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Processing of large document collections

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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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### 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
- no matter how much storage space or transmission bandwidth is available, someone always finds ways to fill it with

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 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

- 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 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 methods

- Most text compression methods can be placed in one of two classes:
- symbolwise methods
- dictionary 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 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

- 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 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

Models

- 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

Models

- 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

- 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

- 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 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 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 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 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 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

- 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

- 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 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 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 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 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 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 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 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 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

- 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

- 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 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 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 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-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 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 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 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 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 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 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

LZ77

- 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

LZ77

- 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

LZ77

- 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

LZ77

- 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

- for the text from the current point ahead:

LZ77

- 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

LZ77

- 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

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