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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

what is variations in raman spectra for nanomaterials
Jyoti Reply
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Crow Reply
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RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
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Smarajit 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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