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Because of the AEP  [link] ,

Pr x 1 l : - log ( P r ( x 1 l ) ) l 0 - H > ϵ 0 .

Therefore, for a typical input Pr ( x 1 l ) > 2 - l 0 ( H + ϵ ) .

Recall that the interval length is l 0 = log ( n w ) H + 2 ϵ , and so the probability that an interval cannot be found in the history is

2 - l 0 ( H + 2 ϵ ) 2 - l 0 ( H + ϵ ) = 2 - l 0 ϵ 0 .

For a long enough interval, this probability goes to zero.

Redundancy of parsing schemes

There are many Ziv-Lempel style parsing algorithms  [link] , [link] , [link] , and each of the variantshas different details, but the key idea is to find the longest match in a window of length n w . The length of the match is L , where we remind the reader that L log ( n w ) H .

Now, encoding L requires log ( n w ) bits, and so the per-symbol compression ratio is log ( n w ) L , which in the limit of large n w approaches the entropy rate H .

However, the encoding of L must also desribe its length, and often the symbol that follows the match. These require length log ( L ) log ( log ( n w ) ) , and the normalized (per-symbol) cost is

log ( log ( n w ) ) L = O log ( log ( n w ) ) log ( n w ) .

Therefore, the redundancy of Ziv-Lempel style compression algorithms is proportional to O log ( log ( n ) ) log ( n ) , which is much greater than the O log ( n ) n that we have seen for parametric sources. The fundamental reason why the redundancy is greater is that the class of non-parametric sources is much richer. Detailedredundancy analyses appear in a series of papers by Savari (c.f.  [link] ).

Parsing for lossy compression

The parsing schemes that we have seen can also be adapted to lossy compression. Let us describe several approaches along these lines.

Fixed length: The first scheme, due to Gupta et al.  [link] , constructs a codebook of size 2 L R ( D ) codewords, where L is the length of the phrase being matched and R ( D ) is the rate distortion function. The algorithm cannot search for perfect matches of the phrase, because this is lossy compression. Instead, it seeks the codeword that matches our input phrase most closely. It turns out that for large L the expected distortion of the lossy match will be approximately D per symbol.

Variable length: Another approach, due to Gioran and Kontoyiannis  [link] , constructs a single long database string, and searches for the longest match whose distortion w.r.t. the input is approximately D ; the location and length of the approximate match are encoded. Seeing that the database is of length 2 L R , encoding the location requires L R bits, and the D -match (a match with distortion D w.r.t. the input string) is typically of length L , giving a per-symbol rate of R ( D ) bits.

An advantage of the latter scheme by Gioran and Kontoyiannis  [link] is reduced memory use. The database is a string of length 2 L R ( D ) , instead of a codebook comprised of 2 L R ( D ) codewords, each of length L . On the other hand, the Gupta et al. algorithm  [link] has better R D performance, because it does not need to spend log ( L ) bits per phrase to describe its length. An improved algorithm, dubbed the hybrid algorithm by Gioran and Kontoyiannis, constructs a single database and performs fixed length coding for the best match of length L in the database. Therefore, it combines the memory usage of a single database approach with the R D performance of fixed length coding.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
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Source:  OpenStax, Universal algorithms in signal processing and communications. OpenStax CNX. May 16, 2013 Download for free at http://cnx.org/content/col11524/1.1
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