1. Processing of large document collections
Fall 2002, Part 3

2. 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

3. 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

4. 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

5. 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)

6. 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

7. Text compression methods
Most text compression methods can be placed in one of two classes:
symbolwise methods
dictionary methods

8. 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)

9. 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

10. 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

11. Symbolwise methods Modeling Coding 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

12. 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

13. Models The set of all possible symbols is called the alphabet
the probability distribution provides an estimated probability for each symbol in the alphabet

14. 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

15. 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

16. 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:

17. 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

18. 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

19. 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 bits

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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’

26. 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

27. 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

28. 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 %

29. 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%

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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

48. 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