Jaak Vilo MTAT.03.190 TextAlgorithms 1 - ut Lecture Compres… · • Shannon's experiments with human predictors show an information rate of between .6 and ... resume 0resume retail

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Text Algorithms (4AP)Lecture: Compressionp

Jaak Vilo

2008 fall

1MTAT.03.190 Text Algorithms Jaak Vilo

Problem

• Compress

– Text

– Images, video, sound, …

• Reduce space, efficient communication, etc…

• Exact compression/decompression

• Lossy compression

• Managing Gigabytes: Compressing and Indexing Documents and Images

• Ian H. Witten, Alistair Moffat, Timothy C. Bell

• Hardcover: 519 pages Publisher:Morgan Kaufmann; 2nd Revised edition edition (11 May 1999) Language English ISBN‐10:1558605703

Links

• http://datacompression.info/

• http://en.wikipedia.org/wiki/Data_compression

• Data Compression Debra A. Lelewer and Daniel S. Hirschberg– http://www ics uci edu/~dan/pubs/DataCompression htmlhttp://www.ics.uci.edu/ dan/pubs/DataCompression.html

• Compression FAQ http://www.faqs.org/faqs/compression‐faq/

• Information Theory Primer With an Appendix on Logarithms by Tom Schneider http://www.lecb.ncifcrf.gov/~toms/paper/primer/

• http://www.cbloom.com/algs/index.html

Problem

• Information transmission

• Information storage

• The data sizes are huge and growing – fax ‐ 1.5 x 106 bit/page

– photo: 2M pixels x 24bit = 6MB

X ray image: ~ 100 MB?– X‐ray image: ~ 100 MB?

– Microarray scanned image: 30‐100 MB

– Tissue‐microarray ‐ hundreds of images, each tens of MB

– Large Hardon Collider (CERN) ‐ The device will produce few peta (1015) bytes of stored data in a year.

– TV (PAL) 2.7 ∙ 108 bit/s

– CD‐sound, super‐audio, DVD, ...

What it’s about?

• Elimination of redundancy

• Being able to predict…

• Compression and decompression – Represent data in a more compact way

D i t i i l f– Decompression ‐ restore original form

• Lossy and lossless compression – Lossless ‐ restore in exact copy

– Lossy ‐ restore almost the same information • Useful when no 100% accuracy needed

• voice, image, movies, ...

• Decompression is deterministic (lossy in compression phase)

• Can achieve much more effective results

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Methods covered:

• Code words (Huffman coding)

• Run‐length encoding

• Arithmetic coding

• Lempel‐Ziv family (compress gzip zip pkzip )• Lempel‐Ziv family (compress, gzip, zip, pkzip, ...)

• Burrows‐Wheeler family (bzip2)

• Other methods, including images

• Kolmogorov complexity

• Search from compressed texts

Model

Model Model

Encoder DecoderData Data

Compresseddata

• Let pS be a probability of message S

• The information content can be represented in terms of bits

• I(S) = ‐log( pS ) bits

• If the p=1 then the information content is 0 (no new i f i )information) – If Pr[s]=1 then I(s) = 0.

– In other words, I(death)=I(taxes)=0

• I( heads or tails ) = 1 ‐‐ if the coin is fair

• Entropy H(S) is the average information content of S

– H(S) = pS ∙ I(S) = ‐pS log( pS ) bits

http://en.wikipedia.org/wiki/Information_entropy

• Shannon's experiments with human predictors show an information rate of between .6 and 1.3 bits per character, depending on the experimental setup; the PPM compressionexperimental setup; the PPM compression algorithm can achieve a compression ratio of 1.5 bits per character.

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• The data compression world is all abuzz about Marcus Hutter’s recently announced 50,000 euro prize for record‐breaking data compressors. Marcus, of the Swiss Dalle Molle Institute for Artificial Intelligence, apparently in cahoots with Florida compression maven Matt Mahoney, is offering cash prizes for what amounts to the most impressive ability to compress 100 MBytes of Wikipedia data. (Note that nobody is going to win exactly 50,000 euros ‐ the prize amount is prorated based on how well you beat the current record.)

