Text Algorithms (6EAP)

Compression

Jaak Vilo 2012 fall

Jaak Vilo MTAT.03.190 Text Algorithms 1 Problem

• Compress – Text – Images, , sound, …

• Reduce space, efficient communicaon, etc… – Data deduplicaon

• Exact compression/decompression • • Managing Gigabytes: Compressing and Indexing Documents and Images • Ian H. Wien, Alistair Moffat, Timothy C. Bell • Hardcover: 519 pages Publisher: Morgan Kaufmann; 2nd Revised edion edion (11 May 1999) Language English ISBN-10: 1558605703 Links

• hp://datacompression.info/ • hp://en.wikipedia.org/wiki/Data_compression • Debra A. Lelewer and Daniel S. Hirschberg – hp://www.ics.uci.edu/~dan/pubs/DataCompression.html

• Compression FAQ hp://www.faqs.org/faqs/compression-faq/ • Informaon Theory Primer With an Appendix on Logarithms by Tom

Schneider hp://www.lecb.ncifcrf.gov/~toms/paper/primer/ • hp://www.cbloom.com/algs/index.html Problem

• Informaon transmission • Informaon storage • The data sizes are huge and growing – fax - 1.5 x 106 /page – photo: 2M pixels x 24bit = 6MB – 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, ...

– Human genome – 3.2Gbase. 30x sequencing => 100Gbase + quality info (+ raw data) – 1000 genomes, all individual genomes … What it’s about?

• Eliminaon of redundancy • Being able to predict… • Compression and decompression – Represent data in a more compact way – Decompression - restore original form • Lossy and – Lossless - restore in exact copy – Lossy - restore almost the same informaon • Useful when no 100% accuracy needed • voice, image, movies, ... • Decompression is determinisc (lossy in compression phase) • Can achieve much more effecve results Methods covered:

• Code words (Huffman coding) • Run-length encoding • Arithmec coding • Lempel-Ziv family (compress, gzip, zip, pkzip, ...) • Burrows-Wheeler family (bzip2) • Other methods, including images • Kolmogorov • Search from compressed texts Model

Model Model

Compressed Data data Data Encoder Decoder • Let pS be a of message S • The informaon content can be represented in terms of

• I(S) = -log( pS ) bits • If the p=1 then the informaon content is 0 (no new informaon) – 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 informaon content of S

– H(S) = pS · I(S) = -pS log( pS ) bits hp://en.wikipedia.org/wiki/Informaon_entropy

• Shannon's experiments with human predictors show an informaon rate of between .6 and 1.3 bits per character, depending on the experimental setup; the PPM compression algorithm can achieve a compression rao of 1.5 bits per character. • No compression can on average achieve beer 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(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 informaon 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 http://prize.hutter1.net/ • The data compression world is all abuzz about Marcus Huer’s recently announced 50,000 euro prize for record-breaking data compressors. Marcus, of the Swiss Dalle Molle Instute for Arficial Intelligence, apparently in cahoots with Florida compression maven Ma 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 aempt to silence people foolishly claiming to be able to compress random data. Marcus is instead looking for the most effecve way to reproduce the Wiki data, and he’s pung up real money as an incenve. The benchmark that contestants need to beat is that set by Ma Mahoney’s paq8f , the current record holder at 18.3 MB. (Alexander Ratushnyak’s submission of a variant looks to clock in at a dy 17.6 MB, and should soon be confirmed as the new standard.) • So why is an AI guy inserng himself into the world of compression? Well, Marcus realizes that good data compression is all about modeling the data. The beer you understand the data stream, the beer 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 demonstraon of Arficial Intelligence. hp://marknelson.us/2006/08/24/the-huer-prize/#comment-293

Model

Model Model

Compressed Data data Data Encoder Decoder Stac or adapve

• Stac model does not change during the compression • Adapve model can be updated during the process • Symbols not in message cannot have 0-probability • Semi-adapve 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? • Rao (t/p) t: original message length • p: compressed message length • In texts - bits per symbol • The me and memory used for compression • The me and memory used for decompression • error tolerance (e.g. self-correcng code) Shorter code words…

• 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

• hp://michael.dipperstein.com/rle/index.html • The string:

• "aaaabbcdeeeeefghhhij"

• may be replaced with

• "a4b2c1d1e5f1g1h3i1j1". • This is not shorter because 1-leer 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" Alphabecally ordered word-lists

resume 0resume retail 2tail retain 5n retard 4rd retire 3ire

Coding techniques

• Coding refers to techniques used to encode tokens or symbols. • Two of the best known coding algorithms are Huffman Coding and Arithmec Coding. • Coding algorithms are effecve 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 probabilies of symbols: • S = 'aa bbb cccc ddddd eeeeee fffffffgggggggg'

