Cjb0912010 lz algorithms
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This document discusses Lempel-Ziv algorithms for lossless data compression. It introduces the LZ77 and LZ78 algorithms, which use adaptive dictionaries to encode repeated patterns in data. The document describes how LZ77 uses a sliding window to find the longest matching string and represents it with an offset-length pair. It also explains how LZ78 builds an explicit dictionary as it encodes. The document provides examples and discusses improvements made to these original Lempel-Ziv algorithms.
![Explanation
• Offset = search pointer – match pointer (o = 7)
• Length of match = number of consecutive letters matched
(l = 4)
• Code word (c = C(r)), where C(r) is the code word for r
• Encoding triple: <o, l, c> = <7, 4, C(r)>
• If FLC is used and alphabet size is |A|, <o, l, c> can be
encoded with [log2S] + [log2W] + [log2|A|] bits.
Ramaiah School of Advanced Studies –Bangalore
M.S 8](https://image.slidesharecdn.com/cjb0912010lzalgorithms-130124202720-phpapp01/85/Cjb0912010-lz-algorithms-8-320.jpg)
Cjb0912010 lz algorithms
- 1. LIMPEL ZIV ALGORITHMS BY S T RAJAN CSN-CJB0912010 Ramaiah School of Advanced Studies –Bangalore M.S 1
- 2. INTRODUCTION • Lempel-Ziv is a lossless date compression method algorithm • The generalized idea comes from pigeonhole principle . • If N items are placed into M pigeonholes where n>m . Fig: A pigeonhole • An image of pigeons in holes. Here there are n = 10 pigeons in m = 9 holes. Since 10 is greater than 9, the pigeonhole principle says that at least one hole has more than one pigeon • This concept may varies with input.so it cannot applied every time. Ramaiah School of Advanced Studies -Bangalore M.S – 2
- 3. Lempel Ziv Algorithms • Lossless Data compression is technique used to produce the original information from a compressed data. • Like Huffman coding ,run length coding ,Arithmetic coding etc., Lempel- Ziv is a lossless data compression technique used more often. • The Lempel Ziv algorithms belong to yet another category of lossless compression techniques known as dictionary coders. • Abraham Lempel & Jacob ziv together published their first compression method is sometimes referred to as "LZ77," for the year 1977, in which the duo published an article entitled "A Universal Algorithm for Sequential Data Compression" .The pair wrote another paper in 1978 outlining another dictionary approach know as LZ78 algorithm which was modified by Terry Welch in 1984. Ramaiah School of Advanced Studies –Bangalore M.S 3
- 4. Limpel Ziv Algorithm Family LZ77 LZ78 LZJ LZR LZSS LZH LZB LZFG LZT LZW LZC LZMW APPLICATIONS: APPLICATIONS: • ZIP GIF • GZIP V.42 • STACKER COMPRESS Fig 1: Limpel Ziv Algorithm Family Ramaiah School of Advanced Studies –Bangalore M.S 4
- 5. Types of Dictionary • The dictionary holds a list of strings of symbols and it may be static or dynamic (adaptive). • Static dictionary – permanent, sometimes allowing the addition of strings but no deletions • Dynamic dictionary – holding strings previously found in the input stream, allowing for additions and deletions of strings as new input is being read • LZ Algorithms are used in “ADAPTIVE DICTIONARY” • The dictionary is being built in a single pass, while at the same time encoding take places. • It continuously rewrites the dictionary for a file, discarding patterns it previously included and adding new ones when necessary. Ramaiah School of Advanced Studies –Bangalore M.S 5
