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DATA COMPRESSION USING HUFFMAN CODING

Rahul V. Khanwani

Roll No. 47

Department Of Computer Science

HUFFMAN CODING

• Huffman Coding Algorithm— a bottom-up approach.

• The Huffman coding is a procedure to generate a binary code tree. The algorithm invented by David Huffman in 1952 ensures that the probability for the occurrence of every symbol results in its code length.

• Huffman coding could perform effective data compression by reducing the amount of redundancy in the coding of symbols. Rahul Khanvani For More Visit Binarybuzz.wordpress.com

Huffman Coding Algorithm 1. Initialization: Put all symbols on a list sorted according

to their frequency counts.

2. Repeat until the list has only one symbol left:

1. From the list pick two symbols with the lowest frequency counts

2. Form a Huffman sub-tree that has these two symbols as child nodes and create a parent node.

3. Assign the sum of the children’s frequency counts to the parent and insert it into the list such that the order is maintained.

4. Delete the children from the list.

3. Assign a codeword for each leaf based on the path from the root.

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Example:

Symbol Count

A 15

B 7

C 6

D 6

E 5


Constructing A Tree of Nodes Who Has Minimum Occurance

(11)

D(6) E(5)


Constructing A Tree of Nodes Who Has Minimum Occurance

17

C(6) (11)

D(6) E(5)


Re-Constructing A Tree of Nodes Who Has Minimum Occurance

17

(13)

B(7) C(6)

(11)

D(6) E(5)


Re-Constructing A Tree of Nodes Who Has Minimum Occurance

(39)

A(15) (24)

(13)

B(7) C(6)

(11)

D(6) E(5)


Huffman Coding Result

Symbol Count Bits

A 15 0

B 7 100

C 6 101

D 6 110

E 5 111


Comparison Of Huffman And Shanon-Fano Coding Algorithm Symbol Count Shanon-

Fano Bit Size

Huffman Bit Size

Shanon Fano Total Bits

Huffman Total Bits

A 15 2 1 30 15

B 7 2 3 14 21

C 6 2 3 12 18

D 6 3 3 18 18

E 5 3 3 15 15

Total 89 87


Comparison Conclusion

• Shannon-Fano and Huffman coding are close in performance.

• But Huffman coding will always at least equal the efficiency of Shannon-Fano coding, so it has become the predominant coding method of its type.

• both algorithms take a similar amount of processing power.

• it seems sensible to take the one that gives slightly better performance.

• Huffman was able to prove that this coding method cannot be improved on with any other integral bit-width coding stream.


Huffman Coding Types:

• The construction of a code tree for the Huffman coding is based on a certain probability distribution.

• Varies In Three Types:

– static probability distribution

– dynamic probability distribution

– adaptive probability distribution


Static probability distribution

• Coding procedures with static Huffman codes operate with a predefined code tree.

• Provided that the source data correspond to the adopted frequency distribution, an acceptable efficiency of the coding can be achieved.

• It is not necessary to store the Huffman tree or the frequencies within the encoded data.

• It is sufficient to keep them available within the encoder or decoder software.

• Additionally the coding tables do not need to be generated at run-time.


Dynamic probability distribution

• Instead of a static tree being identical for any type of data, a dynamic analysis of the probability distribution could take place.

• Codes generated from these code trees match the real conditions clearly better than standard distributions.

• The major disadvantage of this procedure is, that the information about the Huffman tree has to be embedded into the compressed files or data transmissions.

• A code table or the symbol's frequencies must be part of the header data.


Adaptive probability distribution

• 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.

• each encoded symbol will be used to refine the code tree. This way a continuous adaption will be achieved and local variations will be compensated at run-time.


Adaptive probability distribution

• Adaptive Huffman codes initially using empty trees operate with a special control character identifying new symbols currently not being part of the tree.

• 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.


Extended Huffman Coding

• Extended Alphabet : For alphabet S={s1,s2,...,sn}, if k symbols are grouped together, then the extended alphabet is:

• Problem: If k is relatively large (e.g., k≥3), then for most practical applications where n>1, k implies a huge symbol table that is impractical.


Adaptive(Dynamic) Huffman Coding

• In adaptive Huffman Coding statistics are gathered and up-dated dynamically as the data stream arrives.


Adaptive(Dynamic) Huffman Coding

1. Initial code : assigns symbols with some initially agreed upon codes, without any prior knowledge of the frequency counts.

2. Update tree : constructs an Adaptive Huffman tree. It basically does two things:

1. increments the frequency counts for the symbols (includ-ing any new ones).

2. updates the configuration of the tree.

3. The encoder and decoder must use exactly the same initial code and update tree routines.


Notes on Adaptive Huffman Tree Updating

• Nodes are numbered in order from left to right, bottom to top. The numbers in parentheses indicates the count.

• The tree must always maintain its sibling property.

• When a swap is necessary, the farthest node with count N is swapped with the node whose count has just been increased to N+ 1.


Adaptive Huffman Coding Example: ABCDPAA

9.(9)

7.(4)

5.(2)

1.A: (1) 2.B: (1)

6.(2)

3.C: (1) 4.D: (1)

8.P.(5)



9.(9)

7.(4)

5.(2)

4.D: (1) 2.B: (1)

6.(2)

3.C: (1) 1.A: (2)

8.P.(5)



9.(9)

7.(4)

5.(2)

4.D: (1) 2.B: (1)

6.(2)

3.C: (1) 1.A: (2+1)

8.P.(5)


Adaptive Huffman Coding Example: ABCDPPPPPAA

9.(10)

7.(5+1)

6. (3)

4(2)

4.D: (1) 2.B: (1)

3.C: (1)

5.A: (3)

8.P(5)


Adaptive Huffman Coding Example: ABCDPPPPPAA

9.(11)

7:p(5) 8.(6)

5.A(3) 6(3)

3.C(1) 4.(2)

1.D(1) 2.B(1)


Another Example: Adaptive Huffman Coding • This is to clearly illustrate more implementation

details. We show exactly what bits are sent, as opposed to simply stating how the tree is updated.

• An additional rule: if any character/symbol is to be sent the first time, it must be preceded by a special symbol, NEW.

• The initial code for NEW is 0. The count for NEW is always kept as 0 (the count is never increased);

• hence it is always denoted as NEW:(0)


Initial code assignment for AADCCDD using adaptive Huffman coding.

(1)

NEW:0 A: (1)

(2)

NEW:0 A: (2)



(3)

A : (2) (1)

NEW:0 D: (1)



(4)

A: (2) (2)

(1)

NEW:0 C: (1)

D: (1)



(4)

A: (2) (2)

(1)

NEW:0 C: (1+1)

D: (1)



(4)

A: (2) (2+1)

(1)

NEW:0 D: (1)

C: (2)



(5)

A: (2) (3)

C : (2) (1)

NEW:0 D: (1)



(6)

A: (2) (4)

C : (2) (2)

NEW:0 D: (2)



(6)

A: (2) (4)

C : (2) (2)

NEW:0 D: (2+1)



(7)

D: (3) (4)

C : (2) (2)

NEW:0 A: (2)


Sequence of symbols and codes sent to the decoder Symbol

NEW A A NEW D NEW C C D D

Code 00000000

00000001

00000001

00000000

00000100

00000000

00000011

00000011

00000100

00000100

It is important to emphasize that the code for a particular symbol changes during the adaptive Huffman coding process.


THANK YOU