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

Algoritm Shannon-Fano

• Input: probabilies 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 unl 1 symbol remains Example 1 a 1/2 0 b 1/4 10 c 1/8 110 d 1/16 1110 e 1/32 11110 f 1/32 11111

Example 1

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

Shannon-Fano S = 'aa bbb cccc ddddd eeeeee fffffffgggggggg'

p(s) code g 0.2 00 0.2 0.525 f 0.175 010 0.175 0.325 0.15 e 0.15 011 1 d 0.125 100 space 0.125 101 0.475 c 0.1 110 b 0.075 1110 a 0.05 1111

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 = 117 • Shannon-Fano not always opmal • Somemes 2 equal probable groups cannot be achieved • Usually beer 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 probabilies to form a "new symbol" with the new probability • Prepend new bits in front of old ones. Char Freq Code space 7 111 Huffman example 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 "this is an example of a huffman tree" x 1 10010 • Huffman coding is opmal when the frequencies of input characters are powers of two. Arithmec coding produces slight gains over Huffman coding, but in pracce these gains have not been large enough to offset arithmec coding's higher computaonal complexity and patent royales • (as of November 2001/Jul2006, IBM owns patents on the core concepts of arithmec coding in several jurisdicons). hp://en.wikipedia.org/wiki/Arithmec_coding#US_patents_on_arithmec_coding Properes 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 error • Error usually remains quite local (proof?) • Has been shown, the code is opmal • 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 • 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 heuriscs ... Arithmec (en)coding

• Arithmec coding is a method for lossless data compression. • It is a form of , but where other entropy encoding techniques separate the input message into its component symbols and replace each symbol with a code word, arithmec coding encodes the enre message into a single number, a fracon n where (0.0 n < 1.0). • Huffman coding is opmal for character encoding (one character-one code word) and simple to program. Arithmec coding is beer sll, since it can allocate fraconal bits, but more complicated. • Wikipedia hp://en.wikipedia.org/wiki/Arithmec_encoding • Enre message is a single floang point number, a fracon n where (0.0 n < 1.0). • Every symbol gets a probability based on the model • Probabilies represent non-intersecng intervals • Every text is such an 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 arithmecs • Alternave - blockwise, use integer- arithmecs

• Works, if smallest pi not too small • Best rao • Problem - the speed, and error tolerance, small change has catastrophic effect • Invented by Jorma Rissanen (then at IBM) • Arithmec coding revisited by Alistair Moffat, Radford M. Neal, Ian H. Wien - hp://portal.acm.org/citaon.cfm?id=874788

• Models for arithmec coding - • HMM Hidden Markov Models • ... • Context methods: Abrahamson dependency model • Use the context to maximum, to predict the next symbol • PPM - Predicon by Paral Matching • Several contexts, choose best • Variaons Diconary based compression

• Diconary (symbol table) , list codes • If not in disconary, use escape • Usual heuriscs searches for longest repeat • With fixed table one can search for opmal code • With adapve diconary the opmal coding is NP-complete • Quite good for English language texts, for example Lempel-Ziv family, LZ, LZ-77

• Use the diconary to memorise the previously compressed parts • LZ-77 • Sliding window of previous text; and text to be compressed

/bbaaabbbaaffacbbaa…./...abbbaabab...

• Lookahead - longest prefix that begins within the moving window, is encoded with [posion,length] • In example, [5,6] • Fast! (Commercial soware, e.g. PKZIP, Stacker, DoubleSpace, ) • Several alternave codes for same string (alternave substrings will match) • Lempel-Ziv compression from McGill Univ. hp://www.cs.mcgill.ca/~cs251/OldCourses/1997/topic23/ Original LZ77 • Triples [posion,length,next char] • If output [a,b,c], advance by b+1 posions • For each part of the triple the nr of bits is reserved depending on window length ⎡ log(n-f) ⎤ + ⎡ log(f) ⎤ + 8 where 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

• Diconary • Store strings from processed part of the message • Next symbol is the longest match from diconary, that matches the text to be processed • LZ78 (Ziv and Lempel 1978) • First, diconary is empty, with index 0 • Code [i,c] - refers to diconary (word u at pos. i) and c is the next symbol • Add uc to diconary • Example: ababcabc → [0,a][0,b][1,b][0,c][3,c] LZW

