# 0.4 Universal compression for context tree sources  (Page 2/3)

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We note in past that there is no need to spend a bit to encode leaves of depth $D$ . To see this, consider a procedure for encoding the structure of a tree:

Consider the tree sourced depicted in [link] . In order to encode the structure of this tree, we will utilize the followingprocedure. (Such a procedure has appeared, for example, in  [link] .)

Start from root. [procedure(root)]
1. If node $S$ is of depth $D$ (maximum), then return.
2. If node $S$ is internal node, then {
encode 0
procedure(0S)
procedure(1S)
} else encode 1.
3. return.

Let us now simulate the procedure, the procedure will traverse through the following states of the tree in [link] while outputting the corresponding bits.

 Source root 0 1 01 001 101 11 Encoded symbol 0 1 0 0 1

Returning to tree pruning, following [link] we see that we must initialize $\text{MDL}\left(s\right)=\text{KT}\left({n}_{x}\left(s,0\right),{n}_{x}\left(s,1\right)\right)$ for $s$ of full depth $|s|=D$ without the extra bit.

At the end of the pruning procedure, ${T}_{\left\{\right\}}^{*}$ the maximizing tree for the root, will be the optimal tree for universal coding.

## Burrows wheeler transform

The Burrows Wheeler transform (BWT) was proposed by Burrows and Wheeler in 1994  [link] (see also the analysis by Effros et al.  [link] and references therein). It is an invertible permutation sort that sorts symbols according to their contexts. Thatway, the symbols that were generated by the same state of the context tree are grouped together, which as we will see is advantageous.

To compute the BWT, we first compute all cyclical shifts of the input $x$ . Next, we sort the cyclical shifts.The output of the BWT consists of $y$ , the last column of the matrix of sorted shifts, and $i$ the index of the original version. We illustrate with an example.

Consider the input $x=banana$ . First, we compute the cyclic shifts and their sorts.

 All Shifts Sorted banana abanan abanan anaban nabana ananab anaban banana nanaba nabana ananab nanaba

The output of the BWT consists of $y=nnbaaa$ , the last column of the matrix of sorted shifts (to the right), and the index $i=4$ containing the original input.

Interestingly, we can recover $x$ from $y$ and $i$ . Seeing that $y$ is structured and thus quite compressible, the BWT can be used as a compression system; a building block that illustrates such a system appearsin [link] .

To see that the BWT is invertible, let us work out how to do this by continuing our example.

In the matrix of sorted shifts, column 1 is a sorted version of column  $n$ , which we know.

 Column 1 Column n a n a n a b b a n a n a

Now take column  $n$ and put it before column 1:

 Column n Column 1 n a n a b a a b a n a n

We now sort these rows, which each consist of 2 symbols: $ab$ , $an$ , $an$ , $ba$ , $na$ , and $na$ . Now fill column 2 of the sorted shifts matrix accordingly.

 Columns 1–2 Column  $n$ ab n an n an b ba a na a na a

The entire matrix can be unraveled, and the row containing the original $x$ is indexed by $i$ .

What is the BWT good for? The key property of the BWT is that symbols generated by the same state are grouped together in $y$ . To see this, note how the last column  $n$ can be rotated to a position to the left of column 1, and symbols that came before the same prefix appear together.(To bunch together symbols generated by the same suffix, we can reverse the order of symbols in $x$ before running the BWT.) Therefore, $y$ has the form of a piecewise i.i.d. sequence  [link] , where segments generated by the same state of the context tree are bunched together.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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