Processing of large document collections

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

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

Processing of large document collections