# 5.5 Huffman coding

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A description of the Huffman source encoding algorithm.

One particular source coding algorithm is the Huffman encoding algorithm. It is a source coding algorithm which approaches, and sometimes achieves, Shannon's bound forsource compression. A brief discussion of the algorithm is also given in another module .

## Huffman encoding algorithm

• Sort source outputs in decreasing order of their probabilities
• Merge the two least-probable outputs into a single output whose probability is the sum of the corresponding probabilities.
• If the number of remaining outputs is more than 2, then go to step 1.
• Arbitrarily assign 0 and 1 as codewords for the two remaining outputs.
• If an output is the result of the merger of two outputs in a preceding step, append the current codeword with a 0 and a 1 toobtain the codeword the the preceding outputs and repeat step 5. If no output is preceded by another output in a preceding step, thenstop.

$X\in \{A, B, C, D\}$ with probabilities { $\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{1}{8}$ , $\frac{1}{8}$ }

$\mathrm{Average length}=\frac{1}{2}\times 1+\frac{1}{4}\times 2+\frac{1}{8}\times 3+\frac{1}{8}\times 3=\frac{14}{8}$ . As you may recall, the entropy of the source was also $H(X)=\frac{14}{8}$ . In this case, the Huffman code achieves the lower bound of $\frac{14}{8}\frac{\mathrm{bits}}{\mathrm{output}}$ .

In general, we can define average code length as

$\langle \rangle =\sum_{x\in \stackrel{}{X}} p(X, x)(x)$
where $\stackrel{}{X}$ is the set of possible values of $x$ .

It is not very hard to show that

$H(X)\ge \langle \rangle > H(X)+1$
For compressing single source output at a time, Huffman codes provide nearly optimum code lengths.

The drawbacks of Huffman coding

• Codes are variable length.
• The algorithm requires the knowledge of the probabilities, $p(X, x)$ for all $x\in \stackrel{}{X}$ .
Another powerful source coder that does not have the aboveshortcomings is Lempel and Ziv.

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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
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why?
what school?
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biomolecules are e building blocks of every organics and inorganic materials.
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anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
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how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
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for teaching engĺish at school how nano technology help us
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what is fullerene does it is used to make bukky balls
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what is the actual application of fullerenes nowadays?
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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.
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what is the Synthesis, properties,and applications of carbon nano chemistry
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is Bucky paper clear?
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Do you know which machine is used to that process?
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how to fabricate graphene ink ?
for screen printed electrodes ?
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What is lattice structure?
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or in general
Ebrahim
in general
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Graphene has a hexagonal structure
tahir
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Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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