• This prize differs considerably from my Million Digit Challenge, which is really nothing more than an attempt to silence people foolishly claiming to be able to compress random data. Marcus is instead looking for the most effective way to reproduce the Wiki data, and he’s putting up real money as an incentive. The benchmark that contestants need to beat is that

b h ’ f h d h ld ( l d h k’

http://marknelson.us/2006/08/24/the‐hutter‐prize/#comment‐293

set by Matt Mahoney’s paq8f , the current record holder at 18.3 MB. (Alexander Ratushnyak’s submission of a paq variant looks to clock in at a tidy 17.6 MB, and should soon be confirmed as the new standard.)

• So why is an AI guy inserting himself into the world of compression? Well, Marcus realizes that good data compression is all about modeling the data. The better you understand the data stream, the better you can predict the incoming tokens in a stream. Claude Shannon empirically found that humans could model English text with an entropy of 1.1 to 1.6 0.6 to 1.3 bits per character, which at at best should mean that 100 MB of Wikipedia data could be reduced to 13.75 7.5 MB, with an upper bound of perhaps 20 16.25 MB. The theory is that reaching that 7.5 MB range is going to take such a good understanding of the data stream that it will amount to a demonstration of Artificial Intelligence.

• No compression can on average achieve better compression than the entropy

• Entropy depends on the model (or choice of symbols)

• Let M={ m1, .. mn } be a set of symbols of the model A and let p(m ) be the probability of the symbol mp(mi) be the probability of the symbol mi

• The entropy of the model A, H(M) is ‐∑i=1..n p(mi) ∙ log( p(mi) ) bits

• Let the message S = s1, .. sk, and every symbol si be in the model M. The information content of model A is ‐∑i=1..k log p(si)

• Every symbol has to have a probability, otherwise it cannot be coded if it is present in the data

Static or adaptive

• Static model does not change during the compression

• Adaptive model can be updated during the process

• Symbols not in message cannot have 0‐probabilitySymbols not in message cannot have 0 probability

• Semi‐adaptive model works in 2 stages, off‐line.

• First create the code table, then encode the message with the code table

How to compare compression techniques?

• Ratio (t/p) t: original message length

• p: compressed message length

• In texts ‐ bits per symbol

• The time and memory used for compression

• The time and memory used for decompression

• error tolerance (e.g. self‐correcting code)

• S = 'aa bbb cccc ddddd eeeeee fffffffgggggggg'

• Alphabet of 8

• Length = 40 symbols

• Equal length codewords

• 3‐bit a 000 b 001 c 010 d 011 e 100 f 101 g 110 space 110

• S compressed ‐ 3*40 = 120 bits

Run‐length encoding

• http://michael.dipperstein.com/rle/index.html

• The string:

• "aaaabbcdeeeeefghhhij"

• may be replaced withy p

• "a4b2c1d1e5f1g1h3i1j1".

• This is not shorter because 1‐letter repeat takes more characters...

• "a3b1cde4fgh2ij"

• Now we need to know which characters are followed by run‐length.

• E.g. use escape symbols.

• Or, use the symbol itself ‐ if repeated, then must be followed by run‐length

• "aa2bb0cdee3fghh1ij"

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Coding the alphabetically ordered word‐lists

resume 0resume resume retail retain retard retire

2tail 5n 4rd 3ire

Coding techniques

• Coding refers to techniques used to encode tokens or symbols.

• Two of the best known coding algorithms are Huffman Coding and Arithmetic CodingHuffman Coding and Arithmetic Coding.

• Coding algorithms are effective at compressing data when they use fewer bits for high probability symbols and more bits for low probability symbols.

Variable length encoders

• How to use codes of variable length?

• Decoder needs to know how long is the symbol

• Prefix‐free code: no code can be a prefix of another code

• Calculate the frequencies and probabilities of symbols:

f i ( )freq ratio p(s)a 2 2/40 0.05b 3 3/40 0.075c 4 4/40 0.1d 5 5/40 0.125space 5 5/40 0.125e 6 6/40 0.15f 7 7/40 0.175g 8 8/40 0.2