• Code consists of indices only! • First, diconary has every symbol /alphabet/ • Update diconary like LZ78 • In decoding there is a danger: See abcabca – If abc is in diconary – add abca to diconary – next is abca, output that code – But when decoding, aer abc it is not known that abca is in the diconary • Soluon - if the diconary entry is used immediately aer its creaon, the 1st and last characters have to match

• Many representaons for the diconary. • List, hash, sorted list, combinaon, binary tree, trie, suffix tree, ... LZJ

• Coding - search for longest prefix. • Code - address of the trie node • From the root of the trie, there is a transion on every symbol (like in LZW). • If out of memory, remove these nodes/ branches that have been used only once • In pracce, h=6, diconary has 8192 nodes LZFG

• Effecve LZ method • From LZJ • Create a suffix tree for the window • Code - node address plus nr of characters from teh edge. • The internal and leaf nodes with different codes • small match directly... (?) Burrows-Wheeler

• See FAQ hp://www.faqs.org/faqs/compression-faq/part2/secon-9.html • The method described in the original paper is really a composite of three different algorithms: – the block sorng main engine (a lossless, very slightly expansive preprocessor), – the move-to-front coder (a byte-for-byte simple, fast, locally adapve noncompressive coder) and – a simple stascal compressor (first order Huffman is menoned as a candidate) eventually doing the compression. • Of these three methods only the first two are discussed here as they are what constutes the heart of the algorithm. These two algorithms combined form a completely reversible (lossless) transformaon that - with typical input - skews the first order symbol distribuons to make the data more compressible with simple methods. Intuively speaking, the method transforms slack in the higher order probabilies of the input block (thus making them more even, whitening them) to slack in the lower order stascs. This effect is what is seen in the histogram of the resulng symbol data. • Please, read the arcle by Mark Nelson: • Data Compression with the Burrows-Wheeler Transform Mark Nelson, Dr. Dobb's Journal September, 1996. hp://marknelson.us/1996/09/01/bwt/

Burrows-Wheeler Transform (BWT)

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Example

• Decode: errktreteoe.e

• Hint: . Is the last character, alphabecally first… Syntacc compression

• Context Free Grammar for presenng the syntax tree • Usually for source code • Assumpon - program is syntaccally correct • Comments • Features, constants - group by group

• 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 represenng run- lengths. • Joint Photographic Experts Group JPEG 2000 hp://www.jpeg.org/jpeg2000/ • Image Compression -- JPEG from W.B. Pennebaker, J.L. Mitchell, "The JPEG Sll Image Data Compression Standard", Van Nostrand Reinhold, 1993. • Color image, 8 or 12 bits per pixel per color. • Four modes Sequenal Mode • Lossless Mode • Progressive Mode • Hierarchical Mode • DCT (Discrete Cosine Transform) • from hp://www.utdallas.edu/~aria/mcl/post/ • Lossy signal compression works on the basis of transming the "important" signal content, while oming other parts (Quanzaon). To perform this quanzaon effecvely, a linear de-correlang transform is applied to the signal prior to quanzaon. All exisng 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 and H.263 and its descendants. For a detailed discussion of the theory behind quanzaon and jusficaon of the usage of linear transforms, see reference [1] below. • A brief overview of JPEG compression is as follows. The JPEG encoder parons 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 quanzaon matrix and then rounded to the nearest integer. This operaon is equivalent to applying different uniform quanzers to different frequency bands of the image. The high-frequency image content can be quanzed more coarsely than the low-frequency content, due to two factors. • L9_Compression/lena/ Lena Vector quanzaon

• Vector quanzaon • Diconary-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 funcons oscillang at different frequencies. DCTs are important to numerous applicaons 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 soluon of paral differenal equaons. The use of cosine rather than sine funcons is crical in these applicaons: for compression, it turns out that cosine funcons are much more efficient (as explained below, fewer are needed to approximate a typical signal), whereas for differenal equaons the cosines express a parcular choice of boundary condions.

• hp://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. • In parcular, a DCT is a Fourier-related transform similar to the discrete (DFT), but using only real numbers. DCTs are equivalent to DFTs of roughly twice the length, operang on real data with even symmetry (since the Fourier transform of a real and even funcon is real and even), where in some variants the input and/or output data are shied by half a sample. There are eight standard DCT variants, of which four are common.

