<< Chapter < Page Chapter >> Page >

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] .)

Tree used in Example 10 to demonstrate how the structure of the source is encoded.
Tree used in [link] to demonstrate how the structure of the source is encoded.

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
} 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 MDL ( s ) = KT ( n x ( s , 0 ) , n x ( s , 1 ) ) for s of full depth | s | = D without the extra bit.

At the end of the pruning procedure, T { } * 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 = b a n a n a . 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 = n n b a a a , 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] .

Typical compression system using the Burrows Wheeler transform
Typical compression system using the Burrows Wheeler transform  [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: a b , a n , a n , b a , n a , and n a . 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.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

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
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Universal algorithms in signal processing and communications' conversation and receive update notifications?

Rachel Woolard
Start Quiz
Dewey Compton
Start Exam