Cjb0912010 lz algorithms

LIMPEL ZIV ALGORITHMS

BY
S T RAJAN
CSN-CJB0912010

 Ramaiah School of Advanced Studies –Bangalore
M.S 1

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

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.

M.S 3

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

M.S 4

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.

M.S 5

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

M.S 6

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)
M.S 7

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.

M.S 8

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.

M.S 9

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

M.S 11

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;
}
}
M.S 12

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
M.S 13

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.
M.S 14

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

M.S 15

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_

M.S 16

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

M.S 17

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
M.S 18

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

M.S 19

• 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
M.S 20

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

M.S 21

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

M.S 22

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.

M.S 23

Cjb0912010 lz algorithms

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