- 6. LZ77 General approach • Dictionary is a portion of the previously encoded sequence • Use a sliding window for compression Mechanism • Find the maximum length match for the string pointed to by the search pointer in the search buffer, and encode it Rationale • If patterns tend to repeat locally, we should be able to get more efficient representation Ramaiah School of Advanced Studies –Bangalore M.S 6
- 7. LZ77 • Sliding window is composed of a search buffer and a look ahead buffer (note: window size W = S + LA). Match pointer search pointer a_ _ a br a - a da br a r r a r r a_ look ahead buffer Search buffer (size LA=7) (size S=8) Ramaiah School of Advanced Studies –Bangalore M.S 7
- 8. Explanation • Offset = search pointer – match pointer (o = 7) • Length of match = number of consecutive letters matched (l = 4) • Code word (c = C(r)), where C(r) is the code word for r • Encoding triple: <o, l, c> = <7, 4, C(r)> • If FLC is used and alphabet size is |A|, <o, l, c> can be encoded with [log2S] + [log2W] + [log2|A|] bits. Ramaiah School of Advanced Studies –Bangalore M.S 8
- 9. Possible Cases for Triples • There could be three different possibilities that may be encountered during the coding process: -No match for the next character to be encoded in the window -There is a match -The matched string extends inside the look-ahead buffer • For each of these cases, we have a triple to signal the case to the decoder. Ramaiah School of Advanced Studies –Bangalore M.S 9
- 10. ENCODING • Sequence cabracadabrarrarrad - |cadabrar|rarrad| W = 13, S = 7 |cadabrar|rarrad| - |cabraca|dabrar|rarrad |cadabrar|rarrad| no match for d send <3, 3, C(r)> send <0, 0, C(d)> Could we do better? -|abracad|abrarr|arrad Send <3, 5, C(d)> instead |abracad|abrarr|arrad |abracad|abrarr|arrad |abracad|abrarr|arrad send <7, 4, C(r)> Ramaiah School of Advanced Studies –Bangalore M.S 10
- 11. DECODING • Current input: <0, 0, C(d)> <7, 4, C(r)> <3, 5, C(d)> • Current output: cabraca Decode: <0, 0, C(d)> Decode C(d): c|abracad| Decode: <7, 4, C(r)> Start with the first „a‟, copy four letters: cabra|cadabra Decode C(r): cabrac|adabrar Decode: <3, 5, C(d)> Start with the first „r‟, copy three letters: cabracada|brarrar| Copy two more letters: cabracadabr|arrarar| Decode C(d): cabracadabrarrarard Ramaiah School of Advanced Studies –Bangalore M.S 11
- 12. Algorithm while (lookAheadBuffer not empty) { get a reference (position, length) to longest match; if (length > 0) { output (position, length, next symbol); shift the window length+1 positions along; } else { output (0, 0, first symbol in the lookahead buffer); shift the window 1 character along; } } Ramaiah School of Advanced Studies –Bangalore M.S 12
- 13. Points • For LZ77, we have -Adaptive scheme, no prior knowledge -Asymptotically approaches the source statistics - Assumes that recurring patterns close to each others • Possible improvements -Variable-bit encoding: PKZip, zip, gzip, …, etc., uses a variable-length coder to encode <o, l, c>. -Variable buffer size: larger buffer requires faster searches - Elimination of <0, 0, C(x)> -LZSS sends a flag bit to signal whether the next “token” is an <o, l> pair or the codeword of a symbol Ramaiah School of Advanced Studies –Bangalore M.S 13
