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Abdullah Aldahami(11074595)
April 6, 2010 1
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Huffman Coding is a simple algorithm that generates a set of variable sized codes with the minimum average size.
Huffman codes are part of several data formats as ZIP, MPEG and JPEG.
The code is generated based on the estimated probability of occurrence.
Huffman coding works by creating an optimal binary tree of nodes, that can be stored in a regular array.
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The method starts by building a list of all the alphabet symbols in descending order of their probabilities (frequency of appearance).
It then construct a tree from bottom to top.
Step by step, the two symbols with the smallest probabilities are selected; added to the top.
When the tree is completed, the codes of the symbols are assigned.
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Example: circuit elements in digital computations
Summation of frequencies (Number of events) is 40
Character
Frequency
i 6
t 5
space 4
c 3
e 3
n 3
u 2
l 2
Character
Frequency
m 2
s 2
a 2
o 2
r 1
d 1
g 1
p 1
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Example: circuit elements in digital computations
r 1 d 1 g 1 p 1
2u 2 l 2 m 2 2 a 2 o 2s 2
4 4 4 4c 3 e 3 n 3‘ ‘
4
7t 5 7i 6 7 8
12
13
13
25
40
0 1
0 1
0 1 0 1
0 10 10 1
0 10 1
0 1
0 10 1
0 1 0 1
0 1
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So, the code will be generated as follows:
Character
Frequency
Code CodeLengt
h
TotalLengt
h
i 6 010 3 18t 5 000 3 15
space 4 110 3 12c 3 0010 4 12e 3 0110 4 12n 3 100 3 9u 2 0011
05
10l 2 0111
05
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Character
Frequency
Code CodeLengt
h
TotalLengt
h
m 2 01111
510
s 2 1001 4 8a 2 1110 4 8o 2 1111 4 8r 1 0011
106
6d 1 0011
116
6g 1 1000
05
5p 1 1000
15
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Total is 154 bits with Huffman Coding compared to 240 bits with no compression
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Input
Symbol i t ‘ ’ c e n u l
ProbabilityP(x) 0.15
0.125
0.10.07
50.07
50.07
50.05 0.05
Output
Code 010 000 110 0010 0110 1000011
00111
0Code
length(in bits) (Li)
3 3 3 4 4 3 5 5
Weighted path length
Li ×P(x)0.45
0.375
0.3 0.3 0.30.22
50.25 0.25
Optimality
Probability budget (2-
Li)
1/8 1/8 1/8 1/16 1/16 1/8 1/32 1/32
Information of a
Message I(x)
= – log2 P(x)
2.74 3.00 3.32 3.74 3.74 3.74 4.32 4.32
Entropy H(x)
=-P(x) log2 P(x)
0.411
0.375
0.332
0.280
0.280
0.280
0.216
0.216
• Entropy is a measure defined in information theory that quantifies the information of an information source.•The measure entropy gives an impression about the success of a data compression process.
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Input
Symbol m s a o r d g p Sum
ProbabilityP(x) 0.05 0.05 0.05 0.05 0.025 0.025
0.025
0.025 = 1
Output
Code01111
1001
1110
1111
001110
001111
10000
10001
Code length
(in bits) (Li)5 4 4 4 6 6 5 5
Weighted path length
Li ×P(x)0.25 0.2 0.2 0.2 0.15 0.15
0.125
0.125 3.85
Optimality
Probability budget (2-
Li)
1/32 1/16 1/16 1/16 1/64 1/64 1/32 1/32 = 1
Information of a
Message I(x)
= – log2 P(x)
4.32 4.32 4.32 4.32 5.32 5.32 5.32 5.32
Entropy H(x)
=-P(x) log2 P(x)
0.216
0.216
0.216
0.216
0.133 0.1330.13
30.13
3
3.787
Bit/sym
•The sum of the probability budgets across all symbols is always less than or equal to one. In this example, the sum is equal to one; as a result, the code is termed a complete code.
• Huffman coding approaches the optimum on 98.36% = (3.787 / 3.85) *100
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Static probability distribution (Static Huffman Coding)
Coding procedures with static Huffman codes operate with a predefined code tree,
previously defined for any type of data and is independent from the particular contents.
The primary problem of a static, predefined code tree arises, if the real probability
distribution strongly differs from the assumptions. In this case the compression rate
decreases drastically.
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Adaptive probability distribution (Adaptive Huffman Coding)
The adaptive coding procedure uses a code tree that is permanently adapted to the
previously encoded or decoded data.
Starting with an empty tree or a standard distribution.
This variant is characterized by its minimum requirements for header data, but the
attainable compression rate is unfavourable at the beginning of the coding or for small
files.
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