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As a consequence of the structure of y, it is easy to see that it can be compressed with the following redundancy,

r | T | ( r - 1 ) 2 log ( n | T | + O ( 1 ) ) + | T | log ( n ) ,

where the new | T | log ( n ) term arises from coding the locations of transitions between segments (states of the tree) in the BWT output. Not only is the BWT convenient for compression, but it is amenable to fast computation. Both the BWT and its inverse can be implemented in O ( n ) time. This combination of great compression and speed has made the BWT quite popular in compressors that have appeared since the late 1990s. For example, the bzip2 archiving package is very popular among network administrators.

That said, from a theoretical perspective the BWT suffers from an extraneous redundancy of | T | log ( n ) bits. Until this gap was resolved, the theoretical community still preferred the semi-predictive method or another approach based on mixtures.

Semi-predictive coding using the bwt

Another approach for using the BWT is to use y only for learning the MDL tree source T * . To do so, note that when the BWT is run, it is possible to track the correspondences between contexts and segments of the BWT output. Therefore, information about per-segment symbol count is available, and can be easily applied to perform the tree pruning procedure that we have seen. Not only that, but some BWT computation algorithms (e.g., suffix tree approaches) maintain this information for all context depths and not just bounded D . In short, the BWT allows to compute the minimizing tree T * in linear time  [link] .

Given the minimizing tree T * , it is not obvious how to determine which state generated each character of y (respectively, x ) in linear time. It has been shown by Martín et al.  [link] that this step can also be performed in linear time by developing a state machine whose states include the leaves of T * . The result is a two part code, where the first part computes the optimal T * via BWT, and the second part actually compresses x by tracking which state of T * generated each of the symbols. To summarize, we have a linear complexity algorithm for compressing and decompressing a source while achieving the redundancy bounds for the class of tree sources.

Context tree weighting

We discussed in [link] for the problem of encoding a transition between two known i.i.d. distributions that

1 n i = 1 n p θ i ( x ) > 1 n max i { p θ i ( x ) } .

Therefore, a mixture over all parameter values yields a greater probability (and thus lower coding length) than the maximizing approach. Keep in mind that finding the optimal MDL tree source T * is analogous to the plug-in approach, and it would reduce the coding length if we could assign the probability as a mixture over all possible trees, where we assign trees with fewer leaves a greater weight. That is, ideally we want to implement

Pr ( x ) = T 2 - | code ( T ) | · p T ( x ) ,

where | code ( T ) | is the length of the encoding procedure that we discussed for the tree structure T , and p T ( x ) is the probability for the sequence x under the model T .

Willems et al. showed how to implement such a mixture in a simple way over the class of tree sources of bounded depth D . As before, the algorithm proceeds in a bottom up manner from leaves toward the root. At leaves, the probability p s assigned to symbols that were generated within that context s is the Krichevsky-Trofimov probability, p K T ( s , x )   [link] . For s that is an internal node whose depth is less than D , the approach by Willems et al.  [link] is to mix ( i ) the probabilities of keeping the branches for 0s and 1s and ( ii ) pruning,

p s = 1 2 p K T ( s , x ) + 1 2 p 0 s · p 1 s .

It can be shown that this simple formula allows to implement a mixture over the class of bounded depth context tree sources, thus reducing the coding length w.r.t. the semi-predictive approach.

In fact, Willems later showed how to extend the context tree weighting (CTW) approach to tree sources of unbounded depth  [link] . Unfortunately, while the basic bounded depth CTW has complexity that is comparable to the BWT, the unboundedCTW has potentially higher complexity.

Questions & Answers

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
absolutely yes
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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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