Algoritm Shannon‐Fano

• Input: probabilities of symbols

• Output: Codewords in prefix free coding

1. Sort symbols by frequency

2. Divide to two almost probable groups

3. First group gets prefix 0, other 1

4. Repeat recursively in each group until 1 symbol remains

Example 1

a 1/2 0b 1/4 10c 1/8 110d 1/16 1110e 1/32 11110f 1/32 11111

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Example 1

a 1/2 0b 1/4 10

Code:

c 1/8 110d 1/16 1110e 1/32 11110f 1/32 11111

Shannon‐Fano

S = 'aa bbb cccc ddddd eeeeee fffffffgggggggg'

p(s) codeg 0.2 00f 0 175 010

0.5250.2

0 3250 175f 0.175 010e 0.15 011d 0.125 100space 0.125 101c 0.1 110b 0.075 1110a 0.05 1111

1

0.475

0.3250.15

0.175

Shannon‐Fano

• S = 'aa bbb cccc ddddd eeeeee fffffffgggggggg'

• S in compressed is 117 bits

• 2*4 + 3*4 + 4*3 + 5*3 + 5*3 + 6*3 + 7*3 + 8*2 11= 117

• Shannon‐Fano not always optimal

• Sometimes 2 equal probable groups cannot be achieved

• Usually better than H+1 bits per symbol, when H is entropy.

Huffman code

• Works the opposite way.

• Start from least probable symbols and separate them with 0 and 1 (sufix)

• Add probabilities to form a "new symbol" with the new probabilityprobability

• Prepend new bits in front of old ones.

Huffman exampleChar Freq Code

space 7 111

a 4 010

e 4 000

f 3 1101

h 2 1010

i 2 1000

m 2 0111

n 2 0010

s 2 1011

t 2 0110

l 1 11001

o 1 00110

p 1 10011

r 1 11000

u 1 00111

x 1 10010"this is an example of a huffman tree"

• Huffman coding is optimal when the frequencies of input characters are powers of two. Arithmetic coding produces slight gains over Huffman coding but in practice theseover Huffman coding, but in practice these gains have not been large enough to offset arithmetic coding's higher computational complexity and patent royalties

• (as of November 2001/Jul2006, IBM owns patents on the core concepts of arithmetic coding in several jurisdictions).http://en.wikipedia.org/wiki/Arithmetic_coding#US_patents_on_arithmetic_coding

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Properties of Huffman coding

• Error tolerance quite good

• In case of the loss, adding or change of a single bit, the differences remain local to the place of the errorplace of the error

• Error usually remains quite local (proof?)

• Has been shown, the code is optimal

• Can be shown the average result is H+p+0.086, where H is the entropy and p is the probability of the most probable symbol

Move to Front

• Move to Front (MTF), Least recently used (LRU)

• Keep a list of last k symbols of S

C d• Code

– use the code for symbol.

– if in codebook, move to front

– if not in codebook, move to first, remove the last

• c.f. the handling of memory paging

• Other heuristics ...

Arithmetic (en)coding

• Arithmetic coding is a method for lossless data compression.

• It is a form of entropy encoding, but where other entropy encoding techniques separate the input message into its component symbols and replace each symbol with a code word, arithmetic coding encodes the entire message into a single number, a fraction n where (0.0 n < 1.0).

• Huffman coding is optimal for character encoding (one character one code• Huffman coding is optimal for character encoding (one character‐one code word) and simple to program. Arithmetic coding is better still, since it can allocate fractional bits, but more complicated.

• Wikipedia http://en.wikipedia.org/wiki/Arithmetic_encoding

• Entire message is a single floating point number, a fraction n where (0.0 n < 1.0).

• Every symbol gets a probability based on the model

• Probabilities represent non‐intersecting intervals

• Every text is such an interval

Let P(A)=0.1, P(B)=0.4, P(C)=0.5

A [0,0.1)AA [0 , 0.01)AB [0.01 , 0.05)AC [0.05 , 0.1 )

B [0.1,0.5)BA [0.1 , 0.14)BB [0.14 , 0.3 )BC [0.3 , 0.5 )

C [0.5,1)CA [0.5 , 0.55 )CB [0.55 , 0.75 )CC [0.75 , 1 )

• Add a EOF symbol.