- 14. Improvements • LZR The Lempel - Ziv - Renau modification allows pointers to reference anything that has been encoded without being limited by the length of the search. • LZSS The popular modification by Storer and Szymanski (1982) which is used for the mandatory inclusion of the next non-matching symbol into each codeword will lead to situations in which the symbol is being explicitly coded despite the possibility of it being part of the next match. • LZB LZB uses an elaborate scheme for encoding the references and lengths with varying sizes. • LZH The LZH implementation employs Huffman coding to compress the pointers. Ramaiah School of Advanced Studies –Bangalore M.S 14
- 15. LZ78 • LZ78 improvements from LZ77 -No search buffer – explicit dictionary instead -Encoder/decoder must build dictionary in sync - Encoding: <i, c> i = index in dictionary table c = code of the following character • Example: encode the following contents wabba_wabba_wabba_wabba_woo_woo_woo Ramaiah School of Advanced Studies –Bangalore M.S 15
- 16. EXAMPLE-1 • Input: wabba_wabba_wabba_wabba_woo_woo_woo • Dictionaries: Final Dictionary Initial dictionary is empty Encoder output index entry <0,c(w)> 1 w index entry <0,c(a)> 2 a <0,c(b)> 3 b <3,c(a)> 4 ba <0,c(_)> 5 _ <1,c(a)> 6 wa <3,c(b)> 7 bb <2,c(_)> 8 a_ Ramaiah School of Advanced Studies –Bangalore M.S 16
- 17. EXAMPLE(Continue..) Encoder index entry output <6,c(b)> 9 wab <4,c(_)> 10 ba_ <9,c(b)> 11 wabb <8,c(w)> 12 a_w <0,c(o)> 13 o <13,c(_)> 14 o_ <13,c(o)> 15 wo <1,c(w)> 16 o_w <13,c(o)> 17 oo Ramaiah School of Advanced Studies –Bangalore M.S 17
- 18. Remarks • Observation If we keep on encoding, the dictionary will keep on growing • Possible solutions Stop growing the dictionary Effectively switch to a static dictionary Prune it Based on usage statistics Reset it Start all over again • The best solution depends on the knowledge of the source Ramaiah School of Advanced Studies –Bangalore M.S 18
- 19. Improvements • LZ78 has limitation as it grows explicitly. • LZW was developed by Terry Welch • The dictionary has to be initialized with all the symbols of the input alphabet and this initial dictionary needs to be made known to the decoder. • IDEA • Instead of <i, c>, encode i only Ramaiah School of Advanced Studies –Bangalore M.S 19
- 20. • Input: wabba_wabba_wabba_wabba_woo_woo_woo • OUTPUT: 5 2 3 3 2 1 6 8 10 12 9 11 7 16 5 4 4 11 21 23 4 Final • Dictionaries: INDEX ENTRY INDEX ENTRY Intial Dictionary 1 _ 14 a_w 2 a 15 wabb INDEX ENTRY 3 b 16 ba_ 1 _ 4 o 17 _wa 2 a 5 w 18 abb 3 b 6 wa 19 ba_w 4 o 7 ab 20 wo 5 w 8 bb 21 oo 9 ba 22 o_ 10 a_ 23 _wo 11 _w 24 oo_ 12 wab 25 _woo 12 bba Ramaiah School of Advanced Studies –Bangalore M.S 20
- 21. Algorithm while (!done) read next symbol into a if (p*a) is in dictionary // Note: „*‟ stands for concatenation p = p*a else send out index of p add p*a to the dictionary p=a end Ramaiah School of Advanced Studies –Bangalore M.S 21
- 22. APPLICATION:COMPRESS • An early implementation of LZW • Adaptive dictionary, starts with 2^bmax–1entries • Dictionary grows up to double in size (2bmax) • User can configure max codeword length bmax = 9~16 • When dictionary reaches 2bmax entries, it becomes a static dictionary encoder • If compression ratio falls below a threshold, dictionary is reset. APPLICATION : GIF IMAGES PNG IMAGES Ramaiah School of Advanced Studies –Bangalore M.S 22
- 23. References • BELL, T. C., CLEARY, J. G., AND WITTEN, I. H. Text Compression. Prentice Hall, Upper Sadle River, NJ, 1990. • SAYOOD, K. Introduction to Data Compression. Academic Press, San Diego, CA, 1996, 2000. • ZIV, J., AND LEMPEL, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23 (1977), 337 • ZIV, J., AND LEMPEL, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24 (1978),530–536. Ramaiah School of Advanced Studies –Bangalore M.S 23