• Digital Image Processing: hp://www-ee.uta.edu/dip/ Block Diagram of JPEG Baseline From Wallace’s JPEG tutorial (1993) JPEG tutorial Wallace’s From

ENEE631 Digital Image Lec13 – Transf. Coding & JPEG [65] Processing (Spring'04) s EE330 (Princeton) (Princeton) s EE330 ’ From Liu 475 x 330 x 3 = 157 KB luminance ENEE631 Digital Image Lec13 – Transf. Coding & JPEG [66] Processing (Spring'04) RGB Components s EE330 (Princeton) (Princeton) s EE330 ’ From Liu

ENEE631 Digital Image Lec13 – Transf. Coding & JPEG [67] Processing (Spring'04) Y U V (Y Cb Cr) Components s EE330 (Princeton) (Princeton) s EE330 ’ From Liu Assign more bits to Y, less bits to Cb and Cr ENEE631 Digital Image Lec13 – Transf. Coding & JPEG [68] Processing (Spring'04) JPEG Compression (Q=75%) s EE330 (Princeton) (Princeton) s EE330 ’

From Liu 45 KB, compression ration ~ 4:1

ENEE631 Digital Image Lec13 – Transf. Coding & JPEG [69] Processing (Spring'04) JPEG Compression (Q=75%) s EE330 (Princeton) (Princeton) s EE330 ’

From Liu 45 KB, compression ration ~ 4:1

ENEE631 Digital Image Lec13 – Transf. Coding & JPEG [70] Processing (Spring'04) JPEG Compression (Q=75% & 30%) s EE330 (Princeton) (Princeton) s EE330 ’

From Liu 45 KB 22 KB

ENEE631 Digital Image Lec13 – Transf. Coding & JPEG [71] Processing (Spring'04) Uncompressed (100KB)

JPEG 75% (18KB)

UMCP ENEE408G Slides (created by M.Wu & R.Liu2002)& © ENEE408G Slides (created byM.Wu UMCP JPEG 50% (12KB)

JPEG 30% (9KB)

JPEG 10% (5KB) ENEE631 Digital Image Lec13 – Transf. Coding & JPEG [72] Processing (Spring'04) 1.4-billion-pixel digital camera

• Monday, November 24, 2008 hp://www.technologyreview.com/compung/21705/page1/ • Giant Camera Tracks Asteroids • The 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 innovave 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 compression

• group at Waterloo hp://links.uwaterloo.ca/.home.html • A "Hitchhiker's Guide to Fractal Compression" For Beginners p://links.uwaterloo.ca/pub/Fractals/Papers/ Waterloo/vr95.pdf • Encode using fractals. • Search for regions that with a simple transformaon can be similar to each other. • Compressin rao 20-80 • Moving Pictures Experts Group ( hp://www.chiariglione.org/mpeg/ )

• MPEG Compression : hp://www.cs.cf.ac.uk/Dave/Mulmedia/node255.html • Screen divided into 256 blocks, where the changes and movements are tracked • Only differences from previous frame are shown • Compression rao 50-100 • When one compresses files, can we sll use fast search techniques without decompressing frst? • Somemes, yes • e.g. Udi Manber has developed a method • Approximate Matching of Run-Length Compressed Strings Veli 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 compung the longest common subsequence (LCS) between two strings of length m and n , compressed to m' and n' runs. We extend an exisng algorithm for the LCS to the 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 operaons can be chosen arbitrarily. This approach also gives an algorithm for approximate searching of a paern of m leers (m' runs) in a text of n leers (n' runs) in O(mm'n') me. 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, Chain, ... • What is the compressed version of sequence '1234567891011121314151617181920212223242526...' ? • Every symbol appears almost equally frequently, almost "random" 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, oen noted K(S) • Condional complexity : K(S|T). Reproduce S given T. • hp://en.wikipedia.org/wiki/Algorithmic_informaon_theory • Algorithmic informaon theory is a field of study which aempts to capture the concept of complexity using tools from theorecal . 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 input, outputs that string. Strings that can be produced by short programs are considered to be not very complex. This noon is surprisingly deep and can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halng problem. • The field was developed by and Gregory Chain starng in the late 1960s. Kolmogorov complexity: size of circle in bits...

Model Model

Compressed Data data Data Encoder Decoder G J Chain

• hp://www.cs.umaine.edu/~chain/ • Distance using K. • d(S,T) = ( K(S|T) + K(T|S) ) / ( K(S) + K(T) ) • We cannot calculate K, but we can approximate it • E.g. by compression LZ, BWT, etc • d(S,T) = ( C(ST) + C(TS) ) / ( C(S) + C(T) ) • Use of Kolmogorov Distance Idenficaon of Web Page Authorship, Topic and Domain David Parry (PPT) in OSWIR 2005, 2005 workshop on Open Source Web Informaon Retrieval • Informatsioonikaugus by Mart Sõmermaa, Fall 2003 (in Data Mining Research seminar) hp://www.egeen.ee/u/vilo/edu/2003-04/DM_seminar_2003_II/Raport/P06/main.pdf