• Problem with infinite precision arithmetics

• Alternative ‐ blockwise, use integer‐i h iarithmetics

• Works, if smallest pi not too small

• Best ratio

• Problem ‐ the speed, and error tolerance, small change has catastrophic effect

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• Arithmetic coding revisited by Alistair Moffat, Radford M. Neal, Ian H. Witten ‐ http://portal.acm.org/citation.cfm?id=874788

• Models for arithmetic coding ‐

HMM Hidd M k M d l• HMM Hidden Markov Models

• ...

• Context methods: Abrahamson dependency model

• Use the context to maximum, to predict the next symbol

• PPM ‐ Prediction by Partial Matching

• Several contexts, choose best

• Variations

Dictionary based

• Dictionary (symbol table) , list codes

• If not in disctionary, use escape

• Usual heuristics searches for longest repeat

• With fixed table one can search for optimal code

• With adaptive dictionary the optimal coding is NP‐complete

• Quite good for English language texts, for example

Lempel‐Ziv, LZ, LZ‐77

• Use the dictionary to memorise the previously compressed parts

• LZ‐77 pere

• Sliding window of previous text; and text to be compressed

/bbaaabbbaaffacbbaa…./...abbbaabab...

• Lookahead ‐ longest prefix that begins within the moving window, is encoded with [position,length]

• In example, [5,6]

• Fast! (Commercial software, e.g. PKZIP, Stacker, DoubleSpace, )

• Several alternative codes for same string (alternative substrings will match)

• Lempel‐Ziv compression from McGill Univ. http://www.cs.mcgill.ca/~cs251/OldCourses/1997/topic23/

Original LZ77

• Triples [position,length,next char]

• If output [a,b,c], advance by b+1 positions

• For each part of the triple the nr of bits is d d di i d l hreserved depending on window length

log(n‐f) + log(f) + 8where n is window size, and f is lookahead size

• Example: abbbbabbc [0,0,a] [0,0,b] [1,3,a] [3,2,c]

• In example the match actually overlaps with lookahead window

LZ‐78

• Dictionary

• Store strings from processed part of the message

• Next symbol is the longest match from dictionary, that matches the text to be processed

LZ78 (Zi d L l 1978)• LZ78 (Ziv and Lempel 1978)

• First, dictionary is empty, with index 0

• Code [i,c] ‐ refers to dictionary (word u at pos. i) and c is the next symbol

• Add uc to dictionary

• Example: ababcabc → [0,a][0,b][1,b][0,c][3,c]

LZW

• Code consists of indices only! • First, dictionary has every symbol /alphabet/

• Update dictionary like LZ78

• In decoding there is a danger: See abcabca – If abc is in dictionary y

– add abca to dictionary

– next is abca, output that code

– But when decoding, after abc it is not known that abca is in the dictionary

• Solution ‐ if the dictionary entry is used immediately after its creation, the 1st and last characters have to match

• Many representations for the dictionary.

• List, hash, sorted list, combination, binary tree, trie, suffix tree, ...

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LZJ

• Coding ‐ search for longest prefix.

• Code ‐ address of the trie node

• From the root of the trie, there is a transition b l (lik i )on every symbol (like in LZW).

• If out of memory, remove these nodes/branches that have been used only once

• In practice, h=6, dictionary has 8192 nodes

LZFG

• Effective LZ method

• From LZJ

• Create a suffix tree for the window

• Code ‐ node address plus nr opf characters from teh edge.

• The internal and leaf nodes with different codes

• small match directly... (?)

Burrows‐Wheeler

• See FAQ http://www.faqs.org/faqs/compression‐faq/part2/section‐9.html

• The method described in the original paper is really a composite of three different algorithms:

– the block sorting main engine (a lossless, very slightly expansive preprocessor),

– the move‐to‐front coder (a byte‐for‐byte simple, fast, locally adaptive noncompressive coder) and

– a simple statistical compressor (first order Huffman is mentioned as a candidate) eventually doing the compression.

• Of these three methods only the first two are discussed here as they are what constitutes the heart of the algorithm. These two algorithms combined form a completely reversible (lossless) transformation that ‐ with typical input ‐ skews the first order symbol distributions to make the data more compressible with simple methods. Intuitively speaking, the method transforms slack in the higher order probabilities of the input block (thus making them more even, whitening them) to slack in the lower order statistics. This effect is what is seen in the histogram of the resulting symbol data.

