decode before transmitting the encoded symbols
  • this is called semi-static modelling
semi static models23
Semi-static models
  • Semi-static modelling has the advantage that the model is invariably better suited to the input than a static one, but the penalty paid is
    • having to transmit the model first,
    • as well as the preliminary pass over the data to accumulate symbol probabilities
adaptive models
Adaptive models
  • Adaptive model begins with a bland probability distribution and gradually alters it as more symbols are encountered
  • as an example, assume a zero-order model, i.e., no context is used to predict the next symbol
adaptive models25
Adaptive models
  • Assume that a encoder has already encoded a long text and come to a sentence: It migh
  • now the probability that the next character is ’t’ is estimated to be 49,983/768,078 = 6.5 %, since in the previous text, 49,983 characters of the total of 768,078 characters were ’t’
adaptive models26
Adaptive models
  • Using the same system, ’e’ has the probability 9.4 % and ’x’ has probability 0.11 %
  • the model provides this estimated probability distribution to an encoder
  • the decoder is able to generate the same model since it has the same probability estimates as the encoder
adaptive models27
Adaptive models
  • For a higher-order model, such as a first-order model, the probability is estimated by how often that character has occurred in the current context
  • in a zero-order model earlier, a symbol ’t’ occurred in a context: It migh , but the model made no use of the characters of the phrase
adaptive models28
Adaptive models
  • A first-order model would use the final ’h’ as a context with which to condition the probability estimates
  • the letter ’h’ has occurred 37,525 times in the prior text, and 1,133 of these times it was followed by a ’t’
  • the probability of ’t’ occurring after an ’h’ can be estimated to be 1,133/37,525=3.02 %
adaptive models29
Adaptive models
  • For ’t’, a prediction of 3.2% is actually worse than in the zero-order model because ’t’ is rare in this context (’e’ follows ’h’ much more often)
  • second-order model would use the relative frequency that the context ’gh’ is followed by ’t’, which is the case in 64,6%
adaptive models30
Adaptive models
  • Good: robust, reliable, flexible
  • Bad: not suitable for random access to compressed files
    • a text can be decoded only from the beginning: the model used for coding a particular part of the text is determined from all the preceding text
    • -> not suitable for full-text retrieval
  • Coding is the task of determining the output representation of a symbol, based on a probability distribution supplied by a model
  • general idea: the coder should output short codewords for likely symbols and long codewords for rare ones
  • symbolwise methods depend heavily on a good coder to achieve compression
huffman coding
Huffman coding
  • A phrase is coded by replacing each of its symbols with the codeword given by a table
  • Huffman coding generates codewords for a set of symbols, given some probability distribution for the symbols
  • the type of code is called prefix-free code
    • no codeword is the prefix of another symbol’s codeword
huffman coding33
Huffman coding
  • The codewords can be stored in a tree (a decoding tree)
  • Huffman’s algorithm works by constructing the decoding tree from the bottom up
huffman coding34
Huffman coding
  • Algorithm
    • create for each symbol a leaf node containing the symbol and its probability
    • two nodes with the smallest probabilities become siblings under a new parent node, which is given a probability equal to the sum of its two children’s probabilities
    • the combining operation is repeated until there is only one node without a parent
    • the two branches from every nonleaf node are then labeled 0 and 1
huffman coding35
Huffman coding
  • Huffman coding is generally fast for both encoding and decoding, provided that the probability distribution is static
    • adaptive Huffman coding is possible, but needs either a lot of memory or is slow
  • coupled with a word-based model (rather than character-based model), gives a good compression
dictionary models
Dictionary models
  • Dictionary-based compression methods use the principle of replacing substrings in a text with a codeword that identifies that substring in a dictionary
  • dictionary contains a list of substrings and a codeword for each substring
  • often fixed codewords used
    • reasonable compression is obtained even if coding is simple
dictionary models37
Dictionary models
  • The simplest dictionary compression methods use small dictionaries
  • for instance, digram coding
    • selected pairs of letters are replaced with codewords
    • a dictionary for the ASCII character set might contain the 128 ASCII characters, as well as 128 common letter pairs
dictionary models38
Dictionary models
  • Digram coding…
    • the output codewords are eight bits each
    • the presence of the full ASCII character set ensures that any (ASCII) input can be represented
    • at best, every pair of characters is replaced with a codeword, reducing the input from 7 bits/character to 4 bits/characters
    • at worst, each 7 bit character will be expanded to 8 bits
dictionary models39
Dictionary models
  • Natural extension:
    • put even larger entries in the dictionary, e.g. common words like ’and’, ’the’,… or common components of words like ’pre’, ’tion’…
  • a predefined set of dictionary phrases make the compression domain-dependent
    • or very short phrases have to be used -> good compression is not achieved
dictionary models40
Dictionary models
  • One way to avoid the problem of the dictionary being unsuitable for the text at hand is to use a semi-static dictionary scheme
    • constuct a new dictionary for every text that is to be compressed
    • overhead of transmitting or storing the dictionary is significant
    • decision of which phrases should be included is a difficult problem
dictionary models41
Dictionary models
  • Solution: use an adaptive dictionary scheme
  • Ziv-Lempel coders (LZ77 and LZ78)
  • a substring of text is replaced with a pointer to where it has occurred previously
  • dictionary: all the text prior to the current position
  • codewords: pointers
dictionary models42
Dictionary models
  • Ziv-Lempel…
    • the prior text makes a very good dictionary since it is usually in the same style and language as upcoming text
    • the dictionary is transmitted implicitly at no extra cost, because the decoder has access to all previously encoded text
  • Key benefits:
    • relatively easy to implement
    • decoding can be performed extremely quickly using only a small amount of memory
  • suitable when the resources required for decoding must be minimized, like when data is distributed or broadcast from a central source to a number of small computers
  • The output of an encoder consists of a sequence of triples, e.g. <3,2,b>
    • the first component of a triple indicates how far back to look in the previous (decoded) text to find the next phrase
    • the second component records how long the phrase is
    • the third component gives the next character from the input
  • The components 1 and 2 constitute a pointer back into the text
  • the component 3 is actually necessary only when the character to be coded does not occur anywhere in the previous input
  • Encoding
    • for the text from the current point ahead:
      • search for the longest match in the previous text
      • output a triple that records the position and length of the match
      • the search for a match may return a length of zero, in which case the position of the match is not relevant
    • search can be accelerated by indexing the prior text with a suitable data structure
  • limitations on how far back a pointer can refer and the maximum size of the string referred to
  • e.g. for English text, a window of a few thousand characters
  • the length of the phrase e.g. maximum of 16 characters
  • otherwise too much space wasted without benefit
  • The decoding program is very simple, so it can be included with the data at very little cost
  • in fact, the compressed data is stored as part of the decoder program, which makes the data self-expanding
  • common way to distribute files