compression 4

8/9/2019 Compression 4

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CPS 296.3:Algorithms in the Real World

Data Compression III


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Compression Outline

Introduction: Lossy s. Lossless! "en#hmar$s! %Information Theory: &ntropy! et#.

Probability Coding: 'u((man ) Arithmeti# Coding

Applications of Probability Coding: PP* ) others

Lempel-Ziv Algorithms:

+ L,--! gip!

+ L,-/! #ompress 01ot #oered in #lass

Other Lossless Algorithms "urros4WheelerLossy algorithms for images5P&! *P&! ...

Compressing graphs and meshes ""7


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Lempel4,i Algorithms

LZ!!0Sliding Windo"ariants: L,SS 0Lempel4,i4Storer4Symans$i

Applications: gzip! S8ueee! L'A! P7,P! ,OO

LZ!#0i#tionary "ased"ariants: L,W 0Lempel4,i4Wel#h! L,C

Applications: compress! ;! CC


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L,--: Sliding Windo Lempel4,i

Dictionaryand buffer>indos? are (i@ed lengthand slide ith the cursor$epeat:

Output %p& l& c'herep position o( the longest mat#h that starts in

the di#tionary 0relatie to the #ursorl length o( longest mat#hc ne@t #har in =u((er =eyond longest mat#h

Adan#e indo =y l) B

a a c a a c a b c a b a b a c

Dictionary

(previously coded)Lookahead

u!!er

"ursor


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L,-- e#oding

e#oder $eeps same di#tionary indo as en#oder.;or ea#h message it loo$s it up in the di#tionary and

inserts a #opy at the end o( the string

What i( l pD 0only part o( the message is in the

di#tionary.&.g. di#t abcd! #odeord (2,9,e)

E Simply #opy (rom le(t to rightfor (i = 0; i < lengt; i!!)

o"t#c"rsor!i$ = o"t#c"rsor%offset!i$

E Out abcdcdcdcdcdce


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Optimiations used =y gzip0#ont.

B. 'u((man #ode the positions! lengths and #hars2. 1on greedy: possi=ly use shorter mat#h so that

ne@t mat#h is =etter

3. Fse a hash ta=le to store the di#tionary.

+ 'ash $eys are all strings o( length 3 in thedi#tionary indo.

+ ;ind the longest mat#h ithin the #orre#thash =u#$et.

+ Puts a limit on the length o( the sear#h ithina =u#$et.

+ Within ea#h =u#$et store in order o( position


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Comparison to Lempel4,i -/

"oth L,-- and L,-/ and their ariants $eep a>di#tionary? o( re#ent strings that hae =een seen.


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Lempel4,i Algorithms Summary

Adapts ell to #hanges in the (ile 0e.g. a se#ond pass? and #ompress mu#h =etter.


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Compression Outline

Introduction: Lossy s. Lossless! "en#hmar$s! %Information Theory: &ntropy! et#.

Probability Coding: 'u((man ) Arithmeti# Coding

Applications of Probability Coding: PP* ) others

Lempel-Ziv Algorithms: L,--! gip! #ompress! %

Other Lossless Algorithms

+ "urros4Wheeler

+ AC"Lossy algorithms for images5P&! *P&! ...

Compressing graphs and meshes ""7


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"urros 4Wheeler

Currently near =est >=alan#ed? algorithm (or te@t"rea$s (ile into (i@ed4sie =lo#$s and en#odes ea#h

=lo#$ separately.

(or each bloc)

+ Sort ea#h #hara#ter =y its (ull #onte@t.


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"urros Wheeler: &@ample

LetJs en#ode: decode

Conte@t >raps? around. Last #har is most signi(i#ant.

Sort

Context

All rotations o( input


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"urros Wheeler e#oding

7ey dea: Can #onstru#t entire sorted ta=le (rom sorted#olumn aloneK ;irst: sorting the output gies last #olumn o(#onte@t:

Conte*t Output

c od e

d e

e c

e d

o d


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1o sort pairs in last #olumn o( #onte@t and output #olumnto (orm last to #olumns o( #onte@t:

Conte*t Output

c o

d e

d e

e ce d

o d

Conte*t Output

ec o

ed e

od e

de c

de d

co d


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Repeat until entire ta=le is #omplete. Pointer to (irst#hara#ter proides uni8ue de#oding.

*essage as d in (irst position! pre#eded in rapped(ashion =y e#ode: de#ode.


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Optimiation: onJt really hae to re=uild the hole #onte@tta=le.

What #hara#ter #omes a(terthe (irst #hara#ter! dBD

5ust hae to (ind dBin last

#olumn o( #onte@t and see hat(ollos it: eB.

O=seration: instan#es o( same#hara#ter o( output appear insame order in last #olumn o(#onte@t. 0Proo( is an e@er#ise.


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"urros4Wheeler: e#oding

Outputo

e

ec

d

d

Conte*tc

d

de

e

o

$an)

4

1

2

3

ran$? is the positiono( a #hara#ter i( it eresorted using a sta=lesort.


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"urros4Wheeler e#ode

;un#tion "We#ode0n! Start! nS *oe


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e#ode &@ample

+ $an)%+'

oI 6

e2 I

e6

#3 B

dB 2

d 3


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Oerie o(


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AC" 0Asso#iate Coder o( "uyanos$y

"onte(t "ontents

decode

dec odedecode

decode

decode

deco de

7eep di#tionary sorted =y #onte@t0the last #hara#ter is the mostsigni(i#ant

E ;ind longest mat#h (or #onte@t

E ;ind longest mat#h (or #ontentsE Code

E istan#e =eteen mat#hes inthe sorted order

E Length o( #ontents mat#h

'as aspe#ts o( "urros4Wheeler!and L,--

compression 4

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