• Please, read the article by Mark Nelson:

• Data Compression with the Burrows‐Wheeler TransformMark Nelson, Dr. Dobb's Journal September, 1996. http://marknelson.us/1996/09/01/bwt/

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CODE: t: hat acts like this:<13><10><1

t: hat buffer to the constructor

t: hat corrupted the heap, or wo

W: hat goes up must come down<13

t h t h it i 't lik lt: hat happens, it isn't likely

w: hat if you want to dynamicall

t: hat indicates an error.<13><1

t: hat it removes arguments from

t: hat looks like this:<13><10><

t: hat looks something like this

t: hat looks something like this

t: hat once I detect the mangled

Syntactic compression

• Context Free Grammar for presenting the syntax tree

• Usually for source code

i i i ll• Assumption ‐ program is syntactically correct

• Comments

• Features, constants ‐ group by group

Image compression

• Many images, photos, sound, video, ...

Fax group 3

• Fax/group 3

• Black/white, 0/1 code

• Run‐length: 000111001000 → 3,3,2,1,3

• Variable‐length codes for representing run‐lengths.

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• Joint Photographic Experts Group JPEG 2000http://www.jpeg.org/jpeg2000/

• Image Compression ‐‐ JPEG from W.B. Pennebaker, J.L. Mitchell, "The JPEG Still Image Data Compression Standard", Van Nostrand Reinhold, 1993.

• Color image, 8 or 12 bits per pixel per color.

• Four modes Sequential Mode

• Lossless Mode

• Progressive Mode

• Hierarchical Mode

• DCT (Discrete Cosine Transform)

• from http://www.utdallas.edu/~aria/mcl/post/

• Lossy signal compression works on the basis of transmitting the "important" signal content, while omitting other parts (Quantization). To perform this quantization effectively, a linear de‐correlating transform is applied to the signal prior to quantization. All existing image and video coding standards use this approach. The most commonly used transform is the Discrete Cosine Transform (DCT) used in JPEG MPEG 1 MPEG 2 H 261 d H 263 d i d d F d il dJPEG, MPEG‐1, MPEG‐2, H.261 and H.263 and its descendants. For a detailed discussion of the theory behind quantization and justification of the usage of linear transforms, see reference [1] below.

• A brief overview of JPEG compression is as follows. The JPEG encoder partitions the image into 8x8 blocks of pixels. To each of these blocks it applies a 2‐dimensional DCT. The transform matrix is normalized (element‐wise) by a 8x8 quantization matrix and then rounded to the nearest integer. This operation is equivalent to applying different uniform quantizers to different frequency bands of the image. The high‐frequency image content can be quantized more coarsely than the low‐frequency content, due to two factors.

• L9_Compression/lena/

Vector quantization

• Vector quantization

• Dictionary‐meetod

• 2‐dimensional blocks

Discrete cosine transform

• A discrete cosine transform (DCT) expresses a sequence of finitely many data points in terms of a sum of cosine functions oscillating at different frequencies. DCTs are important to numerous applications in science and engineering, from lossy compression of audio and images (where small high‐frequency components can be discarded), to spectral methods for the numerical solution of partial differential equations. The use of cosine p qrather than sine functions is critical in these applications: for compression, it turns out that cosine functions are much more efficient (as explained below, fewer are needed to approximate a typical signal), whereas for differential equations the cosines express a particular choice of boundary conditions.

• http://en.wikipedia.org/wiki/Discrete_cosine_transform

2d DCT (type II) compared to the DFT. For both transforms, there is the magnitude of the spectrum on left and the histogram on right; both spectra are cropped to 1/4, to zoom the behaviour in the lower frequencies. The DCT concentrates most of the power on the lower frequencies.

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• In particular, a DCT is a Fourier‐related transformsimilar to the discrete Fourier transform (DFT), but using only real numbers. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), where in some variants the input and/or output data are shifted by half a sample. There are eight standard DCT variants, of which four are common.

• Digital Image Processing: http://www‐ee.uta.edu/dip/

Block Diagram of JPEG Baseline

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ENEE631 Digital Image Processing (Spring'04)

Lec13 – Transf. Coding & JPEG [62]

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JPEG Compression (Q=75%)



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1.4‐billion‐pixel digital camera

• Monday, November 24, 2008http://www.technologyreview.com/computing/21705/page1/

• Giant Camera Tracks Asteroids

• The camera will offer sharper broader viewsThe camera will offer sharper, broader views of the sky.

• The focal plane of each camera contains an almost complete 64 x 64 array of CCD devices, each containing approximately 600 x 600 pixels, for a total of about 1.4 gigapixels. The CCDs themselves employ the innovative technology called "orthogonal transfer", which is described below.

Diagram showing how an OTA chip is made up of 64 OTCCD cells, each of which has 600 x 600 pixels

Fractal compression

• Fractal Compression group at Waterloohttp://links.uwaterloo.ca/fractals.home.html

• A "Hitchhiker's Guide to Fractal Compression" For BeginnersFor Beginnersftp://links.uwaterloo.ca/pub/Fractals/Papers/Waterloo/vr95.pdf

• Encode using fractals.

• Search for regions that with a simple transformation can be similar to each other.

• Compressin ratio 20‐80

• Moving Pictures Experts Group ( http://www.chiariglione.org/mpeg/ )

• MPEG Compression : http://www.cs.cf.ac.uk/Dave/Multimedia/node255.html

S di id d i 2 6 bl k h h• Screen divided into 256 blocks, where the changes and movements are tracked

• Only differences from previous frame are shown

• Compression ratio 50‐100

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• When one compresses files, can we still use fast search techniques without decompressing frst?

• Sometimes, yes

• e.g. Udi Manber has developed a method

• Approximate Matching of Run‐Length Compressed StringsVeli Mäkinen, Gonzalo Navarro, Esko Ukkonen

• We focus on the problem of approximate matching of strings that have been compressed using run‐length encoding. Previous studies have concentrated on the problem of computing the longest common subsequence (LCS) between two strings of length m and n , compressed to m' and n' runs. We extend an existing algorithm for the LCS to the Levenshtein distance achieving O(m'n+n'm) complexity. Furthermore, we extend this algorithm to a weighted edit distance model, where the weights of the three basic edit operations can be chosen arbitrarily. This approach also gives an algorithm for approximate searching of a pattern of m letters (m' runs) in a text of n letters (n' runs) in O(mm'n') time. Then we propose improvements for a greedy algorithm for the LCS, and conjecture that the improved algorithm has O(m'n') expected case complexity. Experimental results are provided to support the conjecture.

Kolmogorov (or Algorithmic) complexity

• Kolmogorov, Chaitin, ...

• What is the compressed version of sequence '1234567891011121314151617181920212223242526...' ?

• Every symbol appears almost equally frequently, almost "random" by entropyrandom by entropy

• for i=1 to n do print i ;

• Algorithmic complexity (or Kolmogorov complexity) for string S is the length of the shortest program that reproduces S, often noted K(S)

• Conditional complexity : K(S|T). Reproduce S given T.

• http://en.wikipedia.org/wiki/Algorithmic_information_theory

• Algorithmic information theory is a field of study which attempts to capture the concept of complexity using tools from theoretical computer science. The chief idea is to define the complexity (or Kolmogorov complexity) of a string as the length of the shortest program which, when run without any g p g , yinput, outputs that string. Strings that can be produced by short programs are considered to be not very complex. This notion is surprisingly deep and can be used to state and prove impossibility results akin to Gödel's incompleteness theoremand Turing's halting problem.

• The field was developed by Andrey Kolmogorov and Gregory Chaitin starting in the late 1960s.

G J Chaitin

• http://www.cs.umaine.edu/~chaitin/

• Distance using K.

• d(S,T) = ( K(S|T) + K(T|S) ) / ( K(S) + K(T) )

• We cannot calculate K, but we can i iapproximate it

• E.g. by compression LZ, BWT, etc

• d(S,T) = ( C(ST) + C(TS) ) / ( C(S) + C(T) )

• Use of Kolmogorov Distance Identification of Web Page Authorship, Topic and DomainDavid Parry (PPT) in OSWIR 2005, 2005 workshop on Open Source Web Informationworkshop on Open Source Web Information Retrieval

• Informatsioonikaugus by Mart Sõmermaa, Fall 2003 (in Data Mining Research seminar)http://www.egeen.ee/u/vilo/edu/2003‐04/DM_seminar_2003_II/Raport/P06